The Partial Inner Product Space Method: A Quick Overview

Abstract

Many families of function spaces play a central role in analysis, in particular, in signal processing (e.g., wavelet or Gabor analysis). Typical are 𝐿^𝑝 spaces, Besov spaces, amalgam spaces, or modulation spaces. In all these cases, the parameter indexing the family measures the behavior (regularity, decay properties) of particular functions or operators. It turns out that all these space families are, or contain, scales or lattices of Banach spaces, which are special cases of partial inner product spaces (PIP-spaces). In this context, it is often said that such families should be taken as a whole and operators, bases, and frames on them should be defined globally, for the whole family, instead of individual spaces. In this paper, we will give an overview of PIP-spaces and operators on them, illustrating the results by space families of interest in mathematical physics and signal analysis. The interesting fact is that they allow a global definition of operators, and various operator classes on them have been defined.

1. Motivation

In the course of their curriculum, physics and mathematics students are usually taught the basics of Hilbert space, including operators of various types. The justification of this choice is twofold. On the mathematical side, Hilbert space is the example of an infinite-dimensional topological vector space that more closely resembles the familiar Euclidean space and thus it offers the student a smooth introduction into functional analysis. On the physics side, the fact is simply that Hilbert space is the daily language of quantum theory; therefore, mastering it is an essential tool for the quantum physicist.

However, the tool in question is actually insufficient. A pure Hilbert space formulation of quantum mechanics is both inconvenient and foreign to the daily behavior of most physicists, who stick to the more suggestive version of Dirac, although it lacks a rigorous formulation. On the other hand, the interesting solutions of most partial differential equations are seldom smooth or square integrable. Physically meaningful events correspond to changes of regime, which mean discontinuities and/or distributions. Shock waves are a typical example. Actually this state of affairs was recognized long ago by authors like Leray or Sobolev, whence they introduced the notion of weak solution. Thus it is no coincidence that many textbooks on PDEs begin with a thorough study of distribution theory [1–4].

All this naturally leads to the introduction of Rigged Hilbert Spaces (RHS) [5]. In a nutshell, a RHS is a triplet: Φ↪ℋ↪Φ^×, (1.1) where ℋ is a Hilbert space, Φ is a dense subspace of the ℋ, equipped with a locally convex topology, finer than the norm topology inherited from ℋ, and Φ^× is the space of continuous conjugate linear functionals on Φ, endowed with the strong dual topology. By duality, each space in (1.1) is dense in the next one and all embeddings are linear and continuous. In addition, the space Φ is in general required to be reflexive and nuclear. Standard examples of rigged Hilbert spaces are the Schwartz distribution spaces over ℝ or ℝ^𝑛, namely 𝒮⊂𝐿²⊂𝒮^× or 𝒟⊂𝐿²⊂𝒟^× [5–8].

The problem with the RHS (1.1) is that, besides the Hilbert space vectors, it contains only two types of elements: “very good" functions in Φ and “very bad" ones in Φ^×. If one wants a fine control on the behavior of individual elements, one has to interpolate somehow between the two extreme spaces. In the case of the Schwartz triplet, 𝒮⊂𝐿²⊂𝒮^×, a well-known solution is given by a chain of Hilbert spaces, the so-called Hermite representation of tempered distributions [9].

In fact, this is not at all an isolated case. Indeed many function spaces that play a central role in analysis come in the form of families, indexed by one or several parameters that characterize the behavior of functions (smoothness, behavior at infinity, …). The typical structure is a chain or a scale of Hilbert spaces, or a chain of (reflexive) Banach spaces (a discrete chain of Hilbert spaces {ℋ_𝑛}_𝑛∈ℤ is called a scale if there exists a self-adjoint operator 𝐵⩾1 such that ℋ_𝑛=𝐷(𝐵^𝑛),forall𝑛∈ℤ, with the graph norm ‖𝑓‖_𝑛=‖𝐵^𝑛𝑓‖. A similar definition holds for a continuous chain {ℋ_𝛼}_𝛼∈ℝ.). Let us give two familiar examples. (i)First, consider the Lebesgue the Lebesgue 𝐿^𝑝 spaces on a finite interval, for example, ℐ={𝐿^𝑝([0,1],𝑑𝑥),1⩽𝑝⩽∞}: 𝐿^∞⊂⋯⊂𝐿^𝑞⊂𝐿^𝑟⊂⋯⊂𝐿²⊂⋯⊂𝐿^𝑟⊂𝐿^𝑞⊂⋯⊂𝐿¹, (1.2) where 1<𝑞<𝑟<2. Here 𝐿^𝑞 and 𝐿^𝑞 are dual to each other (1/𝑞+1/𝑞=1), and similarly are 𝐿^𝑟,𝐿^𝑟(1/𝑟+1/𝑟=1). By the Hölder inequality, the (𝐿²) inner product ⟨𝑓∣𝑔⟩=¹₀𝑓(𝑥)𝑔(𝑥)𝑑𝑥 (1.3) is well defined if 𝑓∈𝐿^𝑞,𝑔∈𝐿^𝑞. However, it is not well defined for two arbitrary functions 𝑓,𝑔∈𝐿¹. Take, for instance, 𝑓(𝑥)=𝑔(𝑥)=𝑥^−1/2∶𝑓∈𝐿¹, but 𝑓𝑔=𝑓²∉𝐿¹. Thus, on 𝐿¹, (1.3) defines only a partial inner product. The same result holds for any compact subset of ℝ instead of [0,1].(ii)As a second example, take the scale of Hilbert spaces built on the powers of a positive self-adjoint operator 𝐴⩾1 in a Hilbert space ℋ₀. Let ℋ_𝑛 be 𝐷(𝐴^𝑛), the domain of 𝐴^𝑛, equipped with the graph norm ‖𝑓‖_𝑛=‖𝐴^𝑛𝑓‖,𝑓∈𝐷(𝐴^𝑛), for 𝑛∈ℕ or 𝑛∈ℝ⁺, and ℋ_𝑛∶=ℋ_−𝑛=ℋ^×_𝑛 (conjugate dual) 𝒟^∞(𝐴)∶=_𝑛ℋ_𝑛⊂…⊂ℋ₂⊂ℋ₁⊂ℋ₀⊂ℋ₁⊂ℋ₂⋯⊂𝒟_∞(𝐴)∶=_𝑛ℋ_𝑛. (1.4)

Note that, in the second example (ii), the index 𝑛 could also be taken as real, the link between the two cases being established by the spectral theorem for self-adjoint operators. Here again the inner product of ℋ₀ extends to each pair ℋ_𝑛,ℋ_−𝑛, but on 𝒟_∞(𝐴) it yields only a partial inner product. The following examples are standard: (i)(𝐴_p𝑓)(𝑥)=(1+𝑥²)𝑓(𝑥) in 𝐿²(ℝ,𝑑𝑥), (ii)(𝐴_m𝑓)(𝑥)=(1−𝑑²/𝑑𝑥²)𝑓(𝑥) in 𝐿²(ℝ,𝑑𝑥), (iii)(𝐴_osc𝑓)(𝑥)=(1+𝑥²−𝑑²/𝑑𝑥²)𝑓(𝑥) in 𝐿²(ℝ,𝑑𝑥).

(The notation is suggested by the operators of position, momentum and harmonic oscillator energy in quantum mechanics, resp.). Note that both 𝒟^∞(𝐴_p)∩𝒟^∞(𝐴_m) and 𝒟^∞(𝐴_osc) coincide with the Schwartz space 𝒮(ℝ) of smooth functions of fast decay, and 𝒟_∞(𝐴_osc) with the space 𝒮^×(ℝ) of tempered distributions (considered here as continuous conjugate linear functionals on 𝒮). As for the operator 𝐴_m, it generates the scale of Sobolev spaces 𝐻^𝑠(ℝ),𝑠∈ℤorℝ.

However, a moment's reflection shows that the total-order relation inherent in a chain is in fact an unnecessary restriction; partially ordered structures are sufficient, and indeed necessary in practice. For instance, in order to get a better control on the behavior of individual functions, one may consider the lattice built on the powers of 𝐴_p and 𝐴_m simultaneously. Then the extreme spaces are still 𝒮(ℝ) and 𝒮^×(ℝ). Similarly, in the case of several variables, controlling the behavior of a function in each variable separately requires a nonordered set of spaces. This is in fact a statement about tensor products (remember that 𝐿²(𝑋×𝑌)≃𝐿²(𝑋)⊗𝐿²(𝑌)). Indeed the tensor product of two chains of Hilbert spaces, {ℋ_𝑛}⊗{𝒦_𝑚}, is naturally a lattice {ℋ_𝑛⊗𝒦_𝑚} of Hilbert spaces. For instance, in the example above, for two variables 𝑥,𝑦, that would mean considering intermediate Hilbert spaces corresponding to the product of two operators, (𝐴_m(𝑥))^𝑛(𝐴_m(𝑦))^𝑚.

Thus the structure to analyze is that of lattices of Hilbert or Banach spaces, interpolating between the extreme spaces of an RHS, as in (1.1). Many examples can be given, for instance, the lattice generated by the spaces 𝐿^𝑝(ℝ,𝑑𝑥), the amalgam spaces 𝑊(𝐿^𝑝,ℓ^𝑞), the mixed-norm spaces 𝐿_𝑚^𝑝,𝑞(ℝ,𝑑𝑥), and many more. In all these cases, which contain most families of function spaces of interest in analysis and in signal processing, a common structure emerges for the “large" space 𝑉, defined as the union of all individual spaces. There is a lattice of Hilbert or reflexive Banach spaces 𝑉_𝑟, with an (order-reversing) involution 𝑉_𝑟↔𝑉_𝑟, where 𝑉_𝑟=𝑉^×_𝑟 (the space of continuous conjugate linear functionals on 𝑉_𝑟), a central Hilbert space 𝑉_𝑜≃𝑉_𝑜, and a partial inner product on 𝑉 that extends the inner product of 𝑉_𝑜 to pairs of dual spaces 𝑉_𝑟,𝑉_𝑟.

Moreover, many operators should be considered globally, for the whole scale or lattice, instead of on individual spaces. In the case of the spaces 𝐿^𝑝(ℝ), such are, for instance, operators implementing translations (𝑥↦𝑥−𝑦) or dilations (𝑥↦𝑥/𝑎), convolution operators, Fourier transform, and so forth. In the same spirit, it is often useful to have a common basis for the whole family of spaces, such as the Haar basis for the spaces 𝐿^𝑝(ℝ),1<𝑝<∞. Thus we need a notion of operator and basis defined globally for the scale or lattice itself.

This state of affairs prompted A. Grossmann and one of us (the first author) to systematize this approach, and this led to the concept of partial inner product space or PIP-space [10–13]. After many years and various developments, we devoted a full monograph [14] to a detailed survey of the theory. The aim of this paper is to present the formalism of PIP-spaces, which indeed answers these questions. In a first part, the structure of PIP-space is derived systematically from the abstract notion of compatibility and then particularized to the examples listed above. In a second part, operators on PIP-spaces are introduced and illustrated by several operators commonly used in Gabor or wavelet analysis. Finally we describe a number of applications of PIP-spaces in mathematical physics and in signal processing. Of course, the treatment is sketchy, for lack of space. For a complete information, we refer the reader to our monograph [14].

2. Partial Inner Product Spaces

2.1. Basic Definitions

The basic question is how to generate PIP-spaces in a systematic fashion. In order to answer, we may reformulate it as follows: given a vector space 𝑉 and two vectors 𝑓,𝑔∈𝑉, when does their inner product make sense? A way of formalizing the answer is given by the idea of compatibility.

Definition 2.1. A linear compatibility relation on a vector space 𝑉 is a symmetric binary relation 𝑓#𝑔 which preserves linearity: 𝑓#𝑔⇔𝑔#𝑓,∀𝑓,𝑔∈𝑉,𝑓#𝑔,𝑓#ℎ⇒𝑓#(𝛼𝑔+𝛽ℎ),∀𝑓,𝑔,ℎ∈𝑉,∀𝛼,𝛽∈ℂ. (2.1)As a consequence, for every subset 𝑆⊂𝑉, the set 𝑆^#={𝑔∈𝑉∶𝑔#𝑓,forall𝑓∈𝑆} is a vector subspace of 𝑉 and one has 𝑆^##=(𝑆^#)^#⊇𝑆,𝑆^###=𝑆^#. (2.2) Thus one gets the following equivalences: 𝑓#𝑔⇔𝑓∈{𝑔}^#⇔{𝑓}^##⊆{𝑔}^#⇔𝑔∈{𝑓}^#⇔{𝑔}^##⊆{𝑓}^#. (2.3) From now on, we will call assaying subspace of 𝑉 a subspace 𝑆 such that 𝑆^##=𝑆 and denote by ℱ(𝑉,#) the family of all assaying subsets of 𝑉, ordered by inclusion. Let 𝐹 be the isomorphy class of ℱ, that is, ℱ is considered as an abstract partially ordered set. Elements of 𝐹 will be denoted by 𝑟,𝑞,…, and the corresponding assaying subsets by 𝑉_𝑟,𝑉_𝑞,…. By definition, 𝑞⩽𝑟 if and only if 𝑉_𝑞⫅𝑉_𝑟. We also write 𝑉_𝑟=𝑉^#_𝑟,𝑟∈𝐹. Thus the relations (2.3) mean that 𝑓#𝑔 if and only if there is an index 𝑟∈𝐹 such that 𝑓∈𝑉_𝑟,𝑔∈𝑉_𝑟. In other words, vectors should not be considered individually, but only in terms of assaying subspaces, which are the building blocks of the whole structure.

It is easy to see that the map 𝑆↦𝑆^## is a closure, in the sense of universal algebra, so that the assaying subspaces are precisely the “closed" subsets. Therefore one has the following standard result.

Theorem 2.2. The family ℱ(𝑉,#)≡{𝑉_𝑟,𝑟∈𝐹}, ordered by inclusion, is a complete involutive lattice, that is, it is stable under the following operations, arbitrarily iterated: (i)involution:𝑉_𝑟↔𝑉_𝑟=(𝑉_𝑟)^#,(ii)infimum: 𝑉_𝑝∧𝑞≡𝑉_𝑝∧𝑉_𝑞=𝑉_𝑝∩𝑉_𝑞,(𝑝,𝑞,𝑟∈𝐹),(iii)supremum: 𝑉_𝑝∨𝑞≡𝑉_𝑝∨𝑉_𝑞=(𝑉_𝑝+𝑉_𝑞)^##.

The smallest element of ℱ(𝑉,#) is 𝑉^#=⋂_𝑟𝑉_𝑟 and the greatest element is 𝑉=⋃_𝑟𝑉_𝑟. By definition, the index set 𝐹 is also a complete involutive lattice; for instance, (𝑉_𝑝∧𝑞)^#=𝑉_𝑝∧𝑞=𝑉_𝑝∨𝑞=𝑉_𝑝∨𝑉_𝑞. (2.4)

Definition 2.3. A partial inner product on (𝑉,#) is a Hermitian form ⟨⋅∣⋅⟩ defined exactly on compatible pairs of vectors. A partial inner product space (PIP-space) is a vector space 𝑉 equipped with a linear compatibility and a partial inner product.

Note that the partial inner product is not required to be positive definite.

The partial inner product clearly defines a notion of orthogonality: 𝑓⟂𝑔if and only if 𝑓#𝑔 and ⟨𝑓∣𝑔⟩=0.

Definition 2.4. The PIP-space (𝑉,#,⟨⋅∣⋅⟩) is nondegenerate if (𝑉^#)^⟂={0}, that is, if ⟨𝑓∣𝑔⟩=0 for all 𝑓∈𝑉^# implies that 𝑔=0.

We will assume henceforth that our PIP-space (𝑉,#,⟨⋅∣⋅⟩) is nondegenerate. As a consequence, (𝑉^#,𝑉) and every couple (𝑉_𝑟,𝑉_𝑟),𝑟∈𝐹, are dual pairs in the sense of topological vector spaces [15]. We also assume that the partial inner product is positive definite.

Now one wants the topological structure to match the algebraic structure, in particular, the topology 𝜏_𝑟 on 𝑉_𝑟 should be such that its conjugate dual be 𝑉_𝑟: (𝑉_𝑟[𝜏_𝑟])^×=𝑉_𝑟,forall𝑟∈𝐹. This implies that the topology 𝜏_𝑟 must be finer than the weak topology 𝜎(𝑉_𝑟,𝑉_𝑟) and coarser than the Mackey topology 𝜏(𝑉_𝑟,𝑉_𝑟): 𝜎(𝑉_𝑟,𝑉_𝑟)⪯𝜏_𝑟⪯𝜏(𝑉_𝑟,𝑉_𝑟). (2.5) From here on, we will assume that every 𝑉_𝑟 carries its Mackey topology 𝜏(𝑉_𝑟,𝑉_𝑟). This choice has two interesting consequences. First, if 𝑉_𝑟[𝜏_𝑟] is a Hilbert space or a reflexive Banach space, then 𝜏(𝑉_𝑟,𝑉_𝑟) coincides with the norm topology. Next, 𝑟<𝑠 implies that 𝑉_𝑟⊂𝑉_𝑠, and the embedding operator 𝐸_s𝑟∶𝑉_𝑟→𝑉_𝑠 is continuous and has dense range. In particular, 𝑉^# is dense in every 𝑉_𝑟.

2.2. Examples

2.2.1. Sequence Spaces

Let 𝑉 be the space 𝜔 of all complex sequences 𝑥=(𝑥_𝑛) and define on it (i) a compatibility relation by 𝑥#𝑦⇔∑^∞_𝑛=1|𝑥_𝑛𝑦_𝑛|<∞ and (ii) a partial inner product ⟨𝑥∣𝑦⟩=∑^∞_𝑛=1𝑥_𝑛𝑦_𝑛.

Then 𝜔^#=𝜑, the space of finite sequences, and the complete lattice ℱ(𝜔,#) consists of Köthe's perfect sequence spaces [15, § 30]. Among these, typical assaying subspaces are the weighted Hilbert spaces ℓ²(𝑟)={(𝑥_𝑛)∶^∞_𝑛=1|𝑥_𝑛|²𝑟_𝑛⁻²<∞}, (2.6) where 𝑟=(𝑟_𝑛),𝑟_𝑛>0, is a sequence of positive numbers. The involution is ℓ²(𝑟)↔ℓ²(𝑟)=ℓ²(𝑟)^×, where 𝑟_𝑛=1/𝑟_𝑛. In addition, there is a central, self-dual Hilbert space, namely, ℓ²(1)=ℓ²(1)=ℓ², where 1 = (1).

2.2.2. Spaces of Locally Integrable Functions

Let now 𝑉 be 𝐿¹_loc(ℝ,𝑑𝑥), the space of Lebesgue measurable functions, integrable over compact subsets, and define a compatibility relation on it by 𝑓#𝑔⇔∫_ℝ|𝑓(𝑥)𝑔(𝑥)|𝑑𝑥<∞ and a partial inner product ⟨𝑓∣𝑔⟩=∫_ℝ𝑓(𝑥)𝑔(𝑥)𝑑𝑥.

Then 𝑉^#=𝐿^∞_𝑐(ℝ), the space of bounded measurable functions of compact support. The complete lattice ℱ(𝐿¹_loc,#) consists of Köthe function spaces [16, 17]. Here again, typical assaying subspaces are weighted Hilbert spaces 𝐿²(𝑟)={𝑓∈𝐿¹_loc(ℝ,𝑑𝑥)∶_ℝ|𝑓(𝑥)|²𝑟(𝑥)⁻²𝑑𝑥<∞}, (2.7) with 𝑟,𝑟⁻¹∈𝐿²_loc(ℝ,𝑑𝑥),𝑟(𝑥)>0 a.e. The involution is 𝐿²(𝑟)↔𝐿²(𝑟), with 𝑟=𝑟⁻¹, and the central, self-dual Hilbert space is 𝐿²(ℝ,𝑑𝑥).

2.2.3. Nested Hilbert Spaces

This is the original construction of Grossmann [18] for finding an “easy" substitute to distributions, and actually one of the motivations for introducing PIP-spaces. And indeed the two are closely related; see [14, Section 2.4.1]

2.2.4. Rigged Hilbert Spaces

This is the simplest example of PIP-space, but it is a rather poor one. Indeed, in the RHS (1.1), two elements are compatible if both belong to ℋ, or one of them belongs to Φ. Thus the three defining spaces are the only assaying subspaces. The partial inner product is, of course, simply that of ℋ, provided the sesquilinear form that puts Φ and Φ^× in duality has been correctly normalized.

3. Lattices of Hilbert or Banach Spaces

From the previous examples, we learn that ℱ(𝑉,#) is a huge lattice (it is complete!) and that assaying subspaces may be complicated, such as Fréchet spaces, nonmetrizable spaces, and so forth. This situation suggests to choose an involutive sublattice ℐ⊂ℱ, indexed by 𝐼, such that(i)ℐ is generating: 𝑓#𝑔⇔∃𝑟∈𝐼suchthat𝑓∈𝑉_𝑟,𝑔∈𝑉_𝑟, (3.1)(ii)every 𝑉_𝑟,𝑟∈𝐼, is a Hilbert space or a reflexive Banach space,(iii)there is a unique self-dual assaying subspace 𝑉_𝑜=𝑉_𝑜, which is a Hilbert space.

In that case, the structure 𝑉_𝐼∶=(𝑉,ℐ,⟨⋅∣⋅⟩) is called, respectively, a lattice of Hilbert spaces (LHS) or a lattice of Banach spaces (LBS). Both types are particular cases of the so-called indexed PIP-spaces [14]. Note that 𝑉^#,𝑉 themselves usually do not belong to the family {𝑉_𝑟,𝑟∈𝐼}, but they can be recovered as 𝑉^#=_𝑟∈𝐼𝑉_𝑟,𝑉=_𝑟∈𝐼𝑉_𝑟. (3.2)

In the LBS case, the lattice structure takes the following forms: (i)𝑉_𝑝∧𝑞=𝑉_𝑝∩𝑉_𝑞, with the projective norm ‖𝑓‖_𝑝∧𝑞=‖𝑓‖_𝑝+‖𝑓‖_𝑞, (3.3)(ii)𝑉_𝑝∨𝑞=𝑉_𝑝+𝑉_𝑞, with the inductive norm ‖𝑓‖_𝑝∨𝑞=inf_{𝑓=𝑔+ℎ}(‖𝑔‖_𝑝+‖ℎ‖_𝑞),𝑔∈𝑉_𝑝,ℎ∈𝑉_𝑞. (3.4)

These norms are usual in interpolation theory [19]. In the LHS case, one takes similar definitions with squared norms, in order to get Hilbert norms throughout.

In the rest of this section, we will list a series of concrete examples of LHS/LBSs. Some more examples, which are of particular interest in signal processing, will be given in Section 6.2. For simplicity, we will restrict ourselves to one dimension, although most spaces may be defined on ℝ^𝑛,𝑛>1, as well.

3.1. Chains of Hilbert or Banach Spaces

Typical are the two examples described in Section 1. (1)The chain of Lebesgue spaces on a finite interval ℐ={𝐿^𝑝([0,1],𝑑𝑥),1<𝑝<∞}. The chain (1.2) is a (totally ordered) lattice. The corresponding lattice completion is obtained by adding “nonstandard" spaces such as 𝐿^𝑝−=_1<𝑞<𝑝𝐿^𝑞(non-normableFréchet),𝐿^𝑝+=_{𝑝<𝑞<∞}𝐿^𝑞(nonmetrizable). (3.5)(2)The scale (1.4) of Hilbert spaces {ℋ_𝑛,𝑛∈ℤ} built on powers of 𝐴=𝐴^∗⩾1. The lattice completion is similar to the previous one, introducing analogous “nonstandard" spaces [14, Section 5.1].

3.2. Sequence Spaces

3.2.1. A LHS of Weighted ℓ² Spaces

In 𝜔, with the compatibility # and the partial inner product defined in Section 2.2.1, we may take the lattice ℐ={ℓ²(𝑟)} of the weighted Hilbert spaces defined in (2.6), with lattice operations: (i)infimum: ℓ²(𝑝∧𝑞)=ℓ²(𝑝)∧ℓ²(𝑞)=ℓ²(𝑟),𝑟_𝑛=min(𝑝_𝑛,𝑞_𝑛), (ii)supremum: ℓ²(𝑝∨𝑞)=ℓ²(𝑝)∨ℓ²(𝑞)=ℓ²(𝑠),𝑠_𝑛=max(𝑝_𝑛,𝑞_𝑛), (iii)duality: ℓ²(𝑝∧𝑞)↔ℓ²(𝑝∨𝑞),ℓ²(𝑝∨𝑞)↔ℓ²(𝑝∧𝑞).

As a matter of fact, the norms above are equivalent to the projective and inductive norms, respectively. Then, it is easy to show that the lattice ℐ={ℓ²(𝑟)} is generating in ℱ(𝜔,#).

3.2.2. Köthe Perfect Sequence Spaces

We have already noticed that the complete lattice ℱ(𝜔,#) consists precisely of all Köthe perfect sequence spaces. Indeed, these are defined as the assaying subspaces corresponding to the compatibility #, which is called 𝛼-duality [15]. Among these, there is an interesting class, the so-called ℓ_𝜙 spaces associated to symmetric norming functions.

Definition 3.1. A real-valued function 𝜙 defined on the space 𝜑 of finite sequences is said to be a norming function if(n1)𝜙(𝑥)>0for every sequence𝑥∈𝜑,𝑥≠0,(n2)𝜙(𝛼𝑥)=|𝛼|𝜙(𝑥),forall𝑥∈𝜑,forall𝛼∈ℂ,(n3)𝜙(𝑥+𝑦)⩽𝜙(𝑥)+𝜙(𝑦),forall𝑥,𝑦∈𝜑,(n4)𝜙(1,0,0,0,…)=1. A norming function 𝜙 is symmetric if(n5)𝜙(𝑥₁,𝑥₂,…,𝑥_𝑛,0,0,…)=𝜙(|𝑥_𝑗1|,|𝑥_𝑗2|,…,|𝑥_𝑗𝑛|,0,0,…), where 𝑗₁,𝑗₂,…,𝑗_𝑛 is an arbitrary permutation of 1,2,…,𝑛.

From property (n5), it is clear that a symmetric norming function 𝜙 is entirely determined by its values on the set [𝜑] of finite, positive, nonincreasing sequences. Hence, from conditions (n2) and (n4), we deduce that 𝜙_∞(𝑥)⩽𝜙(𝑥)⩽𝜙₁(𝑥),∀𝑥∈𝜑, (3.6) where 𝜙_∞(𝑥)=max_{𝑖=1,…,𝑛}|𝑥_𝑖| and 𝜙₁(𝑥)=∑^𝑛_𝑖=1|𝑥_𝑖|.

To every symmetric norming function 𝜙, one can associate a Banach space ℓ_𝜙 as follows. Given a sequence 𝑥∈𝜔, define its 𝑛th section as 𝑥^(𝑛)=(𝑥₁,𝑥₂,…,𝑥_𝑛,0,0,…). Then the sequence (𝜙(𝑥^(𝑛))) is nondecreasing, so that one can define ℓ_𝜙={𝑥∈𝜔∶sup_𝑛𝜙(𝑥^(𝑛))<∞} (3.7) and then extend the norming function 𝜙 to the whole of ℓ_𝜙 by putting 𝜙(𝑥)=lim_𝑛𝜙(𝑥^(𝑛)). This relation defines a norm 𝜙 on ℓ_𝜙, for which it is complete, hence, a Banach space. In other words, we can also say that ℓ_𝜙={𝑥∈𝜔∶𝜙(𝑥)<∞} is the natural domain of definition of the extended norming function 𝜙. Clearly, one has ℓ_𝜙∞=ℓ^∞ and ℓ_𝜙1=ℓ¹. Similarly, ℓ^𝑝=ℓ_𝜙𝑝, where 𝜙_𝑝(𝑥)=(∑_𝑛|𝑥_𝑛|^𝑝)^1/𝑝. Thus every space ℓ_𝜙 contains ℓ¹ and is contained in ℓ^∞.

In addition, the set of Banach spaces ℓ_𝜙 constitutes a lattice. Given two symmetric norming functions 𝜙 and 𝜓, one defines their infimum and supremum, exactly as for the general case: (i)𝜙∧𝜓∶=max{𝜙,𝜓}, which defines on the space ℓ_𝜙∧𝜓∶=ℓ_𝜙∩ℓ_𝜓 a norm equivalent to 𝜙(𝑥)+𝜓(𝑥),(ii)𝜙∨𝜓∶=min{𝜙,𝜓}, which defines on the space ℓ_𝜙∨𝜓∶=ℓ_𝜙+ℓ_𝜓 a norm equivalent to inf_{𝑥=𝑦+𝑧}{𝜙(𝑦)+𝜓(𝑧)},𝑥∈ℓ_𝜙+ℓ_𝜓,𝑦∈ℓ_𝜙,𝑧∈ℓ_𝜓.

It remains to analyze the relationship of the spaces ℓ_𝜙 with the PIP-space structure of 𝜔. Define, for any finite, positive, nonincreasing sequence 𝑦∈[𝜑], 𝜙(𝑦)∶=max_{𝑥∈[𝜑]}⟨𝑥∣𝑦⟩.𝜙(𝑥) (3.8) The function 𝜙 thus defined is a symmetric norming function; hence, it can be extended to the corresponding Banach space ℓ_𝜙. The function 𝜙 is said to be conjugate to 𝜙 and the space ℓ_𝜙 is the conjugate dual of ℓ_𝜙 with respect to the partial inner product, that is, ℓ_𝜙=(ℓ_𝜙)^#. Clearly one has 𝜙=𝜙; hence, ℓ_𝜙=(ℓ_𝜙)^##=ℓ_𝜙.

In addition, it is easy to show that ℓ_𝜙∧𝜓=ℓ_𝜙∨𝜓 and ℓ_𝜙∨𝜓=ℓ_𝜙∧𝜓. In other words, one gets the following result.

Proposition 3.2. The family of Banach spaces ℓ_𝜙, where 𝜙 is a symmetric norming function, is an involutive sublattice of the lattice ℱ(𝜔,#) and a LBS.

Actually, since every 𝜙 satisfies the inclusions ℓ¹⊂ℓ_𝜙⊂ℓ^∞, the family {ℓ_𝜙} is also an involutive sublattice of the lattice ℱ(ℓ^∞,#) obtained by restricting to ℓ^∞ the PIP-space structure of 𝜔.

These spaces {ℓ_𝜙} may be generalized further to what is called the theory of Banach ideals of sequences. See [14, Section 4.3] for more details.

3.3. Spaces of Locally Integrable Functions

3.3.1. A LHS of Weighted 𝐿² Spaces

In 𝐿¹_loc(ℝ,𝑑𝑥), we may take the lattice ℐ={𝐿²(𝑟)} of the weighted Hilbert spaces defined in (2.7), with (i)infimum: 𝐿²(𝑝∧𝑞)=𝐿²(𝑝)∧𝐿²(𝑞)=𝐿²(𝑟),𝑟(𝑥)=min(𝑝(𝑥),𝑞(𝑥)), (ii)supremum: 𝐿²(𝑝∨𝑞)=𝐿²(𝑝)∨𝐿²(𝑞)=𝐿²(𝑠),𝑠(𝑥)=max(𝑝(𝑥),𝑞(𝑥)), (iii)duality: 𝐿²(𝑝∧𝑞)↔𝐿²(𝑝∨𝑞),𝐿²(𝑝∨𝑞)↔𝐿²(𝑝∧𝑞).

Here too, these norms are equivalent to the projective and inductive norms, respectively.

3.3.2. The Spaces 𝐿^𝑝(ℝ,𝑑𝑥),1<𝑝<∞

The spaces 𝐿^𝑝(ℝ,𝑑𝑥),1<𝑝<∞ do not constitute a scale, since one has only the inclusions 𝐿^𝑝∩𝐿^𝑞⊂𝐿^𝑠,𝑝<𝑠<𝑞. Thus one has to consider the lattice they generate, with the following lattice operations: (i)𝐿^𝑝∧𝐿^𝑞=𝐿^𝑝∩𝐿^𝑞, with projective norm, (ii)𝐿^𝑝∨𝐿^𝑞=𝐿^𝑝+𝐿^𝑞, with inductive norm.

For 1<𝑝,𝑞<∞, both spaces 𝐿^𝑝∧𝐿^𝑞 and 𝐿^𝑝∨𝐿^𝑞 are reflexive Banach spaces and their conjugate duals are, respectively, (𝐿^𝑝∧𝐿^𝑞)^×=𝐿^𝑝∨𝐿^𝑞 and (𝐿^𝑝∨𝐿^𝑞)^×=𝐿^𝑝∧𝐿^𝑞.

It is convenient to introduce the following unified notation: 𝐿^(𝑝,𝑞)=𝐿^𝑝∧𝐿^𝑞𝐿,if𝑝⩾𝑞,_𝑝∨𝐿^𝑞,if𝑝⩽𝑞. (3.9) Then, for 1<𝑝,𝑞<∞, 𝐿^(𝑝,𝑞) is a reflexive Banach space, with conjugate dual 𝐿^(𝑝,𝑞).

Next, if we represent (𝑝,𝑞) by the point of coordinates (1/𝑝,1/𝑞), we may associate all the spaces 𝐿^(𝑝,𝑞)(1⩽𝑝,𝑞⩽∞) in a one-to-one fashion with the points of a unit square J=[0,1]×[0,1] (see Figure 1). Thus, in this picture, the spaces 𝐿^𝑝 are on the main diagonal, intersections 𝐿^𝑝∩𝐿^𝑞 above it and sums 𝐿^𝑝+𝐿^𝑞 below.

Figure 1 

The unit square describing the lattice J.

The space 𝐿^(𝑝,𝑞) is contained in 𝐿^{(𝑝′,𝑞′)} if (𝑝,𝑞) is on the left and/or above (𝑝^′,𝑞^′). Thus the smallest space is 𝑉^#_J=𝐿^(∞,1)=𝐿^∞∩𝐿¹ (3.10) and it corresponds to the upper-left corner, while the largest one is 𝑉_J=𝐿^(1,∞)=𝐿¹+𝐿^∞, (3.11) corresponding to the lower-right corner. Inside the square, duality corresponds to (geometrical) symmetry with respect to the center (1/2,1/2) of the square, which represents the space 𝐿². The ordering of the spaces corresponds to the following rule: 𝐿^(𝑝,𝑞)⊂𝐿^{(𝑝′,𝑞′)}⇔(𝑝,𝑞)⩽(𝑝^′,𝑞^′)⇔𝑝⩾𝑝^′,𝑞⩽𝑞^′. (3.12) With respect to this ordering, J is an involutive lattice with the operations (𝑝,𝑞)∧(𝑝^′,𝑞^′)=(𝑝∨𝑝^′,𝑞∧𝑞^′),(𝑝,𝑞)∨(𝑝^′,𝑞^′)=(𝑝∧𝑝^′,𝑞∨𝑞^′),(𝑝,𝑞)=(𝑝,𝑞), (3.13) where 𝑝∧𝑝^′=min{𝑝,𝑝^′},𝑝∨𝑝^′=max{𝑝,𝑝^′}. It is remarkable that the lattice 𝒥 generated by ℐ={𝐿^𝑝} is obtained at the first “generation". One has, for instance, 𝐿^(𝑟,𝑠)∧𝐿^(𝑎,𝑏)=𝐿^{(𝑟∨𝑎,𝑠∧𝑏)}, both as sets and as topological vector spaces.

3.3.3. Mixed-Norm Lebesgue Spaces 𝐿_𝑚^𝑝,𝑞

An interesting class of function spaces, close relatives to the Lebesgue 𝐿^𝑝 spaces, consists of the so-called 𝐿^𝑃 spaces with mixed norm. Let (𝑋,𝜇) and (𝑌,𝜈) be two 𝜎-finite measure spaces and 1⩽𝑝,𝑞⩽∞ (in the general case, one considers 𝑛 such spaces and 𝑛-tuples 𝑃∶=(𝑝₁,𝑝₂,…,𝑝_𝑛)). Then, a function 𝑓(𝑥,𝑦) measurable on the product space 𝑋×𝑌 is said to belong to 𝐿^(𝑝,𝑞)(𝑋×𝑌) if the number obtained by taking successively the 𝑝-norm in 𝑥 and the 𝑞-norm in 𝑦, in that order, is finite (exchanging the order of the two norms leads in general to a different space). If 𝑝,𝑞<∞, the norm reads ‖𝑓‖_(𝑝,𝑞)=(_𝑌(_𝑋|𝑓(𝑥,𝑦)|^𝑝𝑑𝜇(𝑥))^𝑞/𝑝𝑑𝜈(𝑦))^1/𝑞. (3.14) The analogous norm for 𝑝 or 𝑞=∞ is obvious. For 𝑝=𝑞, one gets the usual space 𝐿^𝑝(𝑋×𝑌).

These spaces enjoy a number of properties similar to those of the 𝐿^𝑝 spaces: (i) each space 𝐿^(𝑝,𝑞) is a Banach space and it is reflexive if and only if 1<𝑝,𝑞<∞; (ii) the conjugate dual of 𝐿^(𝑝,𝑞) is 𝐿^(𝑝,𝑞), where, as usual, 𝑝⁻¹+𝑝⁻¹=1,𝑞⁻¹+𝑞⁻¹=1; thus the topological conjugate dual coincides with the Köthe dual; (iii) the mixed-norm spaces satisfy a generalized Hölder inequality and have nice interpolation properties.

The case 𝑋=𝑌=ℝ^𝑑 with Lebesgue measure is the important one for signal processing [20, Section 11.1]. More generally, one can add a weight function 𝑚 and obtain the spaces 𝐿_𝑚^𝑝,𝑞(ℝ^𝑑) (we switch to a notation more suitable for the applications):‖𝑓‖_𝑚^𝑝,𝑞=(_ℝ𝑑(_ℝ𝑑|𝑓(𝑥,𝜔)|^𝑝𝑚(𝑥,𝜔)^𝑝𝑑𝑥)^𝑞/𝑝𝑑𝜔)^1/𝑞. (3.15) Here the weight function 𝑚 is a nonnegative locally integrable function on ℝ^2𝑑, assumed to be 𝑣-moderate, that is, 𝑚(𝑧₁+𝑧₂)⩽𝑣(𝑧₁)𝑚(𝑧₂), for all 𝑧₁,𝑧₂∈ℝ^2𝑑, with 𝑣 a submultiplicative weight function, that is, 𝑣(𝑧₁+𝑧₂)⩽𝑣(𝑧₁)𝑣(𝑧₂), for all 𝑧₁,𝑧₂∈ℝ^2𝑑. The typical weights are of polynomial growth: 𝑣_𝑠(𝑧)=(1+|𝑧|)^𝑠,𝑠⩾0.

The space 𝐿_𝑚^𝑝,𝑞(ℝ^2𝑑) is a Banach space for the norm ‖⋅‖_𝑚^𝑝,𝑞. The duality property is, as expected, (𝐿_𝑚^𝑝,𝑞)^×=𝐿^𝑝,𝑞_1/𝑚. Of course, things simplify when 𝑝=𝑞: 𝐿_𝑚^𝑝,𝑝(ℝ^2𝑑)=𝐿^𝑝_𝑚(ℝ^2𝑑), a weighted 𝐿^𝑝 space.

Concerning lattice properties of the family of 𝐿_𝑚^𝑝,𝑞 spaces, we cannot expect more than for the 𝐿^𝑝 spaces. Two 𝐿_𝑚^𝑝,𝑞 spaces are never comparable, even for the same weight 𝑚, so one has to take the lattice generated by intersection and duality.

A different type of mixed-norm spaces is obtained if one takes 𝑋=𝑌=ℤ^𝑑, with the counting measure. Thus one gets the space ℓ_𝑚^𝑝,𝑞(ℤ^2𝑑), which consists of all sequences 𝑎=(𝑎_𝑘𝑛),𝑘,𝑛∈ℤ^𝑑, for which the following norm is finite: ‖𝑎‖_ℓ𝑚^𝑝,𝑞∶=(_{𝑛∈ℤ𝑑}(_{𝑘∈ℤ𝑑}|𝑎_𝑘𝑛|^𝑝𝑚(𝑘,𝑛)^𝑝)^𝑞/𝑝)^1/𝑞. (3.16) Contrary to the continuous case, here we do have inclusion relations: if 𝑝₁⩽𝑝₂,𝑞₁⩽𝑞₂ and 𝑚₂⩽𝐶𝑚₁, then ℓ^𝑝1,𝑞1_𝑚1⊆ℓ^𝑝2,𝑞2_𝑚2.

Discrete mixed-norm spaces have been used extensively in functional analysis and signal processing. For instance, they are key to the proof that certain operators are bounded between two given function spaces, such as modulation spaces (see below) or ℓ^𝑝 spaces. In general, a mixed-norm space will prove useful whenever one has a signal consisting of sequences labeled by two indices that play different roles. An obvious example is time-frequency or time-scale analysis: a Gabor or wavelet basis (or frame) is written as {𝜓_𝑗,𝑘,𝑗,𝑘∈ℤ}, where 𝑗 indexes the scale or frequency and 𝑘 the time. More generally, this applies whenever signals are expanded with respect to a dictionary with two indices.

3.3.4. Köthe Function Spaces

The mixed-norm Lebesgue spaces 𝐿_𝑚^𝑝,𝑞 are special cases of a very general class, the so-called Köthe function spaces. These have been introduced (and given that name) by Dieudonné [16] and further studied by Luxemburg-Zaanen [21]. The procedure here is entirely parallel to that used in Section 3.2.2 above for introducing the sequence spaces ℓ_𝜙.

Let (𝑋,𝜇) be a 𝜎-finite measure space and 𝑀⁺ the set of all measurable, non-negative functions on 𝑋, where two functions are identified if they differ at most on a 𝜇-null set. A function norm is a mapping 𝜌∶𝑀⁺→ℝ such that (i)0⩽𝜌(𝑓)⩽∞,forall𝑓∈𝑀⁺ and 𝜌(𝑓)=0 if and only if 𝑓=0, (ii)𝜌(𝑓₁+𝑓₂)⩽𝜌(𝑓₁)+𝜌(𝑓₂),forall𝑓₁,𝑓₂∈𝑀⁺, (iii)𝜌(𝑎𝑓)=𝑎𝜌(𝑓),forall𝑓∈𝑀⁺,forall𝑎⩾0, (iv)𝑓₁⩽𝑓₂⇒𝜌(𝑓₁)⩽𝜌(𝑓₂),forall𝑓₁,𝑓₂∈𝑀⁺.

A function norm 𝜌 is said to have the Fatou property if and only if 0⩽𝑓₁⩽𝑓₂⩽…,𝑓_𝑛∈𝑀⁺ and 𝑓_𝑛→𝑓 pointwise implies that 𝜌(𝑓_𝑛)→𝜌(𝑓).

Given a function norm 𝜌, it can be extended to all complex measurable functions on 𝑋 by defining 𝜌(𝑓)=𝜌(|𝑓|). Denote by 𝐿_𝜌 the set of all measurable 𝑓 such that 𝜌(𝑓)<∞. With the norm ‖𝑓‖=𝜌(𝑓),𝐿_𝜌 is a normed space and a subspace of the vector space 𝑉 of all measurable 𝜇-a.e. finite, functions on 𝑋. Furthermore, if 𝜌 has the Fatou property, 𝐿_𝜌 is complete, that is, a Banach space.

A function norm 𝜌 is said to be saturated if, for any measurable set 𝐸⊂𝑋 of positive measure, there exists a measurable subset 𝐹⊂𝐸 such that 𝜇(𝐹)>0 and 𝜌(𝜒_𝐹)<∞(𝜒_𝐹 is the characteristic function of 𝐹).

Let 𝜌 be a saturated function norm with the Fatou property. Define 𝜌^′(𝑓)=sup{_𝑋|𝑓𝑔|𝑑𝜇∶𝜌(𝑔)⩽1}. (3.17) Then 𝜌^′ is a saturated function norm with the Fatou property and 𝜌^′′≡(𝜌^′)^′=𝜌. Hence, 𝐿_𝜌′ is a Banach space. Moreover, one has also 𝜌^′(𝑓)=sup{|_𝑋𝑓𝑔𝑑𝜇|∶𝜌(𝑔)⩽1}. (3.18) For each 𝜌 as above, 𝐿_𝜌 is a Banach space and 𝐿_𝜌′=(𝐿_𝜌)^#, that is, each 𝐿_𝜌 is assaying. The pair ⟨𝐿_𝜌,𝐿_𝜌′⟩ is actually a dual pair, although ⟨𝑉^#,𝑉⟩ is not. The space 𝐿_𝜌′ is called the Köthe dual or 𝛼-dual of 𝐿_𝜌 and denoted by (𝐿_𝜌)^𝛼.

However, 𝐿_𝜌′ is in general only a closed subspace of the Banach conjugate dual (𝐿_𝜌)^×; thus, the Mackey topology 𝜏(𝐿_𝜌,𝐿_𝜌′) is coarser than the 𝜌-norm topology, which is 𝜏(𝐿_𝜌,(𝐿_𝜌)^×). This defect can be remedied by further restricting 𝜌. A function norm 𝜌 is called absolutely continuous if 𝜌(𝑓_𝑛)↘0 for every sequence 𝑓_𝑛∈𝐿_𝜌 such that 𝑓₁⩾𝑓₂⩾…↘0 pointwise a.e. on 𝑋. For instance, the Lebesgue 𝐿^𝑝-norm is absolutely continuous for 1⩽𝑝<∞, but the 𝐿^∞-norm is not! Also, even if 𝜌 is absolutely continuous, 𝜌^′ need not be. Yet, this is the appropriate concept, in view of the following results: (i)𝐿_𝜌′=(𝐿_𝜌)^𝛼=(𝐿_𝜌)^× if and only if 𝜌 is absolutely continuous; (ii)𝐿_𝜌 is reflexive if and only if 𝜌  and  𝜌^′ are absolutely continuous and 𝜌 has the Fatou property.

Let 𝜌 be a saturated, absolutely continuous function norm on 𝑋, with the Fatou property and such that 𝜌^′ is also absolutely continuous. Then ⟨𝐿_𝜌,𝐿_𝜌′⟩ is a reflexive dual pair of Banach spaces. In addition, the set 𝐽 of all function norms with these properties is an involutive lattice with respect to the following partial order: 𝜌₁⩽𝜌₂ if and only if 𝜌₁(𝑓)⩽𝜌₂(𝑓),foreverymeasurable𝑓. The lattice operations are the following: (i)(𝜌₁∨𝜌₂)(𝑓)=max{𝜌₁(𝑓),𝜌₂(𝑓)},(ii)(𝜌₁∧𝜌₂)(𝑓)=inf{𝜌₁(𝑓₁)+𝜌₂(𝑓₂);𝑓₁,𝑓₂∈𝑀⁺,𝑓₁+𝑓₂=|𝑓|},(iii)involution∶𝜌↔𝜌^′.

For the corresponding Banach spaces, we have the relations 𝐿_{(𝜌1∨𝜌2)}=(𝐿_𝜌1∩𝐿_𝜌2)_proj,𝐿_{(𝜌1∧𝜌2)}=(𝐿_𝜌1+𝐿_𝜌2)_ind. (3.19)

Consider now the usual space 𝑉=𝐿¹_loc(𝑋,𝑑𝜇), with the compatibility and partial inner product defined in Section 2.2.2, so that 𝑉^#=𝐿^∞_𝑐(𝑋,𝑑𝜇). Then the construction outlined above provides 𝐿¹_loc(𝑋,𝑑𝜇) with the structure of a LBS. Indeed, one has the following result.

Proposition 3.3. Let 𝐽 be the set of saturated, absolutely continuous function norms 𝜌 on 𝑋, with the Fatou property and such that 𝜌^′ is also absolutely continuous. Let ℐ denote the set ℐ∶={𝐿_𝜌∶𝜌∈𝐽and𝐿_𝜌⊂𝐿¹_loc}. Then ℐ is a LBS, with the lattice operations defined above.

More general situations may be considered, for which we refer to [14, Section 4.4].

4. Comparing PIP-Spaces

The definition of LBS/LHS given in Section 3 leads to the proper notion of comparison between two linear compatibilities on the same vector space. Namely, we shall say that a compatibility #₁ is finer than #₂, or that #₂ is coarser than #₁, if ℱ(𝑉,#₂)is an involutive cofinal sublattice of ℱ(𝑉,#₁) (given a partially ordered set 𝐹, a subset 𝐾⊂𝐹 is cofinal to 𝐹 if, for any element 𝑥∈𝐹, there is an element 𝑘∈𝐾 such that 𝑥⩽𝑘).

Now, suppose that a linear compatiblity # is given on 𝑉. Then, every involutive cofinal sublattice of ℱ(𝑉,#) defines a coarser PIP-space, and vice versa. Thus coarsening is always possible, and will ultimately lead to a minimal PIP-space, consisting of 𝑉^# and 𝑉 only, that is, the situation of distribution spaces. However, the operation of refining is not always possible; in particular there is no canonical solution, a fortiori no unique maximal solution. There are exceptions, however, for instance, when one is given explicitly a larger set of assaying subspaces that also form, or generate, a larger involutive sublattice. To give an example, the weighted 𝐿² spaces of Section 3.3.1 form an involutive sublattice of the involutive lattice ℐ of Köthe function spaces of Section 3.3.4; thus, ℐ is a genuine refinement of the original LHS.

In the case of a LHS, refining is possible, with infinitely many solutions, by use of interpolation methods or the spectral theorem for self-adjoint operators, which are essentially equivalent in this case. In particular, one may always refine a discrete scale of Hilbert spaces into a (nonunique) continuous one. Indeed, for the scale described in Section 1, Example (ii), one has, by definition, ℋ_𝑛=𝐷(𝐴^𝑛), the domain of 𝐴^𝑛, equipped with the graph norm ‖𝑓‖_𝑛=‖𝐴^𝑛𝑓‖,𝑓∈𝐷(𝐴^𝑛), for 𝑛∈ℕ. Then, for each 0⩽𝛼⩽1, one may define ℋ_𝑛+𝛼∶={𝑓∈ℋ₀∶^∞₁𝑠^2𝑛+2𝛼𝑑⟨𝑓∣𝐸(𝑠)𝑓⟩<∞}, (4.1) where {𝐸(𝑠),1≤𝑠<∞} is the spectral family of 𝐴. With the inner product ⟨𝑓∣𝑔⟩_𝑛+𝛼=⟨𝐴^𝑛+𝛼𝑓∣𝐴^𝑛+𝛼𝑔⟩,𝑓,𝑔∈ℋ_𝑛+𝛼, (4.2)ℋ_𝑛+𝛼 is a Hilbert space and one has the continuous embeddings ℋ_𝑛+1↪ℋ_𝑛+𝛽↪ℋ_𝑛+𝛼↪ℋ_𝑛,0⩽𝛼⩽𝛽⩽1. (4.3) One may go further, as follows. Let 𝜑 be any continuous, positive function on [1,∞) such that 𝜑(𝑡) is unbounded for 𝑡→∞, but increases slower than any power 𝑡^𝛼(0<𝛼1). An example is 𝜑(𝑡)=log𝑡(𝑡⩾1). Then 𝜑(𝐴) is a well-defined self-adjoint operator, with domain 𝐷(𝜑(𝐴))={𝑓∈ℋ₀∶^∞₁(1+𝜑(𝑠))²𝑑⟨𝑓∣𝐸(𝑠)𝑓⟩<∞}. (4.4) With the corresponding inner product ⟨𝑓∣𝑔⟩_𝜑=⟨𝑓∣𝑔⟩+⟨𝜑(𝐴)𝑓∣𝜑(𝐴)𝑔⟩, (4.5)𝐷(𝜑(𝐴)) becomes a Hilbert space ℋ_𝜑. For every 𝛼,0<𝛼⩽1, one has, with proper inclusions and continuous embeddings, ℋ_𝛼↪ℋ_𝜑↪ℋ₀. (4.6) This can be continued as far as one wants, with the result that every scale of Hilbert spaces possesses infinitely many proper refinements which are themselves chains of Hilbert spaces [14, Chapter 5].

Another type of refinement consists in refining a RHS Φ⊂ℋ⊂Φ^×, by inserting a number of intermediate spaces, called interspaces, namely, spaces ℰ such that Φ↪ℰ↪Φ^× (which implies that the conjugate dual ℰ^× is also an interspace). Upon some additional conditions, the most important of which being that Φ be dense in ℰ∩ℱ with its projective topology, for any pair ℰ,ℱ of interspaces, one obtains in that way a proper refining of the original RHS. With this construction, which goes under the name of multiplication framework, one succeeds, for instance, in defining a valid (partial) multiplication between distributions. A thorough analysis may be found in [14, Section 6.3].

5. Operators on PIP-Spaces

5.1. General Definitions

As already mentioned, the basic idea of (indexed) PIP-spaces is that vectors should not be considered individually, but only in terms of the subspaces 𝑉_𝑟 (𝑟∈𝐹 or 𝑟∈𝐼), the building blocks of the structure; see (3.1). Correspondingly, an operator on a PIP-space should be defined in terms of assaying subspaces only, with the proviso that only bounded operators between Hilbert or Banach spaces are allowed. Thus an operator is a coherent collection of bounded operators. More precisely, one has the following.

Definition 5.1. Given a LHS or LBS 𝑉_𝐼={𝑉_𝑟,𝑟∈𝐼}, an operator on 𝑉_𝐼 is a map 𝐴∶𝒟(𝐴)→𝑉, such that (i)𝒟(𝐴)=⋃_{𝑞∈d(𝐴)}𝑉_𝑞, where d(𝐴) is a nonempty subset of 𝐼,(ii)for every 𝑞∈d(𝐴), there exists a 𝑝∈𝐼 such that the restriction of 𝐴 to 𝑉_𝑞 is linear and continuous into 𝑉_𝑝 (we denote this restriction by 𝐴_𝑝𝑞), (iii)𝐴 has no proper extension satisfying (i) and (ii).

The linear bounded operator 𝐴_𝑝𝑞∶𝑉_𝑞→𝑉_𝑝 is called a representative of 𝐴. In terms of the latter, the operator 𝐴 may be characterized by the set j(𝐴)={(𝑞,𝑝)∈𝐼×𝐼∶𝐴_𝑝𝑞exists}. Thus the operator 𝐴 may be identified with the collection of its representatives: 𝐴≃{𝐴_𝑝𝑞∶𝑉_𝑞→𝑉_𝑝∶(𝑞,𝑝)∈j(𝐴)}. (5.1) By condition (ii), the set d(𝐴) is obtained by projecting j(𝐴) on the “first coordinate" axis. The projection i(𝐴) on the “second coordinate" axis plays, in a sense, the role of the range of 𝐴. More precisely, d(𝐴)={𝑞∈𝐼∶thereisa𝑝suchthat𝐴_𝑝𝑞exists},i(𝐴)={𝑝∈𝐼∶thereisa𝑞suchthat𝐴_𝑝𝑞exists}. (5.2) The following properties are immediate see the (see Figure 2): (i)d(𝐴) is an initial subset of 𝐼: if 𝑞∈d(𝐴) and 𝑞^′<𝑞, then 𝑞^′∈d(𝐴), and 𝐴_𝑝𝑞′=𝐴_𝑝𝑞𝐸_𝑞𝑞′, where 𝐸_𝑞𝑞′ is a representative of the unit operator (this is what we mean by a `coherent' collection), (ii)i(𝐴) is a final subset of 𝐼: if 𝑝∈i(𝐴) and 𝑝^′>𝑝, then 𝑝^′∈i(𝐴) and 𝐴_𝑝′𝑞=𝐸_𝑝′𝑝𝐴_𝑝𝑞.(iii)j(𝐴)⊂d(𝐴)×i(𝐴), with strict inclusion in general.

Figure 2 

Characterization of the operator 𝐴, in the case of a scale.

We denote by Op(𝑉_𝐼) the set of all operators on 𝑉_𝐼. Of course, a similar definition may be given for operators 𝐴∶𝑉_𝐼→𝑌_𝐾 between two LHSs or LBSs.

Since 𝑉^# is dense in 𝑉_𝑟, for every 𝑟∈𝐼, an operator may be identified with a separately continuous sesquilinear form on 𝑉^#×𝑉^#. Indeed, the restriction of any representative 𝐴_𝑝𝑞 to 𝑉^#×𝑉^# is such a form, and all these restrictions coincide. Equivalently, an operator may be identified with a continuous linear map from 𝑉^# into 𝑉 (continuity with respect to the respective Mackey topologies).

But the idea behind the notion of operator is to keep also the algebraic operations on operators; namely, we define the following operations:(i) Adjoint: Every 𝐴∈Op(𝑉_𝐼) has a unique adjoint 𝐴^×∈Op(𝑉_𝐼), defined by the relation ⟨𝐴^×𝑥∣𝑦⟩=⟨𝑥∣𝐴𝑦⟩,for𝑦∈𝑉_𝑟,𝑟∈d(𝐴),𝑥∈𝑉_𝑠,𝑠∈i(𝐴), (5.3) that is, (𝐴^×)_𝑟𝑠=(𝐴_𝑠𝑟)^∗(usual Hilbert/Banach space adjoint). It follows that 𝐴^××=𝐴, for every 𝐴∈Op(𝑉_𝐼): no extension is allowed, by the maximality condition (iii) of Definition 5.1.(ii) Partial Multiplication: The product AB is defined if and only if there is a 𝑞∈i(𝐵)∩d(𝐴), that is, if and only if there is a continuous factorization through some 𝑉_𝑞: 𝑉−→_𝑟^𝐵𝑉−→_𝑞^𝐴𝑉_𝑠,thatis,(𝐴𝐵)_𝑠𝑟=𝐴_𝑠𝑞𝐵_𝑞𝑟. (5.4)

It is worth noting that, for a LHS/LBS, the domain 𝒟(𝐴) is always a vector subspace of 𝑉 (this is not true for a general PIP-space). Therefore, Op(𝑉_𝐼) is a vector space and a partial *-algebra [22].

The concept of PIP-space operator is very simple, yet it is a far-reaching generalization of bounded operators. It allows indeed to treat on the same footing all kinds of operators, from bounded ones to very singular ones. By this, we mean the following, loosely speaking. Take 𝑉_𝑟⊂𝑉_𝑜≃𝑉_𝑜⊂𝑉_𝑠(𝑉_𝑜=Hilbertspace). (5.5) Three cases may arise:(i)if 𝐴_𝑜𝑜 exists, then 𝐴 corresponds to a bounded operator 𝑉_𝑜→𝑉_𝑜,(ii)if 𝐴_𝑜𝑜 does not exist, but only 𝐴_𝑜𝑟∶𝑉_𝑟→𝑉_𝑜, with 𝑟<𝑜, then 𝐴 corresponds to an unbounded operator, with domain 𝐷(𝐴)⊃𝑉_𝑟,(iii)if no 𝐴_𝑜𝑟 exists, but only 𝐴_𝑠𝑟∶𝑉_𝑟→𝑉_𝑠, with 𝑟<𝑜<𝑠, then 𝐴 corresponds to a singular operator, with Hilbert space domain possibly reduced to {0}.

5.2. Special Classes of Operators on PIP-Spaces

Exactly as for Hilbert or Banach spaces, one may define various types of operators between PIP-spaces, in particular LBS/LHSs. We discuss briefly the most important classes, namely, regular operators, homomorphisms and isomorphisms, unitary operators, symmetric operators, and orthogonal projections. Further details may be found in the monograph [14].

5.2.1. Regular and Totally Regular Operators

An operator 𝐴 on a nondegenerate PIP-space 𝑉_𝐼, with positive-definite partial inner product, in particular, a LBS/LHS, is called regular if d(𝐴)= i(𝐴)=𝐼 or, equivalently, if 𝐴∶𝑉^#→𝑉^#and𝐴∶𝑉→𝑉 continuously for the respective Mackey topologies. This notion depends only on the pair (𝑉^#,𝑉), not on the particular compatibility #. The set of all regular operators 𝑉_𝐼→𝑉_𝐼 is denoted by Reg(𝑉_𝐼). Thus a regular operator may be multiplied both on the left and on the right by an arbitrary operator. Clearly, the set Reg(𝑉_𝐼) is a *-algebra and can often be identified with an O *-algebra [22, 23].

We give two examples. (1)If 𝑉=𝜔,𝑉^#=𝜑, then Op(𝜔) consists of arbitrary infinite matrices and Reg(𝜔) of infinite matrices with finite rows and finite columns.(2) If 𝑉=𝒮^×,𝑉^#=𝒮, then Op(𝒮^×) consists of arbitrary tempered kernels, while Reg(𝒮^×) contains those kernels that can be extended to 𝒮^× and map 𝒮 into itself. A nice example is the Fourier transform.

An operator 𝐴 is called totally regular if j(𝐴) contains the diagonal of 𝐼×𝐼, that is, 𝐴_𝑟𝑟 exists for every 𝑟∈𝐼 or 𝐴 maps every 𝑉_𝑟 into itself continuously.

5.2.2. Homomorphisms

Let 𝑉_𝐼,𝑌_𝐾 be two LHSs or LBSs. An operator 𝐴∈Op(𝑉_𝐼,𝑌_𝐾) is called a homomorphism if (i)j(𝐴)=𝐼×𝐾 and j(𝐴^×)=𝐾×𝐼,(ii)𝑓#_𝐼𝑔 implies that 𝐴𝑓#_𝐾𝐴𝑔.

We denote by Hom(𝑉_𝐼,𝑌_𝐾) the set of all homomorphisms from 𝑉_𝐼 into 𝑌_𝐾. The following properties are easy to prove: (i)𝐴∈Hom(𝑉_𝐼,𝑌_𝐾) if and only if 𝐴^×∈Hom(𝑌_𝐾,𝑉_𝐼),(ii)if 𝐴∈Hom(𝑉_𝐼,𝑌_𝐾), then j(𝐴^×𝐴) contains the diagonal of 𝐼×𝐼 and j(𝐴𝐴^×) contains the diagonal of 𝐾×𝐾.

The homomorphism 𝑀∈Hom(𝑊_𝐼,𝑌_𝐾) is a monomorphism if 𝑀𝐴=𝑀𝐵 implies that 𝐴=𝐵, for any two elements of 𝐴,𝐵∈Hom(𝑉_𝐼,𝑊_𝐿), where 𝑉_𝐼 is any PIP-space. Typical examples of monomorphisms are the inclusion maps resulting from the restriction of a support. Take for instance, 𝐿¹_loc(𝑋,𝑑𝜇), the space of locally integrable functions on a measure space (𝑋,𝜇). Let Ω be a measurable subset of 𝑋 and Ω^′ its complement, both of nonzero measure, and construct the space 𝐿¹_l𝑜𝑐(Ω,𝑑𝜇), which is a PIP-subspace of 𝐿¹_loc(𝑋,𝑑𝜇) (see Section 5.2.5). Given 𝑓∈𝐿¹_loc(𝑋,𝑑𝜇), define 𝑓^(Ω)=𝑓𝜒_Ω, where 𝜒_Ω is the characteristic function of 𝜒_Ω. Then we obtain an injection monomorphism 𝑀^(Ω)∶𝐿¹_loc(Ω,𝑑𝜇)→𝐿¹_loc(𝑋,𝑑𝜇) as follows: (𝑀^(Ω)𝑓^(Ω)𝑓)(𝑥)=^(Ω)𝑓(𝑥),if𝑥∈Ω,0,if𝑥∉Ω,^(Ω)∈𝐿¹_loc(Ω,𝑑𝜇). (5.6) If we consider the lattice of weighted Hilbert spaces {𝐿²(𝑟)} in this PIP-space, then the correspondence 𝑟↔𝑟^(Ω)=𝑟𝜒_Ω is a bijection between the corresponding involutive lattices.

The homomorphism 𝐴∈Hom(𝑉_𝐼,𝑌_𝐾) is an isomorphism if there exists a homomorphism 𝐵∈Hom(𝑌_𝐾,𝑉_𝐼) such that 𝐵𝐴=1_𝑉,𝐴𝐵=1_𝑌, where 1_𝑉,1_𝑌 denote the identity operators on 𝑉_𝐼,𝑌_𝐾, respectively.

For instance, the Fourier transform is an isomorphism of the Schwartz RHS 𝒮⊂𝐿²⊂𝒮^× onto itself and, similarly, of the Feichtinger triplet (6.16) onto itself. In both cases, the property extends to several scales of lattices interpolating between the two extreme spaces, for instance, the Hilbert chain of the Hermite representation of tempered distributions.

5.2.3. Unitary Operators and Group Representations

The operator 𝑈∈Op(𝑉_𝐼,𝑌_𝐾) is unitary if 𝑈^×𝑈 and 𝑈𝑈^× are defined and 𝑈^×𝑈=1_𝑉,𝑈𝑈^×=1_𝑌. We emphasize that unitary operators need not be homomorphisms ! In fact, unitarity is a rather weak property and it is insufficient for group representations.

Thus a unitary representation of a group 𝐺 into a PIP-space 𝑉_𝐼 is defined as a homomorphism of 𝐺 into the unitary isomorphisms of 𝑉_𝐼. Given such a unitary representation 𝑈 of 𝐺 into 𝑉_𝐼, where the latter has the central Hilbert space ℋ₀, consider the representative 𝑈₀₀(𝑔) of 𝑈(𝑔) in ℋ₀. Then 𝑔↦𝑈₀₀(𝑔) is a unitary representation of 𝐺 into ℋ₀, in the usual sense.

To give an example, let 𝑉_𝐼 be the scale built on the powers of the operator (Hamiltonian) 𝐻=−Δ+𝑣(𝐫) on 𝐿²(ℝ³,𝑑𝐱), where Δ is the Laplacian on ℝ³ and 𝑣 is a (nice) rotation invariant potential. The system admits as symmetry group 𝐺= SO(3) (the full symmetry group might be larger; for instance, the Coulomb potential admits SO(4) as symmetry group for its bound states.) and the representation 𝑈₀₀ is the natural representation of SO(3) in 𝐿²(ℝ³): [𝑈₀₀(𝜌)𝜓](𝐱)=𝜓(𝜌⁻¹𝐱),𝜌∈SO(3). (5.7) Then 𝑈₀₀ extends to a unitary representation 𝑈 by totally regular isomorphisms of 𝑉_𝐼. Angular momentum decompositions, corresponding to irreducible representations of SO(3), extend to 𝑉_𝐼 as well. In addition, this is a good setting also for representations of the Lie algebra 𝔰𝔬(3). Notice that the representation 𝑈 is totally regular, but this need not be the case. For instance, if the potential 𝑣 is not rotation invariant, 𝑈 will no longer be totally regular, although it is still an isomorphism.

5.2.4. Symmetric Operators

An operator 𝐴∈Op(𝑉_𝐼) is symmetric if 𝐴^×=𝐴. Since one has 𝐴^××=𝐴, no extension is allowed, by the maximality condition. Thus symmetric operators behave essentially like bounded self-adjoint operators in a Hilbert space. Yet, they can be very singular, as indicated above, for a chain 𝑉^#⊂⋯⊂𝑉_𝑟⊂𝑉_𝑜=𝑉_𝑜⊂𝑉_𝑠⊂⋯⊂𝑉. (5.8) In this case, the question is whether a symmetric operator 𝐴∈Op(𝑉_𝐼) has a self-adjoint restriction to the central Hilbert space 𝑉_𝑜. In a Hilbert space context, an answer is given by the celebrated KLMN theorem (KLMN stands for Kato, Lax, Lions, Milgram, Nelson). Actually, this classical result already has a distinct PIP-space flavor. Thus is not surprising that the KLMN theorem has a natural generalization to a LHS or a PIP-space with positive-definite partial inner product and central Hilbert space 𝑉_𝑜=𝑉_𝑜, and a quadratic form version as well [14, Section 3.3.5].

An interesting application is a correct description of very singular Hamiltonians in quantum mechanics, typically with 𝛿 or 𝛿^′ interactions. For instance, one can treat in this way the Kronig-Penney crystal model, which consists of a 𝛿 potential at each node of a lattice, in one, two, or three dimensions [24, 25].

5.2.5. Orthogonal Projections, Bases, Frames

An operator 𝑃 on a nondegenerate PIP-space 𝑉, respectively, a LBS/LHS 𝑉_𝐼, is an orthogonal projection if 𝑃∈Hom(𝑉_𝐼) and 𝑃²=𝑃=𝑃^×. It follows immediately from the definition that an orthogonal projection is totally regular, that is, j(𝑃) contains the diagonal 𝐼×𝐼, or still that 𝑃 leaves every assaying subspace invariant. Equivalently, 𝑃 is an orthogonal projection if 𝑃 is an idempotent operator (that is, 𝑃²=𝑃) such that {𝑃𝑓}^#⊇{𝑓}^# for every 𝑓∈𝑉 and ⟨𝑔∣𝑃𝑓⟩=⟨𝑃𝑔∣𝑓⟩ whenever 𝑓#𝑔. We denote by Proj(𝑉) the set of all orthogonal projections in 𝑉 and similarly for Proj(𝑉_𝐼).

These projection operators enjoy several properties similar to those of Hilbert space projectors. Two of them are of special interest in the present context.(i)Given a nondegenerate PIP-space 𝑉, there is a natural notion of subspace, called orthocomplemented, which guarantees that such a subspace 𝑊 of 𝑉 is again a nondegenerate PIP-space with the induced compatibility relation and the restriction of the partial inner product. There are equivalent topological conditions, so that orthocomplemented subspaces are the proper PIP-subspaces [26]. Then the basic theorem about projections states that a subspace 𝑊 of 𝑉 is orthocomplemented if and only if 𝑊 is the range of an orthogonal projection 𝑃∈Proj(𝑉), that is,𝑊=𝑃𝑉. Then 𝑉=𝑊⊕𝑍, where 𝑍 is another orthocomplemented subspace.(ii)An orthogonal projection 𝑃 is of finite rank if and only if 𝑊=Ran𝑃⊂𝑉^# and 𝑊∩𝑊^⟂={0} (this condition is, of course, superfluous when the partial inner product is positive definite).

There is a natural partial order on the set of projections: 𝑃_𝑊≤𝑃_𝑌ifandonlyif𝑊⊆𝑌, (5.9) but the lattice properties of Proj(𝑉) are unknown. Thus we expect that quantum logic may be reformulated in a PIP-space language only under additional restrictions on 𝑉.

Property (ii) has important consequences for the structure of bases. First we recall that a sequence {𝑒_𝑛,𝑛=1,2,…} of vectors in a Banach space 𝐸 is a Schauder basis if, for every 𝑓∈𝐸, there exists a unique sequence of scalar coefficients {𝑐_𝑘,𝑘=1,2,…} such that lim_𝑚→∞‖𝑓−∑^𝑚_𝑘=1𝑐_𝑘𝑒_𝑘‖ = 0. Then one may write 𝑓=^∞_𝑘=1𝑐_𝑘𝑒_𝑘. (5.10) The basis is unconditional if the series (5.10) converges unconditionally in 𝐸 (i.e., it keeps converging after an arbitrary permutation of its terms).

A standard problem is to find, for instance, a sequence of functions that is an unconditional basis for all the spaces 𝐿^𝑝(ℝ),1<𝑝<∞. In the PIP-space language, this statement means that the basis vectors must belong to 𝑉^#=⋂_1<𝑝<∞𝐿^𝑝(ℝ). Also, since (5.10) means that 𝑓 may be approximated by finite sums, the property (ii) of orthogonal projections implies that all the basis vectors must belong to 𝑉^#. Some examples are given in Section 6.2.5.

Actually, in the context of signal processing, orthogonal (in the Hilbert sense) bases are not enough; one needs also biorthogonal bases and, more generally, frames. We recall that a countable family of vectors {𝜓_𝑛} in a Hilbert space ℋ is called a frame if there are two positive constants m, M, with 0<m ⩽ M <∞, such that m‖𝑓‖²⩽^∞_𝑛=1|⟨𝜓_𝑛∣𝑓⟩|²⩽M‖𝑓‖²,∀𝑓∈ℋ. (5.11) The two constants m, M are called frame bounds. If m= M, the frame is said to be tight. Consider the analysis operator 𝐶∶ℋ→ℓ² defined by 𝐶∶𝑓↦{|⟨𝜓_𝑛∣𝑓⟩|} and the frame operator 𝑆=𝐶^∗𝐶. Then the vectors 𝜓_𝑛=𝑆⁻¹𝜓_𝑛 also constitute a frame, called the canonical dual frame, and one has the (strongly converging) expansions 𝑓=^∞_𝑛=1|⟨𝜓_𝑛∣𝑓⟩|𝜓_𝑛=^∞_𝑛=1|⟨𝜓_𝑛∣𝑓⟩|𝜓_𝑛. (5.12) Then the considerations made above for bases should apply to frame vectors as well, that is, the vectors 𝜓_𝑛,𝜓_𝑛 should also belong to 𝑉^#.

6. Applications of PIP-Spaces

6.1. Applications in Mathematical Physics

PIP-spaces have found many applications in mathematical physics, mostly in quantum physics. We will sketch a few of them in this section. Most of what follows is described in detail in [14].

6.1.1. Dirac Formalism in Quantum Mechanics

The mathematical description of a quantum system rests on three basic principles: (i) The superposition principle, which implies that the set of states of the system has a linear structure; (ii) The notion of transition amplitude, given by an inner product: 𝐴(𝜓₁→𝜓₂)=⟨𝜓₂∣𝜓₁⟩, which yields transition probabilities by 𝑃(𝜓₁→𝜓₂)=|⟨𝜓₂∣𝜓₁⟩|²; and (iii) The probabilistic interpretation, which requires that ⟨𝜓∣𝜓⟩=‖𝜓‖²>0, whenever 𝜓≠0.

Combining these basic principles implies that the set of states of the system is a positive-definite inner product space Φ, that is, a pre-Hilbert space. On this basis, Dirac developed a formalism for quantum physics with great computational capacity and broad predictive power. The essential features of Dirac's formalism are the following. (i)Physical observables are represented by linear operators in the space Φ and these operators form an algebra. Therefore, it makes sense to arbitrarily add and multiply operators to form new operators.(ii)For a given quantum physical system, there exist complete systems of commuting observables (CSCO) in the algebra of observables. The system of eigenvectors for a chosen CSCO provides a basis for the space Φ, that is, every vector 𝜙∈Φ can be expanded into the eigenvectors of the CSCO.

In the simplest case, a CSCO consists of only one observable 𝐴, with a mixed spectrum consisting of discrete eigenvalues {𝜆_𝑛}=𝜎_𝑝(𝐴) and a continuous part {𝜆}=𝜎_𝑐(𝐴). The corresponding eigenvectors, written as |𝜆_𝑛⟩,|𝜆⟩, respectively, obey “orthogonality" relations ⟨𝜆_𝑚∣𝜆_𝑛⟩=𝛿_𝑚𝑛,⟨𝜆∣𝜆_𝑛⟩=0,⟨𝜆∣𝜆^′⟩=𝛿(𝜆−𝜆^′). (6.1) Then every 𝜙∈Φ can be expanded as 𝜙=_𝑛|𝜆_𝑛⟩⟨𝜆_𝑛∣𝜙⟩+_{𝜎𝑐(𝐴)}𝑑𝜆|𝜆⟩⟨𝜆∣𝜙⟩. (6.2) Clearly the eigenvectors |𝜆⟩ cannot belong to the pre-Hilbert space Φ, nor to its completion ℋ. Thus Dirac's formalism, while extremely practical and used by physicists on a daily basis, is not mathematically well defined.

For that reason, von Neumann formulated a rigorous version of quantum mechanics, in a pure Hilbert space language. His formulation consists in the following two axioms: (i) Pure states are represented by rays (i.e., one-dimensional subspaces) in a Hilbert space ℋ; and (ii) Observables are represented by self-adjoint operators in ℋ. This formulation is well defined mathematically, but too restrictive. Nonnormalizable eigenvectors, corresponding to points of a continuous spectrum, cannot belong to ℋ, yet they are extremely useful and often have a clear physical meaning (plane waves, for instance). Observables may be unbounded, so that domain considerations must be taken into account. In particular, unbounded operators may not always be multiplied. Thus it is understandable that the large majority of physicists stay with Dirac's formalism.

This had prompted several authors [27, 28, 30, 31] to propose a rigorous version in terms of a RHS Φ⊂ℋ⊂Φ^×. In this scheme, the space Φ is constructed from the basic observables (labeled observables) of the system at hand and is interpreted as the space of all physically preparable states. The conjugate dual Φ^× contains idealized states (probes), identified with measurement devices. In that context, let 𝐴 be an observable, represented by a self-adjoint operator in ℋ such that 𝐴∶Φ→Φ continuously. Then 𝐴 may be transposed by duality to a linear operator 𝐴^×∶Φ^×→Φ^×, which is an extension of 𝐴^†∶=𝐴^∗↾Φ, where 𝐴^∗ is the usual Hilbert space adjoint operator, namely, ⟨𝜙∣𝐴^×𝐹⟩=⟨𝐴𝜙∣𝐹⟩,∀𝜙∈Φ,𝐹∈Φ^×. (6.3) For such an operator, the vector 𝜉_𝜆∈Φ^× is called a generalized eigenvector of 𝐴, with eigenvalue 𝜆∈ℂ, if it satisfies ⟨𝜙∣𝐴^×𝜉_𝜆⟩∶=𝐴^×𝜉_𝜆(𝜙)=𝜆𝜉_𝜆(𝜙)≡𝜆⟨𝜙∣𝜉_𝜆⟩,∀𝜙∈Φ. (6.4) Then the spectral theorem of Gel'fand-Maurin [5, 6] states that 𝐴 possesses in Φ^× a complete orthonormal set of generalized eigenvectors 𝜉_𝜆∈Φ^×,𝜆∈ℝ. This means that, for any two 𝜙,𝜓∈Φ, one has (we split again into the discrete and the continuous part of the spectrum of 𝐴) ⟨𝜙∣𝜓⟩=_𝑛⟨𝜙∣𝜆_𝑛⟩⟨𝜆_𝑛∣𝜓⟩+⟨𝜙∣𝜆⟩⟨𝜆∣𝜓⟩𝜌(𝜆)𝑑𝜆, (6.5) where 𝜌 is a non-negative integrable function. In that way one recovers essentially Dirac's bra-and-ket formalism. This approach is based on a RHS, but the construction is such that a PIP-space version is easily obtained—and is in fact closer to Dirac's spirit. For more details, see [14, Section 7.1.1].

6.1.2. Symmetries, Singular Interactions in Quantum Mechanics

Several other topics in quantum mechanics can be advantageously formulated in a RHS or PIP-space language, for instance, the implementation of symmetries, with the two dual points of view, the active one and the passive one [32]. A symmetry group is represented by a unitary representation in ℋ that extends to a unitary representation 𝑈 in the enveloping PIP-space, in the sense defined in Section 5.2.3. Then, in accordance with the physical interpretation given above, the active point of view corresponds to the action of 𝑈 in 𝑉^#≡Φ, the passive one to the action on 𝑉≡Φ^×.

Another problem is a correct definition of a Hamiltonian with a singular interaction, already mentioned in Section 5.2.4. In the simplest case, the standard definition is 𝐻=−(Δ/2𝑚)+𝑉, where the interaction 𝑉 is given by some reasonable function (potential). However, there are cases where a singular interaction is needed, for instance when 𝑉 is replaced formally by a 𝛿 function or a 𝛿^′ function, with support in a point (or several) or a manifold of lower dimension. Then the usual formulation is based on von Neumann's theory of self-adjoint extensions of symmetric operators, sometimes coupled with Krein's formula [33]. But here the PIP-space approach is a convenient substitute to that approach, as shown in [24, 25] and [14, Section 7.1.3].

6.1.3. Quantum Scattering Theory

In scattering theory, it is common to use scales of Hilbert spaces built on the powers of 𝐴₁∶=(1+|𝐱|²) or 𝐴₂∶=(1+|𝐩|²), and the LHS obtained by combinations of both. This example contains the Sobolev spaces (the scale built on 𝐴₂), the weighted spaces 𝐿²_𝑠 (the scale built on 𝐴₁), and spaces of mixed type. In particular, operators of the form 𝑓(𝐱)𝑔(𝐩), for suitable functions 𝑓,𝑔, play an essential role in the so-called phase-space approach to scattering theory and they may be controlled by this LHS. For instance, their trace ideal properties may be derived in this way and they are used for proving the absence of singular continuous spectrum by the limiting absorption principle.

On the other hand, the Weinberg-van Winter (WVW) formulation of scattering theory [34–36] has a very natural interpretation in terms of a LHS, whose components, including the extreme ones, are Hilbert spaces consisting of functions analytic in a sector; thus the indexing parameter is the opening angle of that sector. This technique has allowed to show that the WVW formalism is a particular case of the Complex Scaling Method [14, Section 7.2.3], a result hitherto unknown.

6.1.4. Quantum Field Theory

Mathematically rigorous formulations of QFT rely heavily on a RHS or a PIP-space approach, primarily Wightman's axiomatic formulation. There, indeed, a quantum field is defined as an operator-valued distribution, which is customarily written in terms of an unsmeared field (field at a point) 𝐴(𝑥), as 𝐴(𝑓)=_ℝ4𝐴(𝑥)𝑓(𝑥)𝑑𝑥,𝑓∈𝒮(ℝ⁴). (6.6) Under quite reasonable assumptions, the unsmeared field can be defined as a map from 𝒮(ℝ⁴) into Op(𝑉), where 𝑉 is a conveniently chosen PIP-space. This allows to give the previous formula a rigorous mathematical meaning [14, Section 7.3.1].

Another PIP-space version of QFT is the Fock construction (tensor algebra) over the RHS 𝒮(𝒱⁺_𝑚)↪𝐿²(𝒱⁺_𝑚,𝑑𝜇)↪𝒮^×(𝒱⁺_𝑚), (6.7) where 𝒱⁺_𝑚 denotes the forward mass shell 𝒱⁺_𝑚={𝑝∈ℝ⁴∶𝑝²=𝑚²,𝑝₀>0} and 𝑑𝜇 the Lorentz invariant measure 𝑑³𝑝/𝑝₀ on it. Write Φ₁=𝒮(𝒱⁺_𝑚), the space of “good" one-particle states. Then define Φ_𝑛=Φ^⨂𝑠𝑛₁, (6.8) where the right-hand side denotes the symmetrized tensor product of 𝑛 copies of Φ₁, corresponding to 𝑛-boson states. Again Φ_𝑛 is reflexive, complete, and nuclear with respect to its natural topology, and it can be described as the end space of a scale of Hilbert spaces. Finally, define Φ=^∞_𝑛=0Φ_𝑛, (6.9) that is, the topological direct sum of the component spaces. Elements of Φ are finite sequences 𝑓={𝑓₀,𝑓₁,…,𝑓_𝑛,…},𝑓₀∈ℂ,𝑓_𝑛∈Φ_𝑛, that is, totally symmetric functions of Schwartz type. The space Φ is reflexive, complete, and nuclear with respect to the direct sum topology. Its dual is the topological product Φ^×=^∞_𝑛=0Φ^×_𝑛. (6.10) Thus we get a suitable RHS, in which the central Hilbert space ℋ is Fock space, that is, the tensor algebra Γ(Φ₁) over Φ₁.

Other examples are the construction of QFT via the Borchers algebra, Nelson's Euclidean field theory, or the precise treatment of unsmeared fields (fields at a point). See [14, Section 7.3] for a detailed presentation.

6.1.5. Representations of Lie Groups and Lie Algebras

Let us return to the situation described in Section 5.2.3. We start with a strongly continuous unitary representation 𝑈₀₀ of a Lie group 𝐺 in a Hilbert space ℋ₀ and seek to build a PIP-space 𝑉_𝐼, with ℋ₀ being its central Hilbert space, such that 𝑈₀₀ extends to a unitary representation 𝑈 into 𝑉_𝐼.

The solution of this problem is well known from Nelson's theory of analytic vectors. Let Δ be the closure of the Nelson operator Δ∶=∑^𝑛_𝑗=1𝑋²_𝑗, where {𝑋_𝑗,𝑗=1,…,𝑛} are the representatives under 𝑈₀₀ of the elements of a basis of the Lie algebra 𝔤 of 𝐺. Δ is essentially self-adjoint on the so-called Gårding domain ℋ^𝐺₀, Δ is self-adjoint, and Δ⩾0. Define 𝑉_𝐼∶={ℋ_𝑛,𝑛∈ℤ} as the canonical scale of Hilbert spaces generated by the operator (Δ+1): 𝑉^#∶=𝒟^∞(Δ)↪ℋ₀↪𝑉∶=𝒟_∞(Δ). (6.11) First, one has 𝒟^∞(Δ)=⋂^∞_𝑛=1𝐷(Δ^𝑛)=ℋ^∞₀, the space of 𝐶^∞-vectors of 𝑈₀₀. Next, for every 𝑔∈𝐺,𝑈₀₀(𝑔) leaves each ℋ_𝑛,𝑛∈ℕ, invariant and its restriction 𝑈_𝑛𝑛(𝑔)∶ℋ_𝑛→ℋ_𝑛 is continuous; thus it can be transposed to a continuous map 𝑈_𝑛𝑛(𝑔⁻¹)∶ℋ_𝑛→ℋ_𝑛. It follows that 𝑈₀₀ extends to a unitary representation 𝑈 in the LHS 𝑉_𝐼. Corresponding to the triplet (6.11), we have three representations of 𝐺, namely, 𝑈₀₀, its restriction 𝑈_∞∞, and the dual 𝑈_∞∞ of the latter, which is an extension of the first two. All three are continuous. Moreover, if one of the three is topologically irreducible (i.e., there is no proper invariant closed subspace), so are the other two.

In addition to the representations of the group 𝐺, the scale 𝑉_𝐼 is the natural tool for studying the properties of the operators representing elements of the Lie algebra 𝔤 or the universal enveloping algebra 𝔘(𝔤) of 𝐺. For every element 𝑥∈𝔤 or 𝐿∈𝔘(𝔤), the representative 𝑈(𝑥), respectively 𝑈(𝐿), originally defined on ℋ^𝐺₀, extends to a regular operator on 𝑉_𝐼. These regular operators have in general no {0,0}-representative, since 𝑥 and 𝐿 are represented in ℋ₀ by unbounded operators. As in the case of the group 𝐺, one gets three *-representations of the enveloping algebra 𝔘(𝔤), and in particular of the Lie algebra 𝔤, in the three spaces of the triplet (6.11). Namely, one has, for every 𝐿,𝐿₁,𝐿₂∈𝔘(𝔤), 𝑈(𝐿₁)𝑈(𝐿₂)=𝑈(𝐿₁𝐿₂),𝑈(𝐿⁺)=𝑈(𝐿)^×, (6.12) where 𝐿↔𝐿⁺ is the involution on 𝔘(𝔤). These representations have the same irreducibility properties as the corresponding ones of the group. See [14, Section 7] for further details.

6.2. Applications in Analysis and Signal Processing

Many families of function spaces of interest in analysis or signal processing are, or contain, lattices of Banach spaces. To quote a few: amalgam spaces, modulation spaces, Besov spaces, 𝛼-modulation spaces, coorbit spaces, which contain many of the previous cases. We shall describe them briefly in succession. For further information about those spaces, we refer to our monograph [14, Chapters 4 and 8].

6.2.1. Amalgam Spaces

A situation intermediate between the mixed-norm spaces 𝐿^𝑝,𝑞(ℝ^2𝑑) (for 𝑚≡1) and the spaces ℓ^𝑝,𝑞(ℤ^2𝑑) is that of the so-called amalgam spaces. They were introduced specifically to overcome the inability of the 𝐿^𝑝 norms to distinguish between the local and the global behavior of functions. The simplest ones are the spaces 𝑊(𝐿^𝑝,ℓ^𝑞) of Wiener [37], which consist of functions on ℝ which are locally in 𝐿^𝑝 with ℓ^𝑞 behavior at infinity, in the sense that the 𝐿^𝑝 norms over the intervals (𝑛,𝑛+1) form an ℓ^𝑞 sequence. It is a Banach space for the norm ‖𝑓‖_𝑝,𝑞={^∞_𝑛=−∞[_𝑛^𝑛+1|𝑓(𝑥)|^𝑝𝑑𝑥]^𝑞/𝑝}^1/𝑞,1⩽𝑝,𝑞<∞. (6.13) The following inclusion relations, with all embeddings continuous, derive immediately from those of the 𝐿^𝑝 and the ℓ^𝑞 spaces.(i)If 𝑞₁⩽𝑞₂, then 𝑊(𝐿^𝑝,ℓ^𝑞1)⊂𝑊(𝐿^𝑝,ℓ^𝑞2). (ii)If 𝑝₁⩽𝑝₂, then 𝑊(𝐿^𝑝2,ℓ^𝑞)⊂𝑊(𝐿^𝑝1,ℓ^𝑞).

Thus the smallest space is 𝑊(𝐿^∞,ℓ¹) and the largest space is 𝑊(𝐿¹,ℓ^∞). As for duality, one has 𝑊(𝐿^𝑝,ℓ^𝑞)^×=𝑊(𝐿^𝑝,ℓ^𝑞),for1⩽𝑝,𝑞<∞.

The interesting fact is that, for 1⩽𝑝,𝑞⩽∞, the set 𝒥 of all amalgam spaces {𝑊(𝐿^𝑝,ℓ^𝑞)} may be represented by the points (𝑝,𝑞) of the same unit square J as in the example of the 𝐿^𝑝 spaces, with the same order structure. However, 𝒥 is not a lattice with respect to the order (3.12). One has indeed 𝑊(𝐿^𝑝,ℓ^𝑞)∧𝑊(𝐿^𝑝′,ℓ^𝑞′)⊃𝑊(𝐿^{𝑝∨𝑝′},ℓ^{𝑞∧𝑞′}),𝑊(𝐿^𝑝,ℓ^𝑞)∨𝑊(𝐿^𝑝′,ℓ^𝑞′)⊂𝑊(𝐿^{𝑝∧𝑝′},ℓ^{𝑞∨𝑞′}), (6.14) where again ∧ means intersection with projective norm and ∨ means vector sum with inductive norm, but equality is not obtained. Thus, as in the previous case, one gets chains by varying either 𝑝 or 𝑞, but not both.

A very useful class of amalgam spaces is the family 𝑊(ℱ𝐿^𝑝,ℓ^𝑞),1⩽𝑝,𝑞⩽∞, where ℱ𝐿^𝑝 denotes the set of Fourier transforms of 𝐿^𝑝 functions (one may even add weights on both spaces). These spaces have, for instance, nice inclusion and convolution properties.

Among these, the most interesting one is 𝒮₀=𝑊(ℱ𝐿¹,ℓ¹), called the Feichtinger algebra. The space 𝒮₀ has many interesting properties; for instance, one has the following. (i)𝒮₀ is a Banach space for the norm ‖𝑓‖_𝒮0=‖𝑉_𝑔0𝑓‖₁, and 𝒮↪𝒮₀↪𝐿², with all embeddings continuous with dense range. Here 𝑔₀ is the Gaussian and 𝑉_𝑔𝑓 denotes the Short-Time Fourier (or Gabor) Transform of 𝑓∈𝐿²(ℝ^𝑑), given in (6.17) below.(ii)𝒮₀ is a Banach algebra with respect to pointwise multiplication and convolution. (iii)Time-frequency shifts 𝑇_𝑥𝑀_𝜔 are isometric on 𝒮₀∶‖𝑇_𝑥𝑀_𝜔𝑓‖_𝒮0=‖𝑓‖_𝒮0, where (𝑇_𝑥𝑔)(𝑦)=𝑔(𝑦−𝑥) (translation) and (𝑀_𝜔ℎ)(𝑦)=𝑒^{2𝜋𝑖𝑦𝜔}ℎ(𝑦) (modulation). 𝒮₀ is continuously embedded in any Banach space with the same property and containing 𝑔₀; thus it is the smallest Banach space with this property. (iv)The Fourier transform is an isometry on 𝒮₀∶‖ℱ𝑓‖_𝒮0=‖𝑓‖_𝒮0.

As for the (conjugate) dual 𝒮^×₀ of 𝒮₀, it is a Banach space with norm ‖𝑓‖_𝒮×0=‖𝑉_𝑔𝑓‖_∞. The space 𝒮^×₀ contains both the 𝛿 function and the pure frequency 𝜒_𝜔(𝑥)=𝑒^{−2𝜋𝑖𝑥𝜔}.

In virtue of (i) above, we have 𝒮↪𝒮₀↪𝐿²↪𝒮^×₀↪𝒮^×, (6.15) where all embeddings are continuous and have dense range. It turns out that the triplet 𝒮₀(ℝ^𝑑)↪𝐿²(ℝ^𝑑)↪𝒮^×₀(ℝ^𝑑) (6.16) is the prototype of a Banach Gel'fand triple, that is, a RHS (or LBS) in which the extreme spaces are (nonreflexive) Banach spaces. This is often a very convenient substitute for Schwartz' RHS and it is widely used in signal processing.

6.2.2. Modulation Spaces and Gabor Analysis (Time-Frequency Analysis)

Modulation spaces are closely linked to, and in fact defined in terms of, the Short-Time Fourier (or Gabor) Transform. Given a 𝐶^∞ window function 𝑔≠0, the Short-Time Fourier Transform (STFT) of 𝑓∈𝐿²(ℝ^𝑑) is defined by (𝑉_𝑔𝑓)(𝑥,𝜔)=⟨𝑀_𝜔𝑇_𝑥𝑔∣𝑓⟩∶=_ℝ𝑑𝑔(𝑦−𝑥)𝑓(𝑦)𝑒^{−2𝜋𝑖𝑦𝜔}𝑑𝑦,𝑥,𝜔∈ℝ^𝑑, (6.17) where, as usual, (𝑇_𝑥𝑔)(𝑦)=𝑔(𝑦−𝑥) (translation) and (𝑀_𝜔ℎ)(𝑦)=𝑒^{2𝜋𝑖𝑦𝜔}ℎ(𝑦) (modulation).

Then, given a 𝑣-moderate weight function 𝑚(𝑥,𝜔), (see Section 3.3.3) the modulation space 𝑀_𝑚^𝑝,𝑞 is defined in terms of a mixed norm of an STFT: 𝑀_𝑚^𝑝,𝑞(ℝ^𝑑)={𝑓∈𝒮^×(ℝ^𝑑)∶𝑉_𝑔𝑓∈𝐿_𝑚^𝑝,𝑞(ℝ^2𝑑)},1⩽𝑝,𝑞⩽∞. (6.18) For 𝑝=𝑞, one writes 𝑀^𝑝_𝑚≡𝑀_𝑚^𝑝,𝑝. The space 𝑀_𝑚^𝑝,𝑞 is a Banach space for the norm ‖𝑓‖_𝑀𝑚^𝑝,𝑞∶=‖𝑉_𝑔𝑓‖_{𝐿𝑚𝑝,𝑞}. (6.19) Actually, there is a slightly more restrictive definition, which uses the weight function 𝑚_𝑠(𝑥,𝜔)≡𝑤_𝑠(𝜔)=(1+|𝜔|)^𝑠,𝑠⩾0, (or, equivalently, 𝑚_𝑠(𝑥,𝜔)=(1+|𝜔|²)^𝑠/2 ), so that the norm reads ‖𝑓‖_𝑀𝑤𝑠^𝑝,𝑞=(_ℝ𝑑(_ℝ𝑑|⟨𝑀_𝜔𝑇_𝑥𝑔∣𝑓⟩|^𝑝𝑑𝑥)^𝑞/𝑝(1+|𝜔|)^𝑠𝑞𝑑𝜔)^1/𝑞. (6.20) Equivalently, one may define a modulation space as the inverse Fourier transform of a Wiener amalgam space: 𝑀_𝑤^𝑝,𝑞_𝑠=ℱ⁻¹(𝑊(𝐿^𝑝,ℓ^𝑞_𝑤𝑠)). (6.21) This space is independent of the choice of window 𝑔, in the sense that different window functions define equivalent norms.

The lattice properties of the family {𝑀_𝑚^𝑝,𝑞,1⩽𝑝,𝑞⩽∞} are, of course, the same as those of the mixed-norm spaces 𝐿_𝑚^𝑝,𝑞. As for duality, one has (𝑀_𝑚^𝑝,𝑞)^×=𝑀^𝑝,𝑞_1/𝑚. Inclusion relations hold, leading again to a lattice structure: if 𝑝₁⩽𝑝₂,𝑞₁⩽𝑞₂, and 𝑚₂⩽𝐶𝑚₁, for some constant 𝐶>0, then 𝑀^𝑝1,𝑞1_𝑚1⊆𝑀^𝑝2,𝑞2_𝑚2. In particular, one has 𝑀¹_𝑣⊆𝑀_𝑚^𝑝,𝑞⊆𝑀^∞_1/𝑣. (6.22) The class of modulation spaces 𝑀_𝑤^𝑝,𝑞_𝑠 contains several well-known spaces, such as the following:(i)The Bessel potential spaces or fractional Sobolev spaces 𝐻^𝑠=𝑀𝑚²_𝑠: 𝐻^𝑠(ℝ^𝑑)=𝑀𝑚²_𝑠(ℝ^𝑑)={𝑓∈𝒮^×∶_ℝ𝑑|𝑓(𝑡)|²(1+|𝑡|²)^𝑠𝑑𝑡<∞},𝑠∈ℝ. (6.23)(ii)𝐿²(ℝ^𝑑)=𝑀²(ℝ^𝑑).(iii)The Feichtinger algebra 𝒮₀=𝑀¹ and its dual 𝒮^×₀=𝑀^∞.

By construction, modulation spaces are function spaces well adapted to Gabor analysis, although they can often be replaced by amalgam spaces. A wealth of information about the spaces and their application in Gabor analysis may be found in the monograph of Gröchenig [20]. Here we just indicate a few relevant points, especially those that are of a PIP-space nature. We consider in particular the action of several types of operators on such spaces. (i)Translation and Modulation Operators (a)Every amalgam space 𝑊(𝐿^𝑝,ℓ^𝑞) and every mixed-norm space 𝐿_𝑚^𝑝,𝑞 are invariant under translation, that is, 𝑇_𝑦 is a totally regular operator in the corresponding PIP-space.(b)Every modulation space 𝑀_𝑚^𝑝,𝑞 is invariant under time-frequency shifts (translation and modulation), that is, 𝑇_𝑦 and 𝑀_𝜉 are totally regular operators.(ii)Fourier Transform (a)For 1⩽𝑝,𝑞⩽2,ℱ maps 𝑊(𝐿^𝑝,ℓ^𝑞) into (𝑊𝐿^𝑞,ℓ^𝑝) continuously, that is, 𝐽(ℱ) contains every pair (𝑝,𝑞),(𝑞,𝑝).(b)If 𝑚(𝜉,−𝑥)⩽𝐶𝑚(𝑥,𝜉), then every space 𝑀^𝑝_𝑚 is invariant under Fourier transform. (iii)Gabor Frame Operators

Given a nonzero window function 𝑔∈𝐿²(ℝ^𝑑) and lattice parameters 𝛼,𝛽>0, the set of vectors 𝒢(𝑔,𝛼,𝛽)={𝑀_𝑛𝛽𝑇_𝑘𝛼𝑔,𝑘,𝑛∈ℤ^𝑑} (6.24) is called a Gabor system. The system 𝒢(𝑔,𝛼,𝛽) is a Gabor frame if there exist two constants m>0 and M<∞ such that m‖𝑓‖²⩽_{𝑘,𝑛∈ℤ𝑑}|⟨M_𝑛𝛽𝑇_𝑘𝛼𝑔∣𝑓⟩|²⩽M‖𝑓‖²,∀𝑓∈𝐿²(ℝ^𝑑). (6.25)

The associated Gabor frame operator 𝑆_𝑔,𝑔 is given by 𝑆_𝑔,𝑔𝑓∶=_{𝑘,𝑛∈ℤ𝑑}⟨𝑀_𝑛𝛽𝑇_𝑘𝛼𝑔∣𝑓⟩𝑀_𝑛𝛽𝑇_𝑘𝛼𝑔. (6.26) The main results of the Gabor time-frequency analysis stem from the following proposition.

Proposition 6.1. If 𝒢(𝑔,𝛼,𝛽) is a Gabor frame, there exists a dual window 𝑔̆=𝑆⁻¹_𝑔𝑔𝑔 such that 𝒢(𝑔,̆𝛼,𝛽) is a frame, called the dual frame. Then one has, for every 𝑓∈𝐿²(ℝ^𝑑), 𝑓=_{𝑘,𝑛∈ℤ𝑑}⟨𝑀_𝑛𝛽𝑇_𝑘𝛼𝑔∣𝑓⟩𝑀_𝑛𝛽𝑇_𝑘𝛼𝑔̆=_{𝑘,𝑛∈ℤ𝑑}⟨𝑀_𝑛𝛽𝑇_𝑘𝛼𝑔̆∣𝑓⟩𝑀_𝑛𝛽𝑇_𝑘𝛼𝑔, (6.27) with unconditional convergence in 𝐿²(ℝ^𝑑).

The outcome of the theory is that the modulation spaces 𝑀_𝑚^𝑝,𝑞 turn out to be the natural class of function spaces for Gabor analysis [20]. Define indeed the following operator, generalizing (6.26) slightly: 𝑆_𝑔,𝑔′𝑓∶=_{𝑘,𝑛∈ℤ𝑑}⟨𝑀_𝑛𝛽𝑇_𝑘𝛼𝑔∣𝑓⟩𝑀_𝑛𝛽𝑇_𝑘𝛼𝑔^′,𝑔,𝑔^′∈𝐿²(ℝ^2𝑑). (6.28) Then one has the following results (they are highly nontrivial and their proof requires deep analysis) (i)If 𝑔,𝑔^′∈𝑊(𝐿^∞,ℓ¹), then the Gabor frame operator 𝑆_𝑔,𝑔′ is bounded on every 𝐿^𝑝(ℝ^2𝑑),1⩽𝑝⩽∞.(ii)If 𝑔,𝑔^′∈𝑀¹_𝑣, then 𝑆_𝑔,𝑔′ is bounded on 𝑀_𝑚^𝑝,𝑞 for all 1⩽𝑝,𝑞⩽∞, all 𝑣-moderate weights 𝑚, and all 𝛼,𝛽.(iii)If 𝑔̆ is a dual window of 𝑔, that is, 𝑆_̆𝑔𝑔,=1 on 𝐿², then the two expansions in (6.27) converge unconditionally in 𝑀_𝑚^𝑝,𝑞 if 𝑝,𝑞<∞.

Clearly statements (i) and (ii) can be translated into PIP-space language, by saying that 𝑆_𝑔,𝑔′ is a totally regular operator in the chain {𝐿^𝑝,1⩽𝑝⩽∞}, respectively, any PIP-space built from modulation spaces.

These results should suffice to convince the reader that the modulation spaces 𝑀_𝑚^𝑝,𝑞 are the “natural” spaces for Gabor analysis. Actually, most of this remains true if one replaces modulation spaces by amalgam spaces 𝑊(𝐿^𝑝,ℓ^𝑞_𝑚). Second, it is obvious that most of the statements have a distinctly PIP-space flavor: it is not some individual space 𝑀_𝑚^𝑝,𝑞 or 𝑊(𝐿^𝑝,ℓ^𝑞_𝑚) that counts, but the whole family, with many operators being regular in the sense of PIP-spaces.

6.2.3. Besov Spaces and Wavelet Analysis (Time-Scale Analysis)

Besov spaces were introduced around 1960 for providing a precise control on the smoothness of solutions of certain partial differential equations. Later on, it was discovered that they are closely linked to wavelet analysis, exactly as the (much more recent) modulation spaces are structurally adapted to Gabor analysis. In fact, there are many equivalent definitions of Besov spaces. We restrict ourselves to a “discrete” formulation, based on a dyadic partition of unity. Other definitions may be found in the literature quoted in [14], in particular [19, 38–40].

Let us consider a weight function 𝜑∈𝒮(ℝ) with the following properties: (i)supp 𝜑={𝜉∶2⁻¹⩽|𝜉|⩽2}, (ii)𝜑(𝜉)>0 for 2⁻¹<|𝜉|<2, (iii)∑^∞_𝑗=−∞𝜑(2^−𝑗𝜉)=1(𝜉≠0).

Then one defines the following functions by their Fourier transform: (i)𝜑_𝑗(𝜉)=𝜑(2^−𝑗𝜉),𝑗∈ℤ∶ high “frequency" for 𝑗>0, low “frequency" for 𝑗<0, (ii)𝜓(𝜉)=1−∑^∞_𝑗=1𝜑(2^−𝑗𝜉)∶ low “frequency" part.

Given the weight function 𝜑, the inhomogeneous Besov space 𝐵^𝑠_𝑝𝑞 is defined as 𝐵^𝑠_𝑝𝑞={𝑓∈𝒮^×∶‖𝑓‖^𝑠_𝑝𝑞<∞}, (6.29) where ‖⋅‖^𝑠_𝑝𝑞 denotes the norm ‖𝑓‖^𝑠_𝑝𝑞∶=‖𝜓∗𝑓‖_𝑝+(^∞_𝑗=1(2^𝑠𝑗‖𝜑_𝑗∗𝑓‖_𝑝)^𝑞)^1/𝑞,𝑠∈ℝ,1⩽𝑝,𝑞⩽∞. (6.30) The space 𝐵^𝑠_𝑝𝑞 is a Banach space and it does not depend on the choice of the weight function 𝜑, since a different choice defines an equivalent norm. Note that 𝐵^𝑠₂₂=𝐻^𝑠, the (fractional) Sobolev space or Bessel potential space.

For 𝑓∈𝐵^𝑠_𝑝𝑞, one may write the following (weakly converging) expansion, known as a dyadic Littlewood-Paley decomposition: 𝑓=𝜓∗𝑓+^∞_𝑗=1𝜑_𝑗∗𝑓. (6.31) Clearly the first term represents the (relatively uninteresting) low-“frequency" part of the function, whereas the second term analyzes in detail the high-“frequency" component.

Besov spaces enjoy many familiar properties (for more details, we refer to the literature, e.g., [19, Section 6.2] or [39, Chapter 2]).

(i)Inclusion Relations
The following relations hold, where all embeddings are continuous: (a)𝒮↪𝐵^𝑠_𝑝𝑞↪𝒮^×, (b)𝐵^𝑠_𝑝𝑞↪𝐿^𝑝, if 1⩽𝑝,𝑞⩽∞ and 𝑠>0, (c)for s₁<𝑠₂,𝐵^𝑠2_𝑝𝑞↪𝐵^𝑠1_𝑝𝑞(1⩽𝑞,𝑝⩽∞), (d)for 1⩽𝑞₁<𝑞₂⩽∞,𝐵^𝑠_𝑝𝑞1↪𝐵^𝑠_𝑝𝑞2(𝑠∈ℝ,1⩽𝑝⩽∞), (e)for 𝑠−1/𝑝=𝑠₁−1/𝑝₁,𝐵^𝑠_𝑝𝑞↪𝐵^𝑠1_𝑝1𝑞1(𝑠,𝑠₁∈ℝ,1⩽𝑝⩽𝑝₁⩽∞,1⩽𝑞⩽𝑞₁⩽∞). In the terminology of Section 4, the first statement means that the spaces 𝐵^𝑠_𝑝𝑞 are interspaces for the RHS 𝒮↪𝐿²↪𝒮^×. The inclusion relations above mean that the family of spaces 𝐵^𝑠_𝑝𝑞 contains again many chains of Banach spaces.

(ii) Interpolation
Besov spaces enjoy nice interpolation properties, in all three parameters 𝑠,𝑝,𝑞.

(iii) Duality
One has (𝐵^𝑠_𝑝𝑞)^×=𝐵^−𝑠_𝑝𝑞(𝑠∈ℝ).

(iv) Translation and Dilation Invariance
Every space 𝐵^𝑠_𝑝𝑞 is invariant under translation and dilation (the unitary dilation operator in 𝐿² reads (𝐷_𝑎𝑓)(𝑥)=𝑎^−1/2𝑓(𝑥/𝑎).)

(v) Regularity Shift
Let 𝐽^𝜎∶𝒮^×→𝒮^× denote the operator 𝐽^𝑠𝑓=ℱ⁻¹{(1+|⋅|²)^𝑠/2ℱ𝑓},𝑠∈ℝ. Then 𝐽^𝜎 is an isomorphism from 𝐵^𝑠_𝑝𝑞 onto 𝐵^𝑠−𝜎_𝑝𝑞. Thus 𝐽^𝜎 is totally regular for 𝜎⩽0, but not for 𝜎>0.

It is also useful to consider the homogeneous Besov space 𝐵̇^𝑠_𝑝𝑞, defined as the set of all 𝑓∈𝒮^× for which ‖𝑓‖̇^𝑠_𝑝𝑞<∞, where the quasinorm ‖⋅‖̇^𝑠_𝑝𝑞 is defined by ‖𝑓‖̇^𝑠_𝑝𝑞∶=(^∞_𝑗=−∞(2^𝑠𝑗‖𝜑_𝑗∗𝑓‖_𝑝)^𝑞)^1/𝑞. (6.32) (This is only a quasi-norm since ‖𝑓‖̇^𝑠_𝑝𝑞=0 if and only if supp𝑓={0}, i.e., 𝑓 is a polynomial.) Note that, if 0∉supp𝑓, then 𝑓∈𝐵̇^𝑠_𝑝𝑞 if and only if 𝑓∈𝐵^𝑠_𝑝𝑞.

The spaces 𝐵̇^𝑠_𝑝𝑞 have properties similar to the previous ones and, in addition, one has 𝐵^𝑠_𝑝𝑞=𝐿^𝑝∩𝐵̇^𝑠_𝑝𝑞 for 𝑠>0,1⩽𝑝,𝑞⩽∞. In particular, every space 𝐵̇^𝑠_𝑝𝑞 is invariant under translation and dilation, which is not surprising, since these spaces are in fact based on the 𝑎𝑥+𝑏 group, consisting precisely of dilations and translations of the real line, via the coorbit space construction (see Section 6.2.4 below).

Besov spaces are well adapted to wavelet analysis, because the definition (6.29) essentially relies on a dyadic partition (powers of 2). Historically, the connection was made with the discrete wavelet analysis, for that reason. Indeed, there exists an equivalent definition given in terms of decay of wavelet coefficients. More precisely, if a function 𝑓 is expanded in a wavelet basis, the decay properties of the wavelet coefficients allow to characterize precisely to which Besov space the function 𝑓 belongs, as we shall see below. In addition, the Besov spaces may also be characterized in terms of the continuous wavelet transform (see [14, Section 8.4] ).

In order to go into details, we have to recall some basic facts about the wavelet transform (for simplicity, we restrict ourselves to one dimension). Whereas the STFT is defined in terms of translation and modulation, the continuous wavelet transform is based on translations and dilations (𝑊_𝜓𝑠)(𝑏,𝑎)=𝑎^−1∞_−∞𝜓(𝑎⁻¹(𝑥−𝑏))𝑠(𝑥)𝑑𝑥,𝑎>0,𝑏∈ℝ,𝑠∈𝐿²(ℝ). (6.33) Note that we use here the so-called 𝐿¹ normalization. It is more frequent to use the 𝐿² normalization, in which the prefactor is 𝑎^−1/2 instead of 𝑎⁻¹. In this relation, the wavelet 𝜓 is assumed to satisfy the admissibility condition 𝑐_𝜓∶=^∞_−∞𝑑𝜔|𝜔|⁻¹|𝜓(𝜔)|²<∞, (6.34) which implies that ∫^∞_−∞𝜓(𝑥)𝑑𝑥=0. This condition is only necessary, but becomes sufficient under some mild restrictions, so that it is commonly used as admissibility condition in practice.

However, discretizing the two parameters 𝑎 and 𝑏 in (6.33) leads in general only to frames. In order to get orthogonal wavelet bases, one relies on the so-called multiresolution analysis of 𝐿²(ℝ). This is defined as an increasing sequence of closed subspaces of 𝐿²(ℝ): ⋯⊂𝒱₋₂⊂𝒱₋₁⊂𝒱₀⊂𝒱₁⊂𝒱₂⊂⋯ (6.35) with ⋂_𝑗∈ℤ𝒱_𝑗={0} and ⋃_𝑗∈ℤ𝒱_𝑗 dense in 𝐿²(ℝ) and such that (1)𝑓(𝑥)∈𝒱_𝑗⇔𝑓(2𝑥)∈𝒱_𝑗+1,(2)there exists a function 𝜙∈𝒱₀, called a scaling function, such that the family {𝜙(⋅−𝑘),𝑘∈ℤ} is an orthonormal basis of 𝒱₀.

Combining conditions (1) and (2), one sees that {𝜙_𝑗𝑘≡2^𝑗/2𝜙(2^𝑗⋅−𝑘),𝑘∈ℤ} is an orthonormal basis of 𝒱_𝑗. The space 𝒱_𝑗 can be interpreted as an approximation space at resolution 2^𝑗. Defining 𝒲_𝑗 as the orthogonal complement of 𝒱_𝑗 in 𝒱_𝑗+1, that is, 𝒱_𝑗⊕𝒲_𝑗=𝒱_𝑗+1, we see that 𝒲_𝑗 contains the additional details needed to improve the resolution from 2^𝑗 to 2^𝑗+1. Thus one gets the decomposition 𝐿²(ℝ)=⊕_𝑗∈ℤ𝒲_𝑗. The crucial theorem then asserts the existence of a function 𝜓, called the mother wavelet, explicitly computable from 𝜙, such that {𝜓_𝑗𝑘≡2^𝑗/2𝜓(2^𝑗⋅−𝑘),𝑘∈ℤ} constitutes an orthonormal basis of 𝒲_𝑗 and thus {𝜓_𝑗𝑘≡2^𝑗/2𝜓(2^𝑗⋅−𝑘),𝑗,𝑘∈ℤ} is an orthonormal basis of 𝐿²(ℝ): these are the orthonormal wavelets. Thus the expansion of an arbitrary function 𝑓∈𝐿² into an orthogonal wavelet basis {𝜓_𝑗𝑘,𝑗,𝑘∈ℤ} reads 𝑓=_{𝑗,𝑘∈ℤ}𝑐_𝑗𝑘𝜓_𝑗𝑘,with𝑐_𝑗𝑘=⟨𝜓_𝑗𝑘∣𝑓⟩. (6.36) Additional regularity conditions can be imposed to the scaling function 𝜙. Given 𝑟∈ℕ, the multiresolution analysis corresponding to 𝜙 is called 𝑟-regular if |𝑑^𝑛𝜙𝑑𝑥^𝑛|⩽𝑐_𝑚(1+|𝑥|^𝑚),forall𝑛⩽𝑟andallintegers𝑚∈ℕ. (6.37) Well-known examples include the Haar wavelets, the B-splines, and the various Daubechies wavelets.

As a result of the “dyadic” definition (6.29)-(6.30), it is natural that Besov spaces can be characterized in terms of an 𝑟-regular multiresolution analysis {𝒱_𝑗}. Let 𝐸_𝑗∶𝐿²→𝒱_𝑗 be the orthogonal projection on 𝒱_𝑗 and 𝐷_𝑗=𝐸_𝑗+1−𝐸_𝑗 that on 𝒲_𝑗. Let 0<𝑠<𝑟 and 𝑓∈𝐿^𝑝(ℝ). Then, 𝑓∈𝐵^s_𝑝𝑞(ℝ) if and only if ‖𝐷_𝑗𝑓‖_𝑝=2^−𝑗𝑠𝛿_𝑗, where (𝛿_𝑗)∈ℓ^𝑞(ℕ), and one has (≍ means equivalence of norms) ‖𝑓‖^𝑠_𝑝𝑞≍‖𝐸₀𝑓‖_𝑝+(_𝑗∈ℤ2^𝑗𝑠𝑞‖𝐷_𝑗𝑓‖^𝑞_𝑝)^1/𝑞. (6.38) Specializing to 𝑝=𝑞=2, one gets a similar result for Sobolev spaces: given 𝑓∈𝐻^−𝑟(ℝ) and |𝑠|<𝑟, 𝑓∈𝐻^𝑠(ℝ) if, and only if, 𝐸₀𝑓∈𝐿²(ℝ) and ‖𝐷_𝑗𝑓‖₂=2^−𝑗𝑠𝜖_𝑗,𝑗∈ℕ, where (𝜖_𝑗)∈ℓ²(ℕ).

But there is more. Indeed, modulation spaces and Besov spaces admit decomposition of elements into wavelet bases and each space can be uniquely characterized by the decay properties of the wavelet coefficients. To be precise, let {𝜓_𝑗𝑘,𝑗,𝑘∈ℤ} be an orthogonal wavelet basis coming from an 𝑟-regular multiresolution analysis based on the scaling function 𝜙. Then the following results are typical [41, Chapters II.9 and VI.10].(i)Inhomogeneous Besov Spaces: 𝑓∈𝐵^𝑠_𝑝𝑞(ℝ) if it can be written as 𝑓(𝑥)=_𝑘∈ℤ𝛽_𝑘𝜙(𝑥−𝑘)+_{𝑗⩾0,𝑘∈ℤ}𝑐_𝑗𝑘𝜓_𝑗𝑘, (6.39) where (𝛽_𝑘)∈ℓ^𝑝 and (∑_𝑘∈ℤ|𝑐_𝑗𝑘|^𝑝)^1/𝑝=2^{−𝑗(𝑠+1/2−1/𝑝)}𝛾_𝑗,with(𝛾_𝑗)∈ℓ^𝑞(ℤ).(ii)Homogeneous Besov Spaces: Let |𝑠|<𝑟. Then, if 𝑓∈𝐵̇^𝑠_𝑝𝑞(ℝ), its wavelet coefficients 𝑐_𝑗𝑘 verify (∑_𝑘∈ℤ|𝑐_𝑗𝑘|^𝑝)^1/𝑝=2^{−𝑗(𝑠+1/2−1/𝑝)}𝛾_𝑗,where(𝛾_𝑗)∈ℓ^𝑞(ℤ). Conversely, if this condition is satisfied, then 𝑓=𝑔+𝑃, where 𝑔∈𝐵̇^𝑠_𝑝𝑞 and 𝑃 is a polynomial.

6.2.4. 𝛼-Modulation Spaces, Coorbit Spaces

The 𝛼-modulation spaces (𝛼∈[0,1]) are spaces intermediate between modulation and Besov spaces, to which they reduce for 𝛼=0 and 𝛼→1, respectively. As for coorbit spaces, they are a far-reaching generalization, based on integrable group representations [42]. They contain most of the previous spaces, but we will refrain from describing them in detail, for lack of space.

6.2.5. Unconditional Bases

We conclude this section with some examples of unconditional wavelet bases, as announced in Section 5.2.5. Actually, the concept of wavelet basis can be further generalized to biorthogonal bases, obtained by considering two scales of the type (6.35) and imposing cross-orthogonality relations [43, Section 8.3]. For precise definitions, we refer to the literature. (i)The Haar wavelet basis is defined by the scaling function 𝜙_𝐻=𝜒_[0,1] and the mother wavelet 𝜓_𝐻=𝜒_[0,1/2]−𝜒_[1/2,1]. It is a standard result that the Haar system is an unconditional basis for every 𝐿^𝑝(ℝ),1<𝑝<∞.(ii) The Lemarié-Meyer wavelet basis is an unconditional basis for all 𝐿^𝑝 spaces, Sobolev spaces, and homogeneous Besov spaces 𝐵̇^𝑠_𝑝𝑞(1⩽𝑝,𝑞<∞) [44].(iii)There is a class of wavelet bases (Wilson bases of exponential decay) that are unconditional bases for every modulation space 𝑀_𝑚^𝑝,𝑞,1⩽𝑝,𝑞<∞, but not for 𝐿^𝑝,1<𝑝<∞,𝑝≠2 [45].

7. Conclusion

Most families of function spaces used in analysis and in signal processing come in scales or lattices and in fact are, or contain, PIP-spaces. The (lattice) indices defining the (partial) order characterize the properties of the corresponding functions or distributions: smoothness, local integrability, decay at infinity. Thus it seems natural to formulate the properties of various operators globally, using the theory of PIP-space operators; in particular the set j(𝐴) of an operator encodes its properties in a very convenient and visual fashion. In addition, it is often possible to determine uniquely whether a function belongs to one of those spaces simply by estimating the (asymptotic) behavior of its Gabor or wavelet coefficients, a real breakthrough in functional analysis [41].

A legitimate question is whether there are instances where a PIP-space is really needed, or a RHS could suffice. The answer is that there are plenty of examples, among the applications we have enumerated in Section 6 (without details, for lack of space). We may therefore expect that the PIP-space formalism will play a significant role in Gabor/wavelet analysis, as well as in mathematical physics.

Concerning the applications in mathematical physics, in almost all cases, the relevant structure is a scale or a chain of Hilbert spaces, which allows a finer control on the behavior of operators. For instance (all details may be found in our monograph [14]) the following are considered. (i)For the singular interactions in quantum mechanics (𝛿 or 𝛿^′ potentials), the approach of Grossmann et al. [18, 24, 25] is definitely the most appropriate; a RHS would be irrelevant.(ii)The very formulation of the WVW approach to quantum scattering theory [34–36] requires a LHS, whose end spaces are in fact Hilbert spaces.(iii)For quantum field theory, the energy bounds of Fredenhagen and Hertel [46] rely in an essential way on the scale generated by the Hamiltonian, and so does Nelson's approach to Euclidean field theory [47].

As for the applications in signal processing, all families of spaces routinely used are, or contain, chains of Banach spaces, which are needed for a fine tuning of elements (usually, distributions) and operators on them. Such are, for instance, 𝐿^𝑝 spaces, amalgam spaces, modulation spaces, Besov spaces, or coorbit spaces, mentioned above. Here again, a RHS, even with Banach end spaces like the Feichtinger algebra and its conjugate dual, is clearly not sufficient. The whole Section 6.2 illustrates the statement.