Compactly Supported Curvelet-Type Systems

Abstract

We study a flexible method for constructing curvelet-type frames. These curvelet-type systems have the same sparse representation properties as curvelets for appropriate classes of smooth functions, and the flexibility of the method allows us to give a constructive description of how to construct curvelet-type systems with a prescribed nature such as compact support in direct space. The method consists of using the machinery of almost diagonal matrices to show that a system of curvelet molecules which is sufficiently close to curvelets constitutes a frame for curvelet-type spaces. Such a system of curvelet molecules can then be constructed using finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay.

1. Introduction

Second-generation curvelets were introduced by Candès and Donoho, who also proved that curvelets give an essentially optimal sparse representation of images (functions) that are 𝐶² except for discontinuities along piecewise 𝐶²-curves [1]. It follows that efficient compression of such images can be archived by thresholding their curvelet expansions. Curvelets form a multiscale system with effective support that follows a parabolic scaling relation 𝑤𝑖𝑑𝑡ℎ≈𝑙𝑒𝑛𝑔𝑡ℎ². Moreover, they also provide an essentially optimal sparse representation of Fourier integral operators [2] and an optimal sparse and well organized solution operator for a wide class of linear hyperbolic differential equations [3]. However, curvelets are band-limited, and contrary to wavelets it is an open question whether compactly supported curvelet-type systems exist.

In this paper we study a flexible method for generating curvelet-type systems with the same sparse representation properties as curvelets (when sparseness is measured in curvelet-type sequence spaces). The method uses a perturbation principle which was first introduced in [4], further generalized in [5], and refined for frames in [6]. We give a constructive description of how to construct curvelet-type system consisting of finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay. This gives the flexibility to construct curvelet-type systems with a prescribed nature (see Section 6) such as compact support in direct space. For the sake of convenience the construction will only be done in ℝ², but it can easily be extended to ℝ^𝑑. The main results can be found in Sections 4 and 5.

The curvelet-type sequence spaces we use are associated with curvelet-type spaces 𝐺^{𝑠𝑝,𝑞} which were introduced in [7]. Here 𝐺^{𝑠𝑝,𝑞} was constructed by applying a curvelet-type splitting of the frequency space to a general construction of decomposition spaces, thereby obtaining a natural family of smoothness spaces for which curvelets constitute frames (see Section 2). Originally, this construction of decomposition spaces based on structured coverings of the frequency space was introduced by Feichtinger and Gröbner [8] and Feichtinger [9]. For example, the classical Triebel-Lizorkin and Besov spaces correspond to dyadic coverings of the frequency space (see [10]).

The outline of the paper is as follows. In Section 2 we define second-generation curvelets and curvelet-type spaces. Furthermore, we introduce curvelet molecules which will be the building blocks for our compactly supported curvelet-type frames. Next, in Section 3 we use the properties of curvelet molecules to show that the “change of frame coefficient” matrix is almost diagonal if the curvelet molecules have sufficient regularity. With the machinery of almost diagonal matrices, we can then in Section 4 show that curvelet molecules which are close enough to curvelets constitute frames for the curvelet-type spaces. Finally, in Section 5 we give a constructive description of how to construct these curvelet molecules from finite linear combinations of shifts and dilates of a single function with sufficient smoothness and decay. We conclude the paper with a short discussion in Section 6 of the possible functions which can be used to construct the curvelet molecules.

2. Second-Generation Curvelets

We begin this section with a brief definition of curvelets and curvelet molecules which will later be used to construct curvelet-type frames. Furthermore, we define the curvelet-type spaces for which curvelets constitute frames. For a much more detailed discussion of the curvelet construction, we refer the reader to [1, 3], and for decomposition spaces, of which the curvelet-type spaces are a subclass, we refer to [7, 8].

Let 𝜈 be an even 𝐶^∞(ℝ) window that is supported on [−𝜋,𝜋] such that its 2𝜋-periodic extension obeys ||𝜈(𝜃)||²+||𝜈(𝜃−𝜋)||²=1,𝜃∈[0,2𝜋).(2.1) Define 𝜈_𝑗,𝑙(𝜃)∶=𝜈(2^⌊𝑗/2⌋𝜃−𝜋𝑙) for 𝑗≥2 and 𝑙=0,1,…,2^⌊𝑗/2⌋−1. Next, with the angular window in place, let 𝑤∈𝐶^∞𝑐(ℝ) obey ||𝑤₀(𝑡)||²+_𝑗≥2||𝑤2^−𝑗𝑡||²=1,𝑡∈ℝ,(2.2) with 𝑤₀∈𝐶^∞𝑐(ℝ) supported in a neighborhood of the origin. We then define 𝜙_𝑗,𝑙(𝜉)∶=𝑤2^−𝑗||𝜉||𝜈_𝑗,𝑙(𝜃)+𝜈_𝑗,𝑙(𝜃+𝜋),𝜉=||𝜉||(cos𝜃,sin𝜃)∈ℝ².(2.3) Notice that the support of 𝑤(2^−𝑗|𝜉|)𝜈_𝑗,0(𝜃) is contained in a rectangle 𝑅_𝑗=𝐼_1𝑗×𝐼_2𝑗 given by 𝐼_1𝑗∶=𝜉₁,𝑡_𝑗≤𝜉₁≤𝑡_𝑗+𝐿_𝑗,𝐼_2𝑗∶=𝜉₂,2||𝜉₂||≤𝑙_𝑗,(2.4) where 𝑡_𝑗 is determined uniquely for a minimal 𝐿_𝑗,𝐿_𝑗∶=𝛿₁𝜋2^𝑗 and 𝑙_𝑗∶=𝛿₂2𝜋2^𝑗/2 (𝛿₁ depends weakly on 𝑗, see [1, Section 2.2]). With 𝐼_1𝑗∶=±𝐼_1𝑗 and 𝑅_𝑗=𝐼_1𝑗×𝐼_2𝑗 the system 𝑒_𝑗,𝑘(𝜉)∶=2^−3𝑗/42𝜋√𝛿₁𝛿₂𝑒^{𝑖((𝑘1+1/2)2−𝑗𝜉1/𝛿1)}𝑒^{𝑖(𝑘22−𝑗/2𝜉2/𝛿2)},𝑘∈ℤ²,(2.5) is an orthonormal basis for 𝐿₂(𝑅_𝑗).

We let 𝑓(𝜉)∶=ℱ(𝑓)(𝜉)∶=(2𝜋)⁻¹∫_ℝ2𝑓(𝑥)𝑒^{−𝑖𝑥⋅𝜉}d𝑥,𝑓∈𝐿₁(ℝ²), and by duality extend it uniquely from 𝒮(ℝ²) to 𝒮^′(ℝ²). Finally, we define𝜂̂_𝜇(𝜉)∶=𝜙_𝑗,𝑙(𝜉)𝑒_𝑗,𝑘𝑅^⊤𝜃_𝜇𝜉,𝜇=(𝑗,𝑙,𝑘),(2.6) where 𝑅_𝜃𝜇 is rotation by the angle 𝜃_𝜇∶=𝜋2^{⌊−𝑗/2⌋}𝑙, and as coarse-scale elements we define 𝜂̂_1,0,𝑘(𝜉)∶=𝛿⁰⁻¹𝜙_1,0(𝜉)𝑒^{𝑖𝑘⋅𝜉/𝛿0}, where 𝜙_1,0(𝜉)∶=𝜔₀(|𝜉|) and 𝛿₀>0 is sufficiently small. The system {𝜂_𝜇}_{𝜇∈𝒥×ℤ}² is called curvelets, 𝒥∶={(𝑗,𝑙)∣𝑗≥1,𝑙=0,1,…,2^⌊𝑗/2⌋−1}. It can be shown that curvelets constitute a tight frame for 𝐿₂(ℝ²) (see [1, Section 2.2]).

To later construct curvelet-type frames, we need a system of functions which share the essential properties of curvelets. As we will see, curvelet molecules, which were introduced in [3] and used there to study hyperbolic differential equations, have all the properties we need. For 𝜅∈ℕ²⁰, we define |𝜅|∶=𝜅₁+𝜅₂, and for suitably differentiable functions we define 𝑓^(𝜅)∶=𝜕^|𝜅|𝑓/(𝜕^𝜅1_𝜉1𝜕^𝜅2_𝜉2).

Definition 2.1. A family of functions {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is said to be a family of curvelet molecules with regularity 𝑅, 𝑅∈ℕ, if for 𝑗≥2 they may be expressed as 𝜓_𝜇(𝑥)=2^3𝑗/4𝑎_𝜇𝐷^2−𝑗𝑅_𝜃𝜇𝑥−𝑘¹𝛿₁,𝑘²𝛿₂,(2.7) where 𝐷₂^−𝑗𝑥=(2^𝑗𝑥₁,2^𝑗/2𝑥₂),𝛿₁,𝛿₂>0 and all functions 𝑎_𝜇 satisfy the following.(i)For |𝜅|≤𝑅 there exist constants 𝐶>0 independent of 𝜇 such that ||𝑎^𝜇(𝜅)(𝑥)||≤𝐶(1+|𝑥|)^−2𝑅.(2.8)(ii)There exist constants 𝐶>0 independent of 𝜇 such that ||𝑎_𝜇̂(𝜉)||≤𝐶min1,2^−𝑗+||𝜉₁||+2^−𝑗/2||𝜉₂||^𝑅.(2.9)The coarse-scale molecules, 𝑗=1, must take the form 𝜓_𝜇(𝑥)=𝑎_𝜇(𝑥−𝑘/𝛿₀),𝛿₀>0, where 𝑎_𝜇 satisfies (2.8).

It can be shown that curvelets constitute a family of curvelet molecules with regularity 𝑅 for any 𝑅∈ℕ (see [3, page 1489]).

To define the curvelet-type spaces which together with the associated sequence spaces will characterize the sparse representation properties of curvelets, we need a suitable partition of unity.

Definition 2.2. Let 𝑄_𝑗,𝑙∶=supp(𝜙_𝑗,𝑙) for (𝑗,𝑙)∈𝒥, where 𝜙_𝑗,𝑙 was defined in (2.3). A bounded admissible partition of unity (BAPU) is a family of functions {Ψ_𝑗,𝑙}_{(𝑗,𝑙)∈𝒥}⊂𝒮∶=𝒮(ℝ²) satisfying:(i)supp(Ψ_𝑗,𝑙)⊆𝑄_𝑗,𝑙,(𝑗,𝑙)∈𝒥; (ii)∑_{(𝑗,𝑙)∈𝒥}Ψ_𝑗,𝑙(𝜉)=1,𝜉∈ℝ²;(iii)sup_{(𝑗,𝑙)∈𝒥}|𝑄_𝑗,𝑙|^1/𝑝−1‖ℱ⁻¹Ψ_𝑗,𝑙‖_{𝐿𝑝(ℝ2)}<∞,𝑝∈(0,1].

An example of a BAPU is {|𝜙_𝑗,𝑙|²}_{(𝑗,𝑙)∈𝒥} which follows from the construction of 𝜙_𝑗,𝑙 (see (2.1) and (2.2)) and curvelets being curvelet molecules with regularity 𝑅 for any 𝑅∈ℕ. We are now ready to define curvelet-type spaces.

Definition 2.3. Let {Ψ_𝑗,𝑙}_{(𝑗,𝑙)∈𝒥} be a BAPU and Ψ_𝑗,𝑙(𝐷)𝑓∶=ℱ⁻¹(Ψ_𝑗,𝑙ℱ𝑓). For 𝑠∈ℝ,0<𝑞<∞ and 0<𝑝≤∞, we define 𝐺^{𝑠𝑝,𝑞}∶=𝐺^{𝑠𝑝,𝑞}(ℝ²) as the set of distributions 𝑓∈𝒮^∶=𝒮^(ℝ²) satisfying ‖𝑓‖_𝐺^{𝑠𝑝,𝑞}∶=_{(𝑗,𝑙)∈𝒥}‖‖2^𝑗𝑠Ψ_𝑗,𝑙(𝐷)𝑓‖‖^𝑞𝐿_𝑝^1/𝑞<∞.(2.10)

It can be shown that 𝐺^{𝑠𝑝,𝑞} is a quasi-Banach space (Banach space for 𝑝,𝑞≥1), and 𝒮 is dense in 𝐺^{𝑠𝑝,𝑞} (see [7, 8]). Furthermore, 𝐺^{𝑠𝑝,𝑞} is independent of the choice of BAPU.

We also need the sequence spaces associated with the curvelet-type spaces. For the sake of convenience, we write ‖𝑓_𝑘‖ instead of ‖{𝑓_𝑘}_𝑘∈𝐾‖ when the index set is clear from the context.

Definition 2.4. For 𝑠∈ℝ,0<𝑞<∞ and 0<𝑝≤∞, we define the sequence space 𝑔^{𝑠𝑝,𝑞} as the set of sequences {𝑧_𝜇}_{𝜇∈𝒥×ℤ}²⊂ℂ satisfying ‖‖𝑧_𝜇‖‖_{𝑔𝑠𝑝,𝑞}∶=‖‖‖‖2^{𝑗(𝑠+(3/2)(1/2−1/𝑝))}_𝑘∈ℤ2||𝑧_𝜇||^𝑝^1/𝑝‖‖‖‖_𝑙𝑞<∞,(2.11) where the 𝑙_𝑝-norm is replaced with the 𝑙_∞-norm if 𝑝=∞.

Notice that the sequence spaces 𝑙_𝑞 are special cases of 𝑔^{𝑠𝑝,𝑞} as we have 𝑔^{−(3/2)(1/2−1/𝑞)𝑞,𝑞}=𝑙_𝑞.

Next, we introduce frames for 𝐺^{𝑠𝑝,𝑞} and use the notation 𝐹≍𝐺 when there exist two constants 0<𝐶₁≤𝐶₂<∞, depending only on “allowable” parameters, such that 𝐶₁𝐹≤𝐺≤𝐶₂𝐹.

Definition 2.5. We say that a family of functions {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² in the dual of 𝐺^{𝑠𝑝,𝑞} is a frame for 𝐺^{𝑠𝑝,𝑞} if for all 𝑓∈𝐺^{𝑠𝑝,𝑞} we have ‖𝑓‖_𝐺^{𝑠𝑝,𝑞}≍‖‖⟨𝑓,𝜓_𝜇⟩‖‖_{𝑔𝑠𝑝,𝑞}.(2.12) The following is called the frame expansion of {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² when it exists: 𝑓=_{𝜇∈𝒥×ℤ2}⟨𝑓,𝑆⁻¹𝜓_𝜇⟩𝜓_𝜇(2.13) in the sense of 𝒮^, where 𝑆 is the frame operator 𝑆𝑓=∑_{𝜇∈𝒥×ℤ2}⟨𝑓,𝜓_𝜇⟩𝜓_𝜇, 𝑓∈𝐺^{𝑠𝑝,𝑞}.

From [7, Lemma 4 and Section 7.3] we have that curvelets (2.6) constitute a frame for the curvelet-type spaces with a frame operator 𝑆 that is equal to the identity, 𝑆=𝐼.

Proposition 2.6. Assume that 𝑠∈ℝ, 0<𝑞<∞ and 0<𝑝≤∞. For any finite sequence {𝑧_𝜇}_{𝜇∈𝒥×ℤ}²⊂ℂ, one has ‖‖‖‖_{𝜇∈𝒥×ℤ2}𝑧_𝜇𝜂_𝜇‖‖‖‖_{𝐺𝑠𝑝,𝑞}≤𝐶‖‖𝑧_𝜇‖‖_{𝑔𝑠𝑝,𝑞}.(2.14) Furthermore, {𝜂_𝜇}_{𝜇∈𝒥×ℤ}² is a frame for 𝐺^{𝑠𝑝,𝑞} with frame operator 𝑆=𝐼, ‖𝑓‖_𝐺^{𝑠𝑝,𝑞}≍‖‖⟨𝑓,𝜂_𝜇⟩‖‖_{𝑔𝑠𝑝,𝑞},𝑓∈𝐺^{𝑠𝑝,𝑞}.(2.15)

Notice that frame expansions for two frames {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² and {𝜂_𝜇}_{𝜇∈𝒥×ℤ}² have the same sparseness when measured in the associated sequence space 𝑔^{𝑠𝑝,𝑞} if {𝑆⁻¹𝜓_𝜇}_{𝜇∈𝒥×ℤ}² and {𝑆⁻¹𝜂_𝜇}_{𝜇∈𝒥×ℤ}² also constitute frames for 𝐺^{𝑠𝑝,𝑞}, ‖‖𝑓,𝑆⁻¹𝜓_𝜇‖‖_𝑔^{𝑠𝑝,𝑞}≍‖𝑓‖_𝐺^{𝑠𝑝,𝑞}_(ℝ²₎≍‖‖⟨𝑓,𝑆⁻¹𝜂_𝜇⟩‖‖_{𝑔𝑠𝑝,𝑞}.(2.16) Hence, to get a curvelet-type system {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² with the same sparse representation properties as curvelets {𝜂_𝜇}_{𝜇∈𝒥×ℤ}², it suffices to prove that {𝑆⁻¹𝜓_𝜇}_{𝜇∈𝒥×ℤ}² constitutes a frame for 𝐺^{𝑠𝑝,𝑞}.

3. Almost Diagonal Matrices

To generate curvelet-type frames in the following sections we introduce the machinery of almost diagonal matrices in this section. Almost diagonal matrices where used in [11] on Besov spaces, and here we find an associated notion of almost diagonal matrices on 𝑔^{𝑠𝑝,𝑞}. The goal is to find a definition so that the composition of two almost diagonal matrices gives a new almost diagonal matrix and almost diagonal matrices are bounded on 𝑔^{𝑠𝑝,𝑞}.

To help us define almost diagonal matrices we use a slight variation of the pseudodistance introduced in [12] which was constructed in [3]. For this we need the center of 𝜂_𝜇 in direct space, 𝑥_𝜇∶=𝑅_𝜃𝜇(𝑘₁2^−𝑗/𝛿₁,𝑘₂2^−𝑗/2𝛿₂), and the “direction” of 𝜂_𝜇,𝜌_𝜇∶=(cos𝜃_𝜇,sin𝜃_𝜇).

Definition 3.1. Given a pair of indices 𝜇=(𝑗,𝑙,𝑘) and 𝜇^=(𝑗^,𝑙^,𝑘^), we define the dyadic-parabolic pseudodistance as 𝜔𝜇,𝜇^∶=2^{|𝑗−𝑗′|}1+min2^𝑗,2^𝑗′𝑑𝜇,𝜇^,(3.1) where 𝑑𝜇,𝜇^∶=||𝜃_𝜇−𝜃_𝜇′||²+||𝑥_𝜇−𝑥_𝜇′||²+||𝜌_𝜇,𝑥_𝜇−𝑥_𝜇′||.(3.2)

The dyadic-parabolic distance was studied in detail in [3], and from there we can deduce the following properties.(i)For 𝛿>0 there exists 𝐶>0 such that _𝑘_∈ℤ2𝜔𝜇,𝜇^^{−3/2−𝛿}≤𝐶.(3.3)(ii)For 𝛿>0 there exists 𝐶>0 such that _(𝑗_{,𝑙)∈𝒥}𝜔𝜇,𝜇^^{−1/2−𝛿}≤𝐶.(3.4)(iii)For 𝑁≥2 and 𝛿>0 there exists 𝐶>0 such that _{𝜇′′∈}_𝒥×ℤ2𝜔𝜇,𝜇^^{−𝑁−𝛿}𝜔𝜇^,𝜇^^{−𝑁−𝛿}≤𝐶𝜔𝜇,𝜇^^{−𝑁−𝛿/2}.(3.5)(iv)Let {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² and {𝜂_𝜇}_{𝜇∈𝒥×ℤ}² be two families of curvelet molecules with regularity 4𝑅, 𝑅∈ℕ. Then there exists 𝐶>0 such that ||𝜓_𝜇,𝜂_𝜇′||≤𝐶𝜔𝜇,𝜇^^−𝑅.(3.6)

These properties lead us to the following definition of almost diagonal matrices on 𝑔^{𝑠𝑝,𝑞}.

Definition 3.2. Assume that 𝑠∈ℝ, 0<𝑞<∞ and 0<𝑝≤∞. Let 𝑟∶=min(1,𝑝,𝑞) and 𝑡∶=𝑠+(3/2)(1/2−1/𝑝). A matrix 𝐀={𝑎_𝜇𝜇′}_{𝜇,𝜇′∈𝒥×ℤ}² is called almost diagonal on 𝑔^{𝑠𝑝,𝑞} if there exists 𝐶,𝛿>0 such that ||𝑎_𝜇𝜇′||≤𝐶2^{(𝑗′−𝑗)𝑡}𝜔𝜇,𝜇^^{−2/𝑟−𝛿}.(3.7)

Remark 3.3. Note that by using (3.5), we get that the composition of two almost diagonal matrices on 𝑔^{𝑠𝑝,𝑞} gives a new almost diagonal matrix on 𝑔^{𝑠𝑝,𝑞}.

We are now ready to show the most important property of almost diagonal matrices; they act boundedly on the curvelet-type spaces.

Proposition 3.4. If 𝐀 is almost diagonal on 𝑔^{𝑠𝑝,𝑞}, then 𝐀 is bounded on 𝑔^{𝑠𝑝,𝑞}.

Proof. We only prove the result for 𝑝<∞ as the result for 𝑝=∞ follows in a similar way with 𝑙_𝑝 replaced by 𝑙_∞. Let 𝜔₀(𝜇,𝜇^)∶=𝜔(𝑗,𝑙,0,𝑗^,𝑙^,0),𝑧∶={𝑧_𝜇}_{𝜇∈𝒥×ℤ}²∈𝑔^{𝑠𝑝,𝑞}, and assume for now that 𝑝≥1. We begin with looking at the 𝑙_𝑝-norm of ‖𝐀𝑧‖_𝑔^{𝑠𝑝,𝑞}. By using Minkowski’s inequality, Hölder’s inequality and (3.3) we get _𝑘∈ℤ2||(𝐀𝑧)_𝜇||^𝑝^1/𝑝≤𝐶⎛⎜⎜⎝_𝑘∈ℤ2⎛⎜⎜⎝_{𝑗′,𝑙′∈𝒥}2^{(𝑗′−𝑗)𝑡}𝜔₀𝜇,𝜇^^{−1/2𝑟−𝛿/2}_𝑘_′∈ℤ2||𝑧_𝜇′||𝜔𝜇,𝜇^^{−3/2𝑟−𝛿/2}⎞⎟⎟⎠^𝑝⎞⎟⎟⎠^1/𝑝≤𝐶_{𝑗′,𝑙′∈𝒥}2^{(𝑗′−𝑗)𝑡}𝜔₀𝜇,𝜇^^{−1/2𝑟−𝛿/2}⎛⎜⎜⎝_𝑘∈ℤ2⎛⎜⎜⎝_{𝑘′∈ℤ2}||𝑧_𝜇′||𝜔𝜇,𝜇^^{−3/2𝑟−𝛿/2}⎞⎟⎟⎠^𝑝⎞⎟⎟⎠^1/𝑝≤𝐶_{𝑗′,𝑙′∈𝒥}2^{(𝑗′−𝑗)𝑡}𝜔₀𝜇,𝜇^^{−1/2𝑟−𝛿/2}×⎛⎜⎜⎝_{𝑘∈ℤ2𝑘}_′∈ℤ2||𝑧_𝜇′||^𝑝𝜔𝜇,𝜇^^{−3/2𝑟−𝛿/2}⎛⎜⎜⎝_{𝑘′∈ℤ2}𝜔𝜇,𝜇^^{−3/2𝑟−𝛿/2}⎞⎟⎟⎠^𝑝−1⎞⎟⎟⎠^1/𝑝≤𝐶_{𝑗′,𝑙′∈𝒥}2^{(𝑗′−𝑗)𝑡}𝜔₀𝜇,𝜇^^{−1/2𝑟−𝛿/2}⎛⎜⎜⎝_{𝑘′∈ℤ2}||𝑧_𝜇′||^𝑝⎞⎟⎟⎠^1/𝑝.(3.8) We then have ‖𝐀𝑧‖_𝑔^{𝑠𝑝,𝑞}≤𝐶⎛⎜⎜⎝_{(𝑗,𝑙)∈𝒥}⎛⎜⎜⎝_{𝑗′,𝑙′∈𝒥}2^𝑗′𝑡𝜔₀𝜇,𝜇^^{−1/2𝑟−𝛿/2}⎛⎜⎜⎝_{𝑘′∈ℤ2}||𝑧_𝜇′||^𝑝⎞⎟⎟⎠^1/𝑝⎞⎟⎟⎠^𝑞⎞⎟⎟⎠^1/𝑞.(3.9) For 𝑞≥1 we use Hölder’s inequality and (3.4) to get ‖𝐀𝑧‖_𝑔^{𝑠𝑝,𝑞}≤𝐶⎛⎜⎜⎝_{(𝑗,𝑙)∈𝒥𝑗′}_{,𝑙′∈𝒥}2^{𝑗′𝑞𝑡}𝜔₀𝜇,𝜇^^{−1/2𝑟−𝛿/2}×⎛⎜⎜⎝_{𝑘′∈ℤ2}||𝑧_𝜇′||^𝑝⎞⎟⎟⎠^𝑞/𝑝⎛⎜⎜⎝_{𝑗′,𝑙′∈𝒥}𝜔₀𝜇,𝜇^^{−1/2𝑟−𝛿/2}⎞⎟⎟⎠^𝑞−1⎞⎟⎟⎠^1/𝑞≤𝐶‖𝑧‖_𝑔^{𝑠𝑝,𝑞}.(3.10) For 𝑞<1 the result follows by a direct estimate. The case 𝑝<1 remains, and here we first observe that 𝐀∶=𝑎̃_𝜇𝜇′_{𝜇,𝜇′∈𝒥×ℤ}²=||𝑎_𝜇𝜇′||^𝑝2^{(𝑗′−𝑗)(𝑡−𝑡𝑝)}_{𝜇,𝜇′∈𝒥×ℤ2}(3.11) is almost diagonal on 𝑔^{𝑠1,𝑞/𝑝}. Furthermore, if we let 𝑣∶={𝑣_𝜇}_{𝜇∈𝒥×ℤ}²∶={|𝑧_𝜇|^𝑝2^{−𝑗(𝑡−𝑡𝑝)}}_{𝜇∈𝒥×ℤ2} we have ‖𝑣‖^𝑔1/𝑝_{𝑠1,𝑞/𝑝}=⎛⎜⎜⎝_{(𝑗,𝑙)∈𝒥}_𝑘∈ℤ22^𝑗𝑡𝑝||𝑧_𝜇||^𝑝^𝑞/𝑝⎞⎟⎟⎠^1/𝑞=‖𝑧‖_𝑔^{𝑠𝑝,𝑞}.(3.12) Before we can put these two observations into use, we need that ||(𝐀𝑧)_𝜇||^𝑝≤_{𝑗′,𝑙′∈𝒥𝑘}_′∈ℤ2||𝑎_𝜇𝜇′||^𝑝||𝑧_𝜇′||^𝑝=2^{𝑗(𝑡−𝑡𝑝)}_(𝑗_{,𝑙′)∈𝒥}_{𝑘′∈ℤ2}𝑎̃_𝜇𝜇′𝑣_𝜇.(3.13) We then have ‖𝐀𝑧‖_𝑔^{𝑠𝑝,𝑞}≤‖‖𝐀𝑣‖‖^𝑔1/𝑝_{𝑠1,𝑞/𝑝}≤𝐶‖𝑣‖^𝑔1/𝑝_{𝑠1,𝑞/𝑝}=𝐶‖𝑧‖_𝑔^{𝑠𝑝,𝑞}.(3.14)

4. Curvelet-Type Frames

In this section we study a family of curvelet molecules {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² which is a small perturbation of curvelets {𝜂_𝜇}_{𝜇∈𝒥×ℤ}². The goal is first to show that if {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is close enough to {𝜂_𝜇}_{𝜇∈𝒥×ℤ}², then {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a frame for 𝐺^{𝑠𝑝,𝑞}. Next to get a frame expansion, we show that {𝑆⁻¹𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is also a frame. The results are inspired by [6] where perturbations of frames were studied in Triebel-Lizorkin and Besov spaces.

Let {𝜓_𝜇}_{𝜇∈𝒥×ℤ}²⊂𝐿₂(ℝ²) be a system that is close to {𝜂_𝜇}_{𝜇∈𝒥×ℤ}² in the sense that there exists 𝜀,𝛿>0 such that for 𝑗≥2𝜂_𝜇(𝑥)−𝜓_𝜇(𝑥)=2^3𝑗/4𝑐_𝜇𝐷^2−𝑗𝑅_𝜃𝜇𝑥−𝑘¹𝛿₁,𝑘²𝛿₂,(4.1) where 𝐷₂^−𝑗𝑥=(2^𝑗𝑥₁,2^𝑗/2𝑥₂), 𝛿₁,𝛿₂>0 and all functions 𝑐_𝜇 satisfy the following.(i)For |𝜅|≤4⌈|𝑡|+2/𝑟⌉+𝛿 we need ||𝑐^𝜇(𝜅)(𝑥)||≤𝜀(1+|𝑥|)^{−8⌈|𝑡|+2/𝑟+𝛿⌉}.(4.2)(ii)Furthermore we need ||𝑐̂_𝜇(𝜉)||≤𝜀min1,2^−𝑗+||𝜉₁||+2^−𝑗/2||𝜉₂||^{4⌈|𝑡|+2/𝑟+𝛿⌉}.(4.3)

We have used the notation from Definition 3.2. The coarse-scale molecules, 𝑗=1, must take the form 𝜂_𝜇(𝑥)−𝜓_𝜇(𝑥)=𝑐_𝜇(𝑥−𝑘/𝛿₀), 𝛿₀>0, where 𝑐_𝜇 satisfies (4.2).

Then {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a family of curvelet molecules with regularity 4⌈|𝑡|+2/𝑟+𝛿⌉ and motivated by {𝜂_𝜇}_{𝜇∈𝒥×ℤ}² being a tight frame for 𝐿₂(ℝ²), we formally define ⟨𝑓,𝜓_𝜇⟩ as 𝑓,𝜓_𝜇^′∶=_{𝜇∈𝒥×ℤ2}𝜂_𝜇,𝜓_𝜇′𝑓,𝜂_𝜇,𝑓∈𝐺^{𝑠𝑝,𝑞}.(4.4) It follows from (3.6) and Proposition 3.4 that ⟨⋅,𝜓_𝜇⟩ is a bounded linear functional on 𝐺^{𝑠𝑝,𝑞}; in fact we have _𝜇∈_𝒥×ℤ2||𝜂_𝜇,𝜓_𝜇′||||𝑓,𝜂_𝜇||≤‖‖‖‖‖⎧⎪⎨⎪⎩_{𝜇∈𝒥×ℤ2}||𝜂_𝜇,𝜓_𝜇′||||𝑓,𝜂_𝜇||⎫⎪⎬⎪⎭_{𝜇′∈𝒥×ℤ2}‖‖‖‖‖_{𝑔𝑠𝑝,𝑞}≤𝐶‖‖𝑓,𝜂_𝜇‖‖_𝑔^{𝑠𝑝,𝑞}≤𝐶‖𝑓‖_𝐺^{𝑠𝑝,𝑞}.(4.5) Furthermore, {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a norming family for 𝐺^{𝑠𝑝,𝑞} as it satisfies ‖⟨𝑓,𝜓_𝜇⟩‖_{𝑔𝑠𝑝,𝑞}≤𝐶‖𝑓‖_𝐺^{𝑠𝑝,𝑞}. This can be used to show that 𝑆 is a bounded operator on 𝐺^{𝑠𝑝,𝑞}, and for small enough 𝜀 this will be the key to showing that {𝜓_𝜇} is a frame for 𝐺^{𝑠𝑝,𝑞}.

Theorem 4.1. There exists 𝜀₀,𝐶₁,𝐶₂>0 such that if {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² satisfies (4.1) for some 0<𝜀≤𝜀₀, then one has 𝐶₁‖𝑓‖_𝐺^{𝑠𝑝,𝑞}≤‖‖⟨𝑓,𝜓_𝜇⟩‖‖_{𝑔𝑠𝑝,𝑞}≤𝐶₂‖𝑓‖_𝐺^{𝑠𝑝,𝑞}𝑓∈𝐺^{𝑠𝑝,𝑞}.(4.6)

Proof. That {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a norming family gives the upper bound; thus we only need to establish the lower bound. For this we use that {𝜀⁻¹(𝜂_𝜇−𝜓_𝜇)}_{𝜇∈𝒥×ℤ}² is also a norming family so we have ‖‖⟨𝑓,𝜂_𝜇−𝜓_𝜇⟩‖‖_{𝑔𝑠𝑝,𝑞}≤𝐶𝜀‖𝑓‖_𝐺^{𝑠𝑝,𝑞}.(4.7) It then follows that ‖𝑓‖_𝐺^{𝑠𝑝,𝑞}≤𝐶‖‖⟨𝑓,𝜂_𝜇⟩‖‖_{𝑔𝑠𝑝,𝑞}≤≤𝐶𝐶‖‖‖‖𝑓𝑓,,𝜓𝜓_𝜇𝜇‖‖_𝑔^𝑠𝜇+‖‖𝑓,𝜂_𝜇−𝜓_𝜇‖‖_𝑔^{𝑠𝑝,𝑞}‖‖_𝑔^{𝑠𝑝,𝑞}+𝜀‖𝑓‖_𝐺^{𝑠𝑝,𝑞}.(4.8) By choosing 𝜀<1/𝐶 we get the lower bound.

As one might guess from Theorem 4.1, the boundedness of the matrix {⟨𝜂_𝜇,𝑆⁻¹𝜓_𝜇′⟩}_{𝜇,𝜇′∈𝒥×ℤ}² on 𝑔^{𝑠𝑝,𝑞} is the key to showing that {𝑆⁻¹𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is also a frame for 𝐺^{𝑠𝑝,𝑞}.

Proposition 4.2. There exists 𝜀₀>0 such that if {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² satisfies (4.1) for some 0<𝜀≤𝜀₀ and furthermore is a frame for 𝐺⁰²²=𝐿₂(ℝ²), then {⟨𝜂_𝜇,𝑆⁻¹𝜓_𝜇⟩}_{𝜇𝜇′∈𝒥×ℤ}² is bounded on 𝑔^{𝑠𝑝,𝑞}.

Proof. The fact that {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a frame for 𝐿₂(ℝ²) ensures that 𝑆⁻¹ is a bounded operator on 𝐿₂(ℝ²). We first show that 𝑆⁻¹ is bounded on 𝐺^{𝑠𝑝,𝑞}. This will follow from showing that ‖(𝐼−𝑆)𝑓‖_𝐺^{𝑠𝑝,𝑞}≤𝐶𝜀‖𝑓‖_𝐺^{𝑠𝑝,𝑞},𝑓∈𝐺^{𝑠𝑝,𝑞},(4.9) choosing 𝜀 small enough and using the Neumann series. Assume for a moment that 𝒟∶={𝑑_𝜇′𝜇}_{𝜇′,𝜇∈𝒥×ℤ2}∶={⟨(𝐼−𝑆)𝜂_𝜇,𝜂_𝜇′⟩}_{𝜇′,𝜇∈𝒥×ℤ2} satisfies ‖𝒟𝑧‖_𝑔^{𝑠𝑝,𝑞}≤𝐶𝜀‖𝑧‖_𝑔^{𝑠𝑝,𝑞}.(4.10) By using that 𝑆 is self-adjoint, we then have ‖(𝐼−𝑆)𝑓‖_𝐺^{𝑠𝑝,𝑞}≤𝐶‖‖(𝐼−𝑆)𝑓,𝜂_𝜇′‖‖_𝑔^{𝑠𝑝,𝑞}=𝐶‖‖𝒟𝑓,𝜂_𝜇‖‖_𝑔^{𝑠𝑝,𝑞}≤𝐶𝜀‖‖{⟨𝑓,𝜂_𝜇⟩}‖‖_{𝑔𝑠𝑝,𝑞}≤𝐶𝜀‖𝑓‖_𝐺^{𝑠𝑝,𝑞}.(4.11) So to show (4.9) it suffices to prove (4.10). Note that ⟨(𝐼−𝑆)𝜂_𝜇,𝜂_𝜇′⟩==_{𝜇′′∈𝒥×ℤ2𝜇′′∈𝒥×ℤ2}⟨𝜂_𝜇,𝜂_𝜇′′⟩⟨𝜂_𝜇′′,𝜂_𝜇′⟩−_{𝜇′∈𝒥×ℤ2}⟨𝜂_𝜇,𝜓_𝜇′′⟩⟨𝜓_𝜇′′,𝜂_𝜇′⟩⟨𝜂_𝜇,𝜂_𝜇′′⟩⟨𝜂_𝜇′′−𝜓_𝜇′′,𝜂_𝜇′⟩+_{𝜇′′∈𝒥×ℤ2}𝜂_𝜇,𝜂_𝜇′′−𝜓_𝜇′′𝜓_𝜇^′′,𝜂_𝜇′.(4.12) By setting 𝒟𝒟₁∶=𝑑_{1(𝜇′)(𝜇′′)}∶=𝜂_𝜇^′′−𝜓_𝜇′′,𝜂_𝜇′₂,𝒟∶=𝑑_{2(𝜇′′)(𝜇)}∶=𝜂_𝜇,𝜂_𝜇′′₃,𝒟∶=𝑑_{3(𝜇′)(𝜇′′)}∶=𝜓_𝜇^′′,𝜂_𝜇′₄,∶=𝑑_{4(𝜇′′)(𝜇)}∶=𝜂_𝜇,𝜂_𝜇′′−𝜓_𝜇′′,(4.13) we have the decomposition 𝒟=𝒟₁𝒟₂+𝒟₃𝒟₄.(4.14) Since {𝜀⁻¹(𝜂_𝜇−𝜓_𝜇)}_{𝜇∈𝒥×ℤ}² is a family of curvelet molecules with regularity 4⌈|𝑡|+2/𝑟+𝛿⌉, we have from (3.6) that 𝜀⁻¹𝒟₁,𝒟₂,𝒟₃,𝜀⁻¹𝒟₄ are almost diagonal on 𝑔^{𝑠𝑝,𝑞}. Next, we use Remark 3.3, and by Proposition 3.4, ‖𝒟𝑧‖_𝑔^{𝑠𝑝,𝑞}≤𝐶𝜀‖𝑧‖_𝑔^{𝑠𝑝,𝑞}.(4.15) Consequently, (4.9) holds, and for sufficiently small 𝜀 the operator 𝑆⁻¹ is bounded on 𝐺^{𝑠𝑝,𝑞}. Finally, let 𝑧∶={𝑧_𝜇}_{𝜇∈𝒥×ℤ}²∈𝑔^{𝑠𝑝,𝑞} and ℎ=∶∑_𝜇𝑧_𝜇𝜂_𝜇. By using (2.9) we have that ℎ∈𝐺^{𝑠𝑝,𝑞}, and as {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a frame for 𝐿₂(ℝ²), we have that 𝑆⁻¹ is self-adjoint which gives _𝜇∈_𝒥×ℤ2⟨𝜂_𝜇,𝑆⁻¹𝜓_𝜇⟩𝑧_𝜇=_{𝜇∈𝒥×ℤ2}⟨𝑆⁻¹𝜂_𝜇,𝜓_𝜇⟩𝑧_𝜇=⟨𝑆⁻¹ℎ,𝜓_𝜇⟩.(4.16) If we combine this with {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² being a norming family (4.5), we get ‖‖‖‖_{𝜇∈𝒥×ℤ2}𝜂_𝜇,𝑆⁻¹𝜓_𝜇′𝑧_𝜇‖‖‖‖_{𝑔𝑠𝑝,𝑞}=‖‖⟨𝑆⁻¹ℎ,𝜓_𝜇′⟩‖‖_{𝑔𝑠𝑝,𝑞}≤𝐶‖‖𝑆⁻¹ℎ‖‖_{𝐺𝑠𝑝,𝑞}≤𝐶‖ℎ‖_𝐺^{𝑠𝑝,𝑞}≤𝐶‖𝑧‖_𝑔^{𝑠𝑝,𝑞}(4.17) which proves that {⟨𝜂_𝜇,𝑆⁻¹𝜓_𝜇⟩}_{𝜇,𝜇∈𝒥×ℤ}² is bounded on 𝑔^{𝑠𝑝,𝑞}.

That {𝑆⁻¹𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a frame for 𝐺^{𝑠𝑝,𝑞} now follows as a consequence of {⟨𝜂_𝜇,𝑆⁻¹𝜓_𝜇⟩}_{𝜇,𝜇∈𝒥×ℤ}² being bounded on 𝑔^{𝑠𝑝,𝑞}. We state the following results without proofs as they follow directly in the same way as in the Besov space case. The proofs can be found in [6]. First, we have the frame expansion.

Lemma 4.3. Assume that {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a family of curvelet molecules with regularity 4⌈|𝑡|+2/𝑟+𝛿⌉ and a frame for 𝐿₂(ℝ²). If {⟨𝜂_𝜇,𝑆⁻¹𝜓_𝜇⟩}_{𝜇,𝜇∈𝒥×ℤ}² is bounded on 𝑔^{𝑠𝑝,𝑞}, then for 𝑓∈𝐺^{𝑠𝑝,𝑞} one has 𝑓=_{𝜇∈𝒥×ℤ2}⟨𝑓,𝑆⁻¹𝜓_𝜇⟩𝜓_𝜇(4.18) in the sense of 𝒮^.

Next, we have that {𝑆⁻¹𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a frame for 𝐺^{𝑠𝑝,𝑞}.

Theorem 4.4. Assume that {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a family of curvelet molecules with regularity 4⌈|𝑡|+2/𝑟+𝛿⌉ and a frame for 𝐿₂(ℝ²). Then {𝑆⁻¹𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a frame for 𝐺^{𝑠𝑝,𝑞} if and only if {⟨𝜂_𝜇,𝑆⁻¹𝜓_𝜇⟩}_{𝜇,𝜇∈𝒥×ℤ}² is bounded on 𝑔^{𝑠𝑝,𝑞}.

It follows from Proposition 4.2, Lemma 4.3, and Theorem 4.4 that if {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a family of curvelet molecules which is close enough to curvelets, then the representation ∑_{𝜇∈𝒥×ℤ2}⟨𝑓,𝑆⁻¹𝜓_𝜇⟩𝜓_𝜇,𝑓∈𝐺^{𝑠𝑝,𝑞}, has the same sparse representation properties as curvelets when measured in 𝑔^{𝑠𝑝,𝑞}. As a final result we also have a frame expansion with {𝑆⁻¹𝜓_𝜇}_{𝜇∈𝒥×ℤ}².

Lemma 4.5. Assume that {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² is a family of curvelet molecules with regularity 4⌈|𝑡|+2/𝑟+𝛿⌉ and a frame for 𝐿₂(ℝ²). If the transpose of {⟨𝜂_𝜇,𝑆⁻¹𝜓_𝜇⟩}_{𝜇,𝜇∈𝒥×ℤ}² is bounded on 𝑔^{𝑠𝑝,𝑞}, then for 𝑓∈𝐺^{𝑠𝑝,𝑞} one has 𝑓=_{𝜇∈𝒥×ℤ2}⟨𝑓,𝜓_𝜇⟩𝑆⁻¹𝜓_𝜇(4.19) in the sense of 𝒮^.

All that remains now is to construct a flexible family of curvelet molecules which is close enough to curvelets in the sense of (4.1).

5. Construction of Curvelet-Type Systems

In this section we construct a flexible curvelet-type system. We do this by showing that finite linear combinations of shifts and dilates of a function 𝑔 with sufficient smoothness and decay can be used to construct a system {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² that satisfies (4.1). From the previous section, we then have that the representation ∑_{𝜇∈𝒥×ℤ2}⟨𝑓,𝑆⁻¹𝜓_𝜇⟩𝜓_𝜇, 𝑓∈𝐺^{𝑠𝑝,𝑞}, has the same sparse representation properties as curvelets when measured in 𝑔^{𝑠𝑝,𝑞}.

First we take 𝑔∈𝐶^𝑀+1(ℝ²), 𝑔̂(0)≠0, which for fixed 𝑁^>2,𝑀>0 satisfies||𝑔^(𝜅)(𝑥)||≤𝐶(1+|𝑥|)^−𝑁′,|𝜅|≤𝑀+1.(5.1) Next, for 𝑚≥1 we define 𝑔_𝑚(𝑥)∶=𝐶_𝑔𝑚²𝑔(𝑚𝑥), where 𝐶_𝑔=∶𝑔̂(0)⁻¹. It then follows that||g^𝑚(𝜅)(𝑥)||≤𝐶𝑚^2+|𝜅|(1+𝑚|𝑥|)^−𝑁′,|𝜅|≤𝑀+1,_ℝ2𝑔_𝑚(𝑥)d𝑥=1.(5.2) We recall that curvelets (2.6) are a family of curvelet molecules for any regularity 𝑅∈ℕ. From the definition of a family of curvelet molecules (Definition 2.1), we have that for 𝑗≥2 curvelet molecules can be expressed as 𝜂_𝜇(𝑥)=2^3𝑗/4𝑎_𝜇𝐷^2−𝑗𝑅_𝜃𝜇𝑥−𝑘¹𝛿₁,𝑘²𝛿₂,(5.3) where 𝑎_𝜇 must satisfy (2.8) and (2.9). So to construct a family of curvelet molecules {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² which satisfy (4.1), we need to construct a family of functions {𝑏_𝜇}_{𝜇∈𝒥×ℤ}² such that 𝑎_𝜇−𝑏_𝜇 satisfy (4.2) and (4.3). We define {𝜓_𝜇}_{𝜇∈𝒥×ℤ}² as 𝜓_𝜇(𝑥)∶=2^3𝑗/4𝑏_𝜇𝐷^2−𝑗𝑅_𝜃𝜇𝑥−𝑘¹𝛿₁,𝑘²𝛿₂(5.4) for 𝑗≥2 and to construct {𝑏_𝜇}_{𝜇∈𝒥×ℤ}² we also need the following set of finite linear combinations:Θ_𝐾,𝑚∶=𝑏_𝜇∶𝑏_𝜇(⋅)=^𝐾_𝑖=1𝑐_𝑖𝑔_𝑚⋅+𝑑_𝑖,𝑐_𝑖∈ℂ,𝑑_𝑖∈ℝ².(5.5) We have omitted the construction of 𝜓_𝜇 for 𝑗=1 as it follow in a similar way.

Proposition 5.1. Let 𝑁^′>𝑁>2,𝑀>0 and 𝑗>0. If 𝑔∈𝐶^𝑀+1(ℝ²),𝑔̂(0)≠0, fulfills (5.1) and 𝑎_𝜇∈𝐿₂(ℝ²)∩𝐶^𝑀+1(ℝ²) fulfills ||𝑎_𝜇̂||𝑎^𝜇(𝜅)(𝑥)||≤𝐶(1+|𝑥|)^−𝑁′,|𝜅|≤𝑀+1,(𝜉)||≤𝐶min1,2^−𝑗+||𝜉₁||+2^−𝑗/2||𝜉₂||^𝑀+1,(5.6) then for any 𝜀>0 there exist 𝐾,𝑚≥1 (𝑚 independent of 𝑗) and 𝑏_𝜇∈Θ_𝐾,𝑚 such that ||||𝑎𝑎_𝜇̂^𝜇(𝜅)(𝑥)−𝑏^𝜇(𝜅)(𝑥)||≤𝜀(1+|𝑥|)^−𝑁,|𝜅|≤𝑀,(𝜉)−𝑏̂_𝜇(𝜉)||≤𝜀min1,2^−𝑗+||𝜉₁||+2^−𝑗/2||𝜉₂||^𝑀.(5.7)

Proof. Let 𝜀>0 and 𝜅,|𝜅|≤𝑀, be given. We construct the approximation of 𝑎_𝜇 in direct space in three steps. First by a convolution operator 𝜔_𝑚=𝑎_𝜇∗𝑔_𝑚, then by 𝜃_𝑞,𝑚 which is the integral in 𝜔_𝑚 taken over a dyadic cube 𝑄, and finally by a discretization over smaller dyadic cubes 𝑏_{𝑙,𝑞,𝑚}. From (5.2) we have 𝑎^𝜇(𝜅)(𝑥)−𝜔^𝑚(𝜅)(𝑥)=_ℝ2𝑎^𝜇(𝜅)(𝑥)−𝑎^𝜇(𝜅)(𝑥−𝑦)𝑔_𝑚(𝑦)d𝑦.(5.8) Define 𝑈∶=𝑚^𝜆/2𝑁, where 𝜆∶=min(1,𝑁^′−𝑁). For |𝑥|≤𝑈, we use the mean value theorem to get ||𝑎^𝜇(𝜅)(𝑥)−𝑎^𝜇(𝜅)(𝑥−𝑦)||≤𝐶min1,||𝑦||.(5.9) Inserting this in (5.8) we have ||𝑎^𝜇(𝜅)(𝑥)−𝜔^𝑚(𝜅)(𝑥)||≤𝐶_ℝ2min1,||𝑦||𝑚²1+𝑚||𝑦||^𝑁′≤𝐶𝑚_−𝜆d𝑦≤𝐶𝑚^−𝜆/2𝑈^𝑁≤𝐶𝑚^−𝜆/2(1+|𝑥|)^𝑁.(5.10) For |𝑥|>𝑈, we split the integral over Ω∶={𝑦∶|𝑦|≤|𝑥|/2} and Ω^𝑐. If 𝑦∈Ω, then |𝑥−𝑦|≥|𝑥|/2, and we have _Ω||𝑎^𝜇(𝜅)(𝑥)−𝑎^𝜇(𝜅)(𝑥−𝑦)||||𝑔_𝑚(𝑦)||d𝑦≤𝐶(1+|𝑥|)^−𝑁′≤𝐶(1+𝑈)^𝜆(1+|𝑥|)^𝑁≤𝐶𝑚^{−𝜆2/2𝑁}(1+|𝑥|)^𝑁.(5.11) Integrating over Ω^𝑐 with |𝑥|>𝑈 gives _Ω𝑐||𝑎≤^𝜇(𝜅)(𝑥)𝐶−𝑎^𝜇(𝜅)(𝑥−𝑦)||||𝑔_𝑚(𝑦)||d𝑦(1+|𝑥|)^𝑁′+_Ω𝑐𝐶𝑚²1+||𝑥−𝑦||^𝑁′1+𝑚||𝑦||^𝑁′≤𝐶d𝑦(1+|𝑥|)^𝑁′+𝐶𝑚^−𝜆(1+|𝑥|)^𝑁≤𝐶𝑚^{−𝜆2/2𝑁}+𝑚^−𝜆(1+|𝑥|)^𝑁.(5.12) So by choosing 𝑚 sufficiently large in (5.10)–(5.12), we get ||𝑎^𝜇(𝜅)(𝑥)−𝜔^𝑚(𝜅)(𝑥)||≤𝜀3(1+|𝑥|)^−𝑁.(5.13) For the next step we fix 𝑚 and choose 𝑞∈ℕ. Let 𝑄 denote the dyadic cube with sidelength 2^𝑞+1, sides parallel with the axes and centered at the origin. We then approximate 𝜔_𝑚 with 𝜃_𝑞,𝑚 defined as 𝜃_𝑞,𝑚(⋅)=_𝑄𝑎_𝜇(𝑦)𝑔_𝑚(⋅−𝑦)d𝑦.(5.14) In which case we have 𝜔^𝑚(𝜅)(𝑥)−𝜃^{(𝜅)𝑞,𝑚}(𝑥)=_𝑄𝑐𝑎_𝜇(𝑦)𝑔^𝑚(𝜅)(𝑥−𝑦)d𝑦,(5.15) and it follows that ||𝜔^𝑚(𝜅)(𝑥)−𝜃^{(𝜅)𝑞,𝑚}(𝑥)||≤_{|𝑦|≥2𝑞}𝐶𝑚^2+|𝜅|1+||𝑦||^𝑁′1+𝑚||𝑥−𝑦||_𝐵^𝑁′d𝑦∶=𝐿.(5.16) We first estimate the integral for |𝑥|≤2^𝑞−1 which gives |𝑦|>|𝑥| and |𝑥−𝑦|≥2^𝑞−1. Hence we obtain 𝐿≤𝐶𝑚^2+|𝜅|(1+|𝑥|)^𝑁′_{|𝑢|≥2𝑞−1}1(1+𝑚|𝑢|)^𝑁d𝑢≤𝐶𝑚^{|𝜅|−𝜆}2^−𝜆𝑞(1+|𝑥|)^𝑁′.(5.17) For |𝑥|>2^𝑞−1, we split the integral over Ω∶={𝑦∶|𝑦|≥2^𝑞}∩{𝑦∶|𝑦|≤|𝑥|/2} and Ω^∶={𝑦∶|𝑦|≥2^𝑞}∩{𝑦∶|𝑦|>|𝑥|/2}. If 𝑦∈Ω, then |𝑥−𝑦|≥|𝑥|/2, and we get _Ω𝑚^2+|𝜅|1+||𝑦||^𝑁′1+𝑚||𝑥−𝑦||^𝑁′d𝑦≤𝐶𝑚^2+|𝜅|(1+𝑚|𝑥|)^𝑁′_{|𝑦|≥2𝑞}11+||𝑦||^𝑁′≤𝐶𝑚_{|𝜅|−𝜆}d𝑦2^−𝜆𝑞(1+|𝑥|)^𝑁.(5.18) Similar for Ω^′ we have _Ω′𝑚^2+|𝜅|1+||𝑦||^𝑁1+𝑚||𝑥−𝑦||^𝑁′d𝑦≤𝐶(1+|𝑥|)^𝑁′_ℝ2𝑚^2+|𝜅|1+𝑚||𝑥−𝑦||^𝑁′≤𝐶𝑚_|𝜅|d𝑦(1+|𝑥|)^𝑁′≤𝑚^|𝜅|2^−𝜆𝑞(1+|𝑥|)^𝑁.(5.19) By choosing 𝑞 sufficiently large in (5.17)–(5.19), we obtain ||𝜔^𝑚(𝜅)(𝑥)−𝜃^{(𝜅)𝑞,𝑚}(𝑥)||≤𝜀3(1+|𝑥|)^−𝑁.(5.20) For the final step we fix 𝑞, choose 𝑙∈ℕ, and approximate 𝜃_𝑞,𝑚 by a discretization 𝑏_{𝑙,𝑞,𝑚}(⋅)=_{𝐼∈𝐻𝑙,𝑞}||𝐼||𝑎_𝜇𝑥_𝐼𝑔_𝑚⋅−𝑥_𝐼,(5.21) where 𝑥_𝐼 is the center of the dyadic cube 𝐼 and 𝐻_𝑙,𝑞 is the set of dyadic cubes with sidelength 2^−𝑙 which together give 𝑄. Note that 𝑏_{𝑙,𝑞,𝑚}∈Θ_𝐾,𝑚, 𝐾=2^𝑞+𝑙+1. We introduce 𝐹(⋅)∶=𝑎_𝜇(⋅)𝑔^𝑚(𝜅)(𝑥−⋅) which gives |||𝜃_{(𝜅)𝑞,𝑚}(𝑥)−𝑏^{(𝜅)𝑙,𝑞,𝑚}(𝑥)|||≤_{𝐼∈𝐻𝑙,𝑞}_𝐼≤||𝑎_𝜇(𝑦)𝑔^𝑚(𝜅)(𝑥−𝑦)−𝑎_𝜇𝑥_𝐼𝑔^𝑚(𝜅)𝑥−𝑥_𝐼||d𝑦_{𝐼∈𝐻𝑙,𝑞}_𝐼||𝐹(𝑦)−𝐹𝑥_𝐼||d𝑦.(5.22) By using the mean value theorem, we then get |||𝜃_{(𝜅)𝑞,𝑚}(𝑥)−𝑏^{(𝜅)𝑙,𝑞,𝑚}(𝑥)|||≤_{𝐼∈𝐻𝑙,𝑞}_𝐼≤𝐶2_2𝑞−𝑙||𝑦−𝑥_𝐼||max_{𝐼|||𝜅′|||𝑧∈𝑙(𝑥,𝑦)≤1}|||𝐹_(𝜅^′₎(𝑧)|||d𝑦max_{′|𝑧|≤2𝑞+1|𝜅|≤|𝜅|+1}|||𝑔_(𝜅^′_)𝑚(𝑥−𝑧)|||,(5.23) where 𝑙(𝑥_𝐼,𝑦) is the line segment between 𝑥_𝐼 and 𝑦. If |𝑥|≤2^𝑞+2 and |𝜅^|≤|𝜅|+1, then we have |||𝑔_(𝜅^′_)𝑚(𝑥−𝑧)|||≤𝐶𝑚^3+|𝜅|≤𝐶𝑚^3+|𝜅|2^𝑞𝑁(1+|𝑥|)^𝑁.(5.24) For |𝑥|>2^𝑞+2 and |𝑧|≤2^𝑞+1, we have |𝑥−𝑧|≥|𝑥|/2, and hence for |𝜅′|≤|𝜅|+1, it follows that |||𝑔_(𝜅^′_)𝑚(𝑥−𝑧)|||≤𝐶𝑚^3+|𝜅|(1+𝑚|𝑥|)^𝑁′≤𝐶𝑚^3+|𝜅|(1+|𝑥|)^𝑁′.(5.25) By choosing 𝑙 sufficiently large, we obtain by combining (5.23)–(5.25) that |||𝜃_{(𝜅)𝑞,𝑚}(𝑥)−𝑏^{(𝜅)𝑙,𝑞,𝑚}(𝑥)|||≤𝜀3(1+|𝑥|)^−𝑁.(5.26) Finally by combining (5.13), (5.20) and (5.26), we get |||𝑎_𝜇(𝜅)(𝑥)−𝑏^{(𝜅)𝑙,𝑞,𝑚}(𝑥)|||≤𝜀(1+|𝑥|)^−𝑁.(5.27) To approximate 𝑎_𝜇 in frequency space we use three steps similar to the approximation in direct space. Note that 𝑏_{𝑙,𝑞,𝑚} still fulfills (5.27) if we choose 𝑙,𝑞,𝑚 even larger. First we use 𝜔_𝑚 to approximate 𝑎_𝜇̂ in which case we have ||𝑎_𝜇̂(𝜉)−𝜔_𝑚(𝜉)||=||||𝑎_𝜇̂(𝜉)^{𝑀/(1+𝑀)}𝑎_𝜇̂(𝜉)^1/(1+𝑀)1−𝐶^𝑔𝑔̂𝜉≤𝐶min1,2_−𝑗𝑚||||+||𝜉₁||+2^−𝑗/2||𝜉₂||^𝑀1+||𝜉||⁻¹||||1−𝐶_𝑔𝑔̂𝜉𝑚||||.(5.28) By choosing 𝜉_𝑔>0 such that 𝐶(1+𝜉_𝑔)⁻¹|1−𝐶_𝑔𝑔̂(𝜉/𝑚)|≤𝜀/3 and 𝑚 such that 𝐶|1−𝐶_𝑔𝑔̂(𝜉/𝑚)|≤𝜀/3 for |𝜉|<𝜉_𝑔, we get ||𝑎_𝜇̂(𝜉)−𝜔_𝑚(𝜉)||≤𝜀3min1,2^−𝑗+||𝜉₁||+2^−𝑗/2||𝜉₂||^𝑀.(5.29) Next, we fix 𝑚, choose 𝑞 and limit the Fourier integral of 𝑎_𝜇 to 𝑄 from the approximation in direct space, 𝜃^𝑞,𝑚(𝜉)=𝑔̂_𝑚(𝜉)_𝑄𝑎_𝜇(𝑥)𝑒^{𝑖𝑥⋅𝜉}d𝑥.(5.30) This gives |||𝜔_𝑚(𝜉)−𝜃^′_𝑞,𝑚(𝜉)|||≤||𝑔̂_𝑚(𝜉)||_|𝑥|>2𝑞||𝑎_𝜇(𝑥)𝑒^{𝑖𝑥⋅𝜉}||d𝑥≤𝐶2^−𝜆𝑞.(5.31) In the last step, we fix 𝑞 and approximate 𝜃^′_𝑞,𝑚 by 𝑏̂_{𝑙,𝑞,𝑚}. We introduce 𝐺(𝑥)∶=𝑎_𝜇(𝑥)𝑒^{𝑖𝑥⋅𝜉} which gives ||𝜃^𝑞,𝑚(𝜉)−𝑏̂_{𝑙,𝑞,𝑚}(𝜉)||≤||𝑔̂_𝑚(𝜉)|||||||_𝑄𝑎_𝜇(𝑥)𝑒^{𝑖𝑥⋅𝜉}d𝑥−_{𝐼∈𝐻𝑙,𝑞}|||||𝐼||𝑎_𝜇≤||𝑔̂_𝑚𝑥_𝐼𝑒^{𝑖𝑥𝐼⋅𝜉}||(𝜉)||_{𝐼∈𝐻𝑙,𝑞}_𝐼≤𝐶2_2𝑞−𝑙||𝐺(𝑥)−𝐺𝑥_𝐼||d𝑥1+||𝜉/𝑚||max^{2|||𝜅′|||𝑥∈ℝ≤1}|||𝐺_(𝜅^′₎(𝑥)|||≤𝐶𝑚2^2𝑞−𝑙.(5.32) By combining (5.29)–(5.32) for sufficiently large l,𝑞,𝑚, we get ||𝑎_𝜇̂(𝜉)−𝑏̂_{𝑙,𝑞,𝑚}(𝜉)||≤𝜀min1,2^−𝑗+||𝜉₁||+2^−𝑗/2||𝜉₂||^𝑀.(5.33) It follows that by choosing 𝑙,𝑞,𝑚 large enough 𝑏_{𝑙,𝑞,𝑚} fulfills both (5.7) and (5.4). Furthermore, we have 𝑏_{𝑙,𝑞,𝑚}∈Θ_𝐾,𝑚, 𝐾=2^𝑞+𝑙+1.

6. Discussion

In this paper we studied a flexible method for generation curvelet-type systems with the same sparse representation properties as curvelets when measured in 𝑔^{𝑠𝑝,𝑞}. With Proposition 4.2, Lemma 4.3, and Theorem 4.4 we proved that a system of curvelet molecules which is close enough to curvelets has these sparse representation properties. Furthermore, with Proposition 5.1 we gave a constructive description of how such a system of curvelet molecules can be constructed from finite linear combinations of shifts and dilates for a single function with sufficient smoothness and decay.

Examples of functions with sufficient smoothness and decay are the exponential function 𝑒^−|⋅|2 and the rational functions (1+|⋅|²)^−𝑁 with 𝑁 sufficiently large. An example with compact support can be constructed by using a spline with compact support. Furthermore as the system is constructed using finite linear combinations of splines, we get a system consisting of modulated compactly supported splines.