A Theory of Summability on a Space of Generalized Functions

Abstract

A theory of summability of orthonormal sets is introduced in multinormed spaces. The approach which is presented caters for infinite sets {𝜙_𝑗}_𝑗∈𝐽, where the index set 𝐽 may be uncountable, and is applied to obtain convergence results in appropriate spaces of test functions and corresponding spaces of generalized functions. These spaces are constructed in a systematic manner that relies heavily on properties of orthonormal bases in Hilbert spaces. A space of almost-periodic generalized functions, in which each generalized function can be expanded in terms of an uncountable basis of exponential functions, is obtained as a special case of our theory.

1. Introduction

In recent papers [1, 2], we investigated the behaviour of certain fractional differential and integral operators on appropriately defined spaces of generalized functions. In each paper, a procedure for producing spaces of test functions and generalized functions, pioneered by Zemanian [3, Chapter 9], played a crucial role. Central to this procedure is the theory of complete orthonormal sequences in 𝐿²-type spaces. Given a complete orthonormal sequence {𝜓_𝑛}^∞_𝑛=0 of eigenfunctions of a suitably restricted symmetric differential operator 𝑇 in 𝐿²(𝐼), where 𝐼 is an interval in ℝ, corresponding spaces of test functions 𝒜(=𝒜(𝐼)) and generalized functions 𝒜^′(=𝒜^′(𝐼)) are obtained in a systematic and constructive manner.

In the theory developed by Zemanian, the operator 𝑇∶𝐿²(𝐼)⊃𝐷(𝑇)→𝐿²(𝐼) is a differential expression of the form 𝑇=𝜃₀𝐷^𝑛1𝜃₁𝐷^𝑛2⋯𝐷^𝑛𝜈𝜃_𝜈, (1) where 𝐷=𝑑/𝑑𝑥, the 𝑛_𝑘 are positive integers, and the 𝜃_𝑘 are smooth functions on 𝐼. Moreover, it is assumed that 𝑇 satisfies the (symmetry) condition 𝑇=𝜃_𝜈(−𝐷)^𝑛𝜈⋯(−𝐷)^𝑛2𝜃₁(−𝐷)^𝑛1𝜃₀ (2) and has smooth eigenfunctions {𝜓_𝑛}^∞_𝑛=0 that form a complete orthonormal basis for 𝐿²(𝐼) and for which the corresponding real eigenvalues {𝜆_𝑛}^∞_𝑛=0 are such that |𝜆_𝑛|→∞ as 𝑛→∞. The domain 𝐷(𝑇) of the operator 𝑇 is 𝐷(𝑇)∶=𝜙∈𝐶^∞(𝐼)∶𝑇^𝑘𝜙∈𝐿²(𝐼),𝑇^𝑘𝜙,𝜓_𝑛2=𝜙,𝑇^𝑘𝜓_𝑛2,∀𝑘,𝑛=0,1,…, (3) where 𝑇⁰ is the identity operator on 𝐿²(𝐼); see [3, Chapter 9]. The space of test functions is then taken to be the vector space 𝐷(𝑇) equipped with the topology generated by the countable multinorm {𝛽_𝑘}^∞_𝑘=0, where 𝛽_𝑘𝜙∶=𝑇^𝑘𝜙₂<∞,∀𝑘=0,1,2,…. (4)

The resulting topological vector space can be shown to be a Fréchet space and is denoted by 𝒜. Note that 𝐷(𝑇) is a dense subspace of 𝐿²(𝐼), since 𝐶^∞₀(𝐼)⊂𝐷(𝑇), and that each eigenfunction 𝜓_𝑛∈𝐷(𝑇). The space of generalized functions 𝒜^′ is the dual of 𝒜 equipped with the weak* topology generated by the multinorm {𝛼_𝜙}, where 𝛼_𝜙𝑓=𝑓,𝜙,𝑓∈𝒜^′,𝜙∈𝒜, (5) where (𝑓,𝜙) is the number assigned to each 𝜙∈𝒜 by 𝑓∈𝒜^′. Since 𝒜^′ is the dual of a Fréchet space, it is sequentially complete with respect to the weak* topology. Also, by identifying functions 𝜑∈𝐿²(𝐼) with regular generalized functions 𝜑∈𝒜^′ via 𝜑,𝜙∶=𝜙,𝜑₂=_𝐼𝜙(𝑥)𝜑(𝑥)𝑑𝑥,𝜙∈𝒜, (6) it can be shown that 𝐿²(𝐼) is continuously imbedded in 𝒜^′. Consequently, we can write 𝒜↪𝐿²(𝐼)↪𝒜^′, (7) where ↪ represents a continuous embedding. Results on the weak* convergence of eigenfunction expansions of generalized functions in the space 𝒜^′ are given in [3, Chapter 9], where it is shown that 𝑓=^∞_𝑛=0𝑓,𝜓_𝑛𝜓_𝑛,∀𝑓∈𝒜^′. (8) It should be noted that the Zemanian theory of eigenfunction expansions has been extended in a number of ways since its introduction; see, for example, [4–7]. In particular, in [5] the differential operator that is used to construct the Fréchet space 𝒜 is replaced by any self-adjoint operator 𝑇 defined on an arbitrary separable Hilbert space 𝐻. The space of “test-functions” is then taken to be the Fréchet space 𝐷_∞∶=^∞_𝑘=1𝐷𝑇^𝑘, (9) in which convergence is defined via the countable multinorm {𝛽_𝑘}^∞_𝑘=0, 𝛽_𝑘(𝜙)∶=‖𝑇^𝑘𝜙‖_𝐻,𝜙∈𝐷_∞. The corresponding space of “generalized functions” is given by 𝐷^′_∞ equipped with the weak* topology. The Zemanian results arise as the special case when 𝐻=𝐿²(𝐼) and 𝑇 is a self-adjoint differential operator whose spectrum contains only eigenvalues; see [5, Remark 3.5]. One of the aims of the present paper is to demonstrate that a Zemanian-type theory of eigenfunction expansions, but possibly involving an uncountable orthonormal basis, can be developed in the more general setting of an arbitrary Hilbert space. This theory will be based on an extension to complete multinormed spaces of the concept of summability in Banach spaces. It is clear that a self-adjoint operator on a separable Hilbert space cannot have an uncountable basis, as all orthonormal bases in a separable Hilbert space are countable; see [8, p.255]. Consequently, in the sequel, we do not assume that that the Hilbert space 𝐻 is separable.

In elementary analysis, the sum of an infinite sequence of terms 𝜙₁,𝜙₂,… in some topological vector space exists if the limit of the corresponding sequence of partial sums exists. To deal with the case of summations of a family of elements {𝜙_𝑗}_𝑗∈𝐽 over a general index set 𝐽, which could be uncountable, requires the notion of summability. There is a well-established theory of summability in a Banach space setting; see, for example, [9, Chapter 6] and [10, Chapter 4]. However, as far as we are aware, a similar theory has not been presented within the framework of a complete multinormed space. Consequently, we begin in Section 2 by giving an appropriate definition of summability in this more general setting and verify that the definition satisfies all the important computational rules.

In Section 3, we establish that the definition of summability given in Section 2 can be used to extend the Zemanian theory of eigenfunction expansions of generalized functions to the case where the orthonormal basis may be uncountable. As the spaces 𝐿²(𝐼) favoured by Zemanian are separable, we work within the more general framework of a Hilbert space 𝐻, which need not be separable, and use results from [5]. As a specific example of an uncountable basis, we go on in Section 4 to examine the nonseparable Hilbert space of 𝐿² almost-periodic functions and show how the theory developed in the previous sections leads nicely to eigenfunction expansions for a class of almost-periodic generalized functions.

2. Infinite Sums in Multinormed Spaces

A multinormed space is a topological vector space in which the topology is defined by a collection of seminorms. Let 𝑋 be a vector space and let 𝑆={𝛾_𝜆}_𝜆∈Λ be a separating collection of seminorms on 𝑋 for some index set Λ. If 𝑆 is countable, then the resulting topological vector space, which we will continue to denote simply by 𝑋, is a countably multinormed space, which, if complete, is a Fréchet space; see [11, p.132].

Let {𝜙_𝑗}_𝑗∈𝐽 be an indexed family in 𝑋 and let ℱ denote the collection of all finite subsets of 𝐽. If we use the binary operation ⊆ to define a partial order on ℱ, then ℱ becomes a directed set. Associated with {𝜙_𝑗}_𝑗∈𝐽 is the indexed set of finite partial sums, given by 𝑆_𝐾∶=_𝑗∈𝐾𝜙_𝑗,𝐾∈ℱ. (10)

The mapping 𝐾→𝑆_𝐾 from ℱ→𝑋, is then a net in 𝑋 and so we can use the theory associated with nets to provide a definition of the sum ∑_𝑗∈𝐽𝜙_𝑗, in terms of a limit of finite partial sums, which is independent of the way in which the terms 𝜙_𝑗 in the sum are ordered.

Definition 1. Let 𝑋 be a complete multinormed space with multinorm {𝛾_𝜆}_𝜆∈Λ. The vectors {𝜙_𝑗}_𝑗∈𝐽⊂𝑋 are said to be summable to 𝜙∈𝑋, or, equivalently, the unordered sum of the indexed set {𝜙_𝑗}_𝑗∈𝐽 converges to 𝜙∈𝑋, if, for each 𝜖>0 and 𝜆∈Λ, there exists 𝐽₀(=𝐽₀(𝜖,𝜆))∈ℱ such that, for all 𝐾∈ℱ with 𝐽₀⊆𝐾, 𝛾_{𝜆𝑗∈𝐾}𝜙_𝑗−𝜙<𝜖. (11) In this case, we write ∑_𝑗∈𝐽𝜙_𝑗=𝜙 and say that the unordered sum converges unconditionally to 𝜙.
In the subsequent work, the phrases “∑_𝑗𝜙_𝑗 converges to 𝜙,” “the vectors {𝜙_𝑗}_𝑗∈𝐽 are summable to 𝜙,” and “{𝜙_𝑗}_𝑗∈𝐽 is summable to 𝜙” will all be used when ∑_𝑗∈𝐽𝜙_𝑗=𝜙 in 𝑋.

Lemma 2. If ∑_𝑗∈𝐽𝜙_𝑗=𝜙, then 𝜙 is unique.

Proof. Let ∑_𝑗∈𝐽𝜙_𝑗=𝜙 and also let ∑_𝑗∈𝐽𝜙_𝑗=𝜓 in 𝑋. Then, from Definition 1, for each 𝜖>0 and 𝜆∈Λ, there exists 𝐽₀(=𝐽₀(𝜖,𝜆))∈ℱ such that whenever 𝐾∈ℱ contains 𝐽₀, we have 𝛾_{𝜆𝑗∈𝐾}𝜙_𝑗<𝜖−𝜙2. (12) Similarly, for 𝜖>0 and 𝜆∈Λ, there exists 𝐾₀(=𝐾₀(𝜖,𝜆))∈ℱ such that whenever 𝑀∈ℱ contains 𝐾₀, 𝛾_{𝜆𝑗∈𝑀}𝜓_𝑗<𝜖−𝜓2. (13)
Now, setting 𝐿₀=𝐾₀∪𝐽₀, let 𝐿 be any set from ℱ containing 𝐿₀. Then 𝐿 contains both 𝐽₀ and 𝐾₀. Hence, from (12) and (13), we obtain 𝛾_𝜆𝜙−𝜓=𝛾_{𝜆𝑗∈𝐿}𝜙_𝑗−𝜓−_𝑗∈𝐿𝜙_𝑗+𝜙≤𝛾_{𝜆𝑗∈𝐿}𝜙_𝑗−𝜓+𝛾_{𝜆𝑗∈𝐿}𝜙_𝑗<𝜖−𝜙2+𝜖2=𝜖, (14) and so 𝛾_𝜆𝜙−𝜓=0. (15) Since this holds for all 𝜆 and {𝛾_𝜆}_𝜆∈Λ is a multinorm, it follows that 𝜙=𝜓.

Lemma 3. Let ∑_𝑗∈𝐽𝜙_𝑗=𝜙 and let ∑_𝑗∈𝐽𝜓_𝑗=𝜓 in 𝑋. Then (i)∑_𝑗∈𝐽(𝜙_𝑗+𝜓_𝑗)=𝜙+𝜓, (ii)∑_𝑗∈𝐽(𝑐𝜙_𝑗)=𝑐𝜙, where 𝑐 is any scalar.

Proof. The proof is a direct consequence of Definition 1 and is similar to the proof of the previous lemma.

The fundamental theorem of this paper is the next one which is used to obtain a number of results in Section 3.

Theorem 4. The unordered sum ∑_𝑗∈𝐽𝜙_𝑗 is convergent in 𝑋 if and only if, for any 𝜖>0 and 𝜆∈Λ, there exists 𝐽₀(=𝐽₀(𝜖,𝜆)) such that 𝛾_𝜆(𝑆_𝐾)<𝜖 for all 𝐾∈ℱ that are disjoint from 𝐽₀.

Proof. To prove this theorem, we shall use the fact that if 𝐾 and 𝐿 are disjoint, finite subsets of 𝐽, then 𝑆_𝐾∪𝐿=𝑆_𝐾+𝑆_𝐿.
Sufficiency. Let 𝜆∈Λ be arbitrarily fixed, and, for each 𝑛=1,2,…, choose a finite subset 𝐽_𝑛 of 𝐽 such that 𝛾_𝜆(𝑆_𝐾)<1/𝑛 for any 𝐾∈ℱ that is disjoint from 𝐽_𝑛. Replacing 𝐽_𝑛 by 𝐽₁∪𝐽₂∪⋯∪𝐽_𝑛, we can assume that 𝐽₁⊂𝐽₂⊂⋯. Then if 𝑚>𝑛, 𝛾_𝜆𝑆_𝐽𝑚−𝑆_𝐽𝑛=𝛾_𝜆𝑆_{𝐽𝑚−𝐽𝑛}<1𝑛as𝐽_𝑚−𝐽_𝑛isdisjointfrom𝐽_𝑛. (16) Hence {𝑆_𝐽𝑛} is a Cauchy sequence in 𝑋, and as we know that 𝑋 is a complete multinormed space, 𝜙=lim_𝑛→∞𝑆_𝐽𝑛 exists. Keeping 𝑛 fixed, let 𝑚→∞ in (16) to obtain 𝛾_𝜆𝜙−𝑆_𝐽𝑛≤1𝑛usingcontinuityof𝛾_𝜆. (17)
Now, for any 𝜖>0, we can choose 𝑛 such that 2/𝑛<𝜖. Then, whenever 𝐾∈ℱ contains 𝐽_𝑛, we obtain, from (16) and (17), 𝛾_𝜆𝜙−𝑆_𝐾=𝛾_𝜆𝜙−𝑆_{𝐾−𝐽𝑛}−𝑆_𝐽𝑛≤𝛾_𝜆𝜙−𝑆_𝐽𝑛+𝛾_𝜆𝑆_{𝐾−𝐽𝑛}<1𝑛+1𝑛<𝜖. (18) Thus, ∑_𝑗𝜙_𝑗 converges to 𝜙 in 𝑋.
Necessity. Suppose ∑_𝑗∈𝐽𝜙_𝑗 converges to 𝜙 in 𝑋. Given 𝜖>0 and 𝜆∈Λ, we can choose 𝐽₀(=𝐽₀(𝜖,𝜆)) such that, whenever 𝑀∈ℱ contains 𝐽₀, 𝛾_𝜆𝑆_𝑀<𝜖−𝜙2. (19)
Then, for any 𝐾∈ℱ that is disjoint from 𝐽₀, we get 𝛾_𝜆𝜙−𝑆_𝐾−𝑆_𝐽0=𝛾_𝜆𝜙−𝑆_𝐾∪𝐽0<𝜖2since𝐾,𝐽₀aredisjoint (20) and also 𝛾_𝜆𝜙−𝑆_𝐽0<𝜖2. (21)
Therefore, 𝛾_𝜆𝑆_𝐾≤𝛾_𝜆𝜙−𝑆_𝐽0+𝛾_𝜆𝜙−𝑆_𝐾−𝑆_𝐽0<𝜖2+𝜖2=𝜖. (22)

Now we state another important result which shows that Definition 1 is equivalent to the usual definition of a convergent series when 𝐽 is countable.

Lemma 5. Let 𝐽 be a countable index set. And let the vectors {𝜙_𝑗}_𝑗∈𝐽 be summable to 𝜙 in 𝑋. Then 𝜙=lim_𝑛→∞𝑆_𝐽𝑛 for any increasing sequence of finite subsets 𝐽_𝑛 of 𝐽 such that ∪_𝑛𝐽_𝑛=𝐽.

Proof. Let {𝜙_𝑗}_𝑗∈𝐽 be summable to 𝜙. Then from Definition 1, for 𝜖>0 and 𝜆∈Λ, there exists 𝐾₀(=𝐾₀(𝜖,𝜆))∈ℱ such that, whenever 𝐾∈ℱ contains 𝐾₀, we have 𝛾_𝜆𝑆_𝐾−𝜙<𝜖. (23) As {𝐽_𝑛} is an increasing sequence of sets in ℱ and ∪_𝑛𝐽_𝑛=𝐽, there exists 𝑁∈ℕ such that 𝐾₀⊂𝐽_𝑁. Moreover 𝐾₀⊂𝐽_𝑛 for all 𝑛≥𝑁, and so 𝛾_𝜆𝑆_𝐽𝑛−𝜙<𝜖whenever𝑛≥𝑁. (24) Hence the result follows.

The importance of Lemma 5 is that when 𝐽 is countable and {𝜙_𝑗}_𝑗∈𝐽 is summable to 𝜙 in 𝑋, then it is often possible to write down explicitly a sequence of partial sums that converge to 𝜙. As a particular case, let 𝐽=ℤ, the set of all integers. Then, we have the following result.

Lemma 6. If ∑_𝑗∈ℤ𝜙_𝑗=𝜙 in 𝑋, then lim_𝑛→∞𝑆_𝐽𝑛=𝜙, where 𝐽_𝑛={𝑚∈ℤ∶|𝑚|≤𝑛}. In this case we can write 𝜙=lim^𝑛_{𝑛→∞𝑗=−𝑛}𝜙_𝑗. (25)

Proof. It follows from Lemma 5.

3. Orthonormal Expansions in Spaces of Test Functions and Generalized Functions

In this section, we demonstrate how the theory of summability developed in Section 2 can be used to extend the distributional eigenfunction-expansion theory of Zemanian to the case when we have an uncountable orthonormal basis. As indicated earlier, all orthonormal bases in 𝐿²(𝐼) are countable, and so, for our extension to be meaningful, we follow the more general approach introduced in [5] of working with a self-adjoint operator 𝑇 on an arbitrary Hilbert space 𝐻. However, unlike [5], where it is assumed that 𝐻 is separable, we shall allow 𝐻 to be nonseparable. Throughout we shall assume that 𝐻 has an (possibly uncountable) orthonormal basis of eigenvectors {𝜓_𝑗}_𝑗∈𝐽 of 𝑇 with corresponding real eigenvalues {𝜆_𝑗}_𝑗∈𝐽. We shall represent the inner product and norm in 𝐻 by ‖⋅‖_𝐻 and (⋅,⋅)_𝐻, respectively, and, as before, we shall denote the collection of all finite subsets of 𝐽 by ℱ.

As discussed in the introduction, 𝐷_∞ is the Fréchet space: 𝐷_∞∶=^∞_𝑘=1𝐷𝑇^𝑘, (26) with multinorm 𝑆={𝛽_𝑘}^∞_𝑘=0, where 𝛽_𝑘𝜙∶=𝑇^𝑘𝜙_𝐻,𝑘=0,1,2,…; (27) see [5, Definition 2.1, Theorem 2.3]. The space of continuous linear functionals on 𝐷_∞ is denoted by 𝐷^′_∞. Convergence in 𝐷^′_∞ is defined by the weak* topology which is generated by the multinorm {𝛼_𝜙}_{𝜙∈𝐷∞}, where 𝛼_𝜙𝑓∶=𝑓,𝜙,𝑓∈𝐷^′_∞,𝜙∈𝐷_∞. (28)

The space 𝐷^′_∞ is sequentially complete, and each 𝜑∈𝐻 generates, uniquely, a continuous linear functional 𝜑∈𝐷^′_∞ via the formula 𝜑,𝜙∶=𝜙,𝜑_𝐻,𝜙∈𝐷_∞; (29) see [5, Theorems 2.6, 2.7]. As a consequence, we can regard 𝐻 as a subspace of 𝐷^′_∞, and so can write 𝐷_∞↪𝐻↪𝐷^′_∞. (30)

Theorem 7. Let 𝜙∈𝐷_∞. Then 𝜙=_𝑗∈𝐽𝜙,𝜓_𝑗𝐻𝜓_𝑗, (31) where the unordered sum is convergent in 𝐷_∞.

Proof. For any 𝐾∈ℱ, we can use the linearity of 𝑇 to write 𝛽_{𝑘𝑗∈𝐾}𝜙,𝜓_𝑗𝐻𝜓_𝑗=−𝜙_𝑗∈𝐾𝜙,𝜓_𝑗𝐻𝑇^𝑘𝜓_𝑗−𝑇^𝑘𝜙_𝐻=_𝑗∈𝐾𝜙,𝜓_𝑗𝐻𝜆^𝑘_𝑗𝜓_𝑗−𝑇^𝑘𝜙_𝐻=_𝑗∈𝐾𝜙,𝑇^𝑘𝜓_𝑗𝐻𝜓_𝑗−𝑇^𝑘𝜙_𝐻=_𝑗∈𝐾𝑇^𝑘𝜙,𝜓_𝑗𝐻𝜓_𝑗−𝑇^𝑘𝜙_𝐻. (32) Since, for each 𝑘∈ℕ₀, {(𝑇^𝑘𝜙,𝜓_𝑗)_𝐻𝜓_𝑗}_𝑗∈𝐽 is summable to 𝑇^𝑘𝜙 in 𝐻, it follows that, for 𝜖>0 and 𝑘∈ℕ₀, there exists 𝐽₀(=𝐽₀(𝜖,𝑘))∈ℱ such that _𝑗∈𝐾𝑇^𝑘𝜙,𝜓_𝑗𝐻𝜓_𝑗−𝑇^𝑘𝜙_𝐻<𝜖 (33) for all 𝐾∈ℱ that contain 𝐽₀. But then 𝛽_{𝑘𝑗∈𝐾}𝜙,𝜓_𝑗𝐻𝜓_𝑗−𝜙<𝜖 (34) for all 𝐾∈ℱ that contain 𝐽₀. Hence {(𝜙,𝜓_𝑗)_𝐻𝜓_𝑗}_𝑗∈𝐽 is summable in 𝐷_∞ to 𝜙.

It is also possible to characterize the unconditional convergence of sums of the form ∑_𝑗∈𝐽𝑎_𝑗𝜓_𝑗, where 𝑎_𝑗∈ℂ for all 𝑗∈𝐽, in the following way.

Theorem 8. Let {𝑎_𝑗}_𝑗∈𝐽 be a family of scalars. Then {𝑎_𝑗𝜓_𝑗}_𝑗∈𝐽 is summable to some 𝜙∈𝐷_∞ if and only if {|𝑎_𝑗|²|𝜆_𝑗|^2𝑘}_𝑗∈𝐽 is summable in ℝ for each 𝑘∈ℕ₀, in which case 𝑎_𝑗=(𝜙,𝜓_𝑗)_𝐻 for all 𝑗.

Proof. For each 𝐾∈ℱ, let 𝒮_𝐾=∑_𝑗∈𝐾𝑎_𝑗𝜓_𝑗. Then for each 𝛽_𝑘, we obtain 𝛽_𝑘(𝒮_𝐾)²=_𝑗∈𝐾𝑎_𝑗𝑇^𝑘𝜓_𝑗²_𝐻=_𝑗∈𝐾𝑎_𝑗²𝜆_𝑗^2𝑘. (35)
Suppose that {|𝑎_𝑗||𝜆_𝑗|^2𝑘}_𝑗∈𝐽 is summable in ℝ for each 𝑘∈ℕ₀, and let 𝜖>0. Then by Theorem 4, there exists 𝐽₀∈ℱ such that ∑_𝑗∈𝐾𝜆_𝑗^2𝑘|𝑎_𝑗|²<𝜖² for any 𝐾∈ℱ that is disjoint from 𝐽₀. But then 𝛽_𝑘(𝒮_𝐾)<𝜖 for any such 𝐾. Since this can be done for each 𝛽_𝑘, {𝑎_𝑗𝜓_𝑗}_𝑗∈𝐽 is summable in 𝐷_∞.
Conversely, if {𝑎_𝑗𝜓_𝑗}_𝑗∈𝐽 is summable in 𝐷_∞, then again, by Theorem 4, for any 𝜖>0 and 𝑘∈ℕ₀, there exists 𝐽₀∈ℱ such that 𝛽_𝑘(𝒮_𝐾)<𝜖 for every 𝐾∈ℱ disjoint from 𝐽₀. Hence, from (35), _𝑗∈𝐾𝜆_𝑗^2𝑘𝑎_𝑗²<𝜖² (36) for any such 𝐾, which implies that {|𝑎_𝑗|²|𝜆_𝑗|^2𝑘}_𝑗∈𝐽 is summable in ℝ for each 𝑘∈ℕ₀.
The final statement in the theorem follows from the fact that, for each fixed 𝑙∈𝐽, 𝜙,𝜓_𝑙𝐻=_𝑗∈𝐽𝑎_𝑗𝑏_𝑗, (37) where 𝑏_𝑙=1 and 𝑏_𝑗=0, 𝑗≠𝑙; see [9, Theorem 6.26].

Corresponding results on the weak* convergence of eigenvector expansions in the space 𝐷^′_∞ can also be established. Once again, these can be regarded as natural extensions of the theory developed in [3, Chapter 9] to the case of an index set 𝐽 which need not be countable. First, note that {𝜓_𝑗}_𝑗∈𝐽 can be regarded as a collection of “regular generalized eigenvectors” in 𝐷^′_∞. As the following theorem shows, each 𝑓∈𝐷^′_∞ can be expressed as a weak*-convergent Fourier series in terms of this collection of regular generalized eigenvectors.

Theorem 9. Each 𝑓∈𝐷^′_∞ can be expressed in the form 𝑓=_𝑗∈𝐽𝑓,𝜓_𝑗𝜓_𝑗, (38) where the unordered sum is convergent in 𝐷^′_∞.

Proof. Let 𝜖>0 and 𝜙∈𝐷_∞. Then, with 𝑆_𝐾=∑_𝑗∈𝐾(𝑓,𝜓_𝑗)𝜓_𝑗, where 𝐾∈ℱ, we have 𝛼_𝜙𝑆_𝐾−𝑓=𝑆_𝐾=−𝑓,𝜙_𝑗∈𝐾𝑓,𝜓_𝑗𝜓_𝑗−=,𝜙𝑓,𝜙𝑓,_𝑗∈𝐾𝜙,𝜓_𝑗𝐻𝜓_𝑗−𝜙≤𝑀_𝑓^𝑘𝑓_𝑟=0𝛼_{𝑟𝑗∈𝐾}𝜙,𝜓_𝑗𝐻𝜓_𝑗−𝜙 (39) (by continuity of 𝑓; see [3, Theorem  1.8-1]).
Since ∑_𝑗∈𝐽(𝜙,𝜓_𝑗)_𝐻𝜓_𝑗=𝜙 in 𝐷_∞ and there are only a finite number of seminorms 𝛼_𝑟, it follows that there exists 𝐽₀∈ℱ such that 𝛼_𝜙𝑆_𝐾−𝑓<𝜖, (40) for all 𝐾∈ℱ that contain 𝐽₀. Hence 𝑓=_𝑗∈𝐽𝑓,𝜓_𝑗𝜓_𝑗. (41)

Theorem 10. Let the null space 𝑁(𝑇) of 𝑇 be finite-dimensional and let {𝑏_𝑗}_𝑗∈𝐽⊂ℂ. Then _𝑗∈𝐽𝑏_𝑗𝜓_𝑗=𝑓 (42) in 𝒜^′ if there exists 𝑞∈ℕ₀ such that _{𝑗∈𝐽∶𝜆𝑗≠0}𝜆_𝑗^−2𝑞𝑏_𝑗² (43) is convergent in ℝ. Moreover, in this case 𝑏_𝑗=(𝑓,𝜓_𝑗) for each 𝑗.

Proof. Let 𝐽^′={𝑗∈𝐽∶𝜆_𝑗≠0} and, for each finite subset 𝐾^′ of 𝐽^′, let 𝑆_𝐾′=_{𝑗∈𝐾′}𝑏_𝑗𝜓_𝑗. (44)
By the convergence assumption and Theorem 4, it follows that, for 𝜖>0, there exists a finite subset 𝐽^′₀ of 𝐽^′ such that _{𝑗∈𝐾′}𝜆_𝑗^−2𝑞𝑏_𝑗²<𝜖 (45) for all finite subsets 𝐾^′ of 𝐽^′ disjoint from 𝐽^′₀. Also, from Theorem 8, _𝑗∈𝐽𝜆_𝑗^2𝑞𝜙,𝜓_𝑗𝐻²<∞. (46)
Hence there exists a finite subset 𝐽^′₁ of 𝐽^′ such that _{𝑗∈𝐾′1}𝜆_𝑗^2𝑞𝜙,𝜓_𝑗𝐻²<𝜖 (47) for all finite subsets 𝐾^′₁ of 𝐽^′ disjoint from 𝐽^′₁. Let 𝐽^′₂=𝐽^′₀∪𝐽^′₁. Then, for any finite subset 𝐾^′ of 𝐽^′ that is disjoint from 𝐽^′₂, we have 𝛼_𝜙𝑆_𝐾′=_{𝑗∈𝐾′}𝑏_𝑗𝜓_𝑗,𝜙=_{𝑗∈𝐾′}𝑏_𝑗𝜙,𝜓_𝑗𝐻=_{𝑗∈𝐾′}𝑏_𝑗𝜆^𝑞_𝑗𝜙,𝜆_𝑗^−𝑞𝜓_𝑗𝐻≤_{𝑗∈𝐾′}𝑏_𝑗𝜆_𝑗^−𝑞𝜆^𝑞_𝑗𝜙,𝜓_𝑗𝐻≤_{𝑗∈𝐾′}𝑏_𝑗𝜆_𝑗^−𝑞2_{𝑗∈𝐾′}𝜆^𝑞_𝑗𝜙,𝜓_𝑗𝐻^21/2<𝜖. (48)
Therefore ∑_{𝑗∈𝐽′}𝑏_𝑗𝜓_𝑗 is convergent in 𝒜^′, which implies that ∑_𝑗∈𝐽𝑏_𝑗𝜓_𝑗 is also summable in 𝒜^′ since 𝑁(𝑇) is finite-dimensional.
To complete the proof, we use the fact that since 𝑓 is a continuous linear functional, we have 𝑓,𝜓_𝑙=_𝑗∈𝐽𝑏_𝑗𝜓_𝑗,𝜓_𝑙=_𝑗∈𝐽𝑏_𝑗𝜓_𝑗,𝜓_𝑙=_𝑗∈𝐽𝑏_𝑗𝜓_𝑙,𝜓_𝑗𝐻=𝑏_𝑙. (49)

4. Almost-Periodic Generalized Functions

To illustrate the theory of expansions of test functions and generalized functions in terms of an uncountable basis of eigenfunctions, we examine the differential operator −𝑖𝐷 on the Hilbert space 𝐻 of 𝐿² almost-periodic functions. As explained in [9, Example 6.19], to obtain this space we consider first the vector space 𝑉 of quasiperiodic functions 𝜂∶ℝ→ℂ of the form 𝜂(𝑥)=^𝑛_𝑘=1𝑎_𝑘𝑒^{𝑖𝜔𝑘𝑥}, (50) where 𝑛∈ℕ,𝑎_𝑘∈ℂ and 𝜔_𝑘∈ℝ are arbitrary constants. This vector space becomes a pre-Hilbert space when we define on it the inner product 𝜂,𝜉_𝐻∶=lim_𝑅→∞12𝑅^𝑅_−𝑅𝜂(𝑥)𝜉(𝑥)𝑑𝑥. (51)

The Hilbert space 𝐻 is then obtained as the completion of this pre-Hilbert space. Elements in 𝐻 are equivalence classes of functions of the form 𝜙(𝑥)=^∞_𝑘=1𝑎_𝑘𝑒^{𝑖𝜔𝑘𝑥}, (52) where ∑^∞_𝑘=1|𝑎_𝑘|²<∞ and the sum in (52) converges with respect to the norm induced by the inner product (51). The set of functions 𝜓_{𝜔𝜔∈ℝ},𝜓_𝜔(𝑥)∶=𝑒^𝑖𝜔𝑥, (53) is an uncountable orthonormal basis for 𝐻, showing that 𝐻 is not separable. Note that each 𝜓_𝜔 can be regarded as an eigenfunction of −𝑖𝐷, defined on the domain 𝑉⊂𝐻, with corresponding eigenvalue 𝜔.

To enable a Fréchet space 𝐷_∞ of test functions and associated space of generalized functions 𝐷^′_∞ to be constructed, we require a self-adjoint extension 𝑇 of −𝑖𝐷. The fact that the set in (53) is an orthonormal basis for 𝐻 enables us to define 𝑇 in an obvious manner as 𝑇𝜙∶=_𝜔∈ℝ𝜔𝜙,𝜓_𝜔𝐻𝜓_𝜔,𝐷(𝑇)∶=𝜙∈𝐻∶_𝜔∈ℝ𝜔𝜙,𝜓_𝜔𝐻²<∞. (54) Note that, for any given 𝜙∈𝐷(𝑇), only countably many terms in the expansions in (54) will be nonzero.

The general theory developed earlier now leads immediately to the following expansion result for generalized functions in 𝐷^′_∞.

Theorem 11. Let 𝐷^′_∞ be the space of generalized functions constructed around the self-adjoint extension of −𝑖𝐷 given by (54). Then each 𝑓∈𝐷^′_∞ can be expressed in the form 𝑓=_𝜔∈ℝ𝑓,𝜓_𝜔𝜓_𝜔, (55) where 𝜓_𝜔(𝑥)=exp(𝑖𝜔𝑥) and the unordered sum is convergent in 𝐷^′_∞.

In this particular case, we can interpret 𝐷^′_∞ as a space of almost-periodic generalized functions.

5. Conclusion

A theory of summability has been developed and applied to obtain important convergence results of an orthonormal set {𝜙_𝑗}_𝑗∈𝐽 in suitably constructed spaces of test functions and generalized functions. As a particular case, a space of almost-periodic generalized functions has been produced, in which each generalized function can be expanded in terms of an uncountable orthonormal basis of exponential functions.