Journal of Function Spaces

Journal of Function Spaces / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 876315 | 18 pages | https://doi.org/10.1155/2012/876315

Compactly Supported Curvelet-Type Systems

Academic Editor: Hans G. Feichtinger
Received08 Dec 2010
Accepted20 Oct 2011
Published06 Feb 2012

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 Candรจs and Donoho, who also proved that curvelets give an essentially optimal sparse representation of images (functions) that are ๐ถ2 except for discontinuities along piecewise ๐ถ2-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 ๐‘ค๐‘–๐‘‘๐‘กโ„Žโ‰ˆ๐‘™๐‘’๐‘›๐‘”๐‘กโ„Ž2. 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 โ„2, but it can easily be extended to โ„๐‘‘. The main results can be found in Sections 4 and 5.

The curvelet-type sequence spaces we use are associated with curvelet-type spaces ๐บ๐‘ ๐‘,๐‘ž which were introduced in [7]. Here ๐บ๐‘ ๐‘,๐‘ž 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 Grรถbner [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 โ€œchange of frame coefficientโ€ 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 ๐œˆ be an even ๐ถโˆž(โ„) window that is supported on [โˆ’๐œ‹,๐œ‹] such that its 2๐œ‹-periodic extension obeys ||||๐œˆ(๐œƒ)2+||||๐œˆ(๐œƒโˆ’๐œ‹)2[=1,๐œƒโˆˆ0,2๐œ‹).(2.1) Define ๐œˆ๐‘—,๐‘™(๐œƒ)โˆถ=๐œˆ(2โŒŠ๐‘—/2โŒ‹๐œƒโˆ’๐œ‹๐‘™) for ๐‘—โ‰ฅ2 and ๐‘™=0,1,โ€ฆ,2โŒŠ๐‘—/2โŒ‹โˆ’1. Next, with the angular window in place, let ๐‘คโˆˆ๐ถโˆž๐‘(โ„) obey ||๐‘ค0||(๐‘ก)2+๎“๐‘—โ‰ฅ2||๐‘ค๎€ท2โˆ’๐‘—๐‘ก๎€ธ||2=1,๐‘กโˆˆโ„,(2.2) with ๐‘ค0โˆˆ๐ถโˆž๐‘(โ„) supported in a neighborhood of the origin. We then define ๐œ™๐‘—,๐‘™๎€ท2(๐œ‰)โˆถ=๐‘คโˆ’๐‘—||๐œ‰||๐œˆ๎€ธ๎€ท๐‘—,๐‘™(๐œƒ)+๐œˆ๐‘—,๐‘™๎€ธ||๐œ‰||(๐œƒ+๐œ‹),๐œ‰=(cos๐œƒ,sin๐œƒ)โˆˆโ„2.(2.3) Notice that the support of ๐‘ค(2โˆ’๐‘—|๐œ‰|)๐œˆ๐‘—,0(๐œƒ) is contained in a rectangle ๐‘…๐‘—=๐ผ1๐‘—ร—๐ผ2๐‘— given by ๐ผ1๐‘—๎€ฝ๐œ‰โˆถ=1,๐‘ก๐‘—โ‰ค๐œ‰1โ‰ค๐‘ก๐‘—+๐ฟ๐‘—๎€พ,๐ผ2๐‘—๎€ฝ๐œ‰โˆถ=2||๐œ‰,22||โ‰ค๐‘™๐‘—๎€พ,(2.4) where ๐‘ก๐‘— is determined uniquely for a minimal ๐ฟ๐‘—,๐ฟ๐‘—โˆถ=๐›ฟ1๐œ‹2๐‘— and ๐‘™๐‘—โˆถ=๐›ฟ22๐œ‹2๐‘—/2 (๐›ฟ1 depends weakly on ๐‘—, see [1, Section 2.2]). With ๎‚๐ผ1๐‘—โˆถ=ยฑ๐ผ1๐‘— and ๎‚๐‘…๐‘—=๎‚๐ผ1๐‘—ร—๐ผ2๐‘— the system ๐‘’๐‘—,๐‘˜2(๐œ‰)โˆถ=โˆ’3๐‘—/4โˆš2๐œ‹๐›ฟ1๐›ฟ2๐‘’๐‘–((๐‘˜1+1/2)2โˆ’๐‘—๐œ‰1/๐›ฟ1)๐‘’๐‘–(๐‘˜22โˆ’๐‘—/2๐œ‰2/๐›ฟ2),๐‘˜โˆˆโ„ค2,(2.5) is an orthonormal basis for ๐ฟ2(๎‚๐‘…๐‘—).

We let ๎๐‘“(๐œ‰)โˆถ=โ„ฑ(๐‘“)(๐œ‰)โˆถ=(2๐œ‹)โˆ’1โˆซโ„2๐‘“(๐‘ฅ)๐‘’โˆ’๐‘–๐‘ฅโ‹…๐œ‰d๐‘ฅ,๐‘“โˆˆ๐ฟ1(โ„2), and by duality extend it uniquely from ๐’ฎ(โ„2) to ๐’ฎโ€ฒ(โ„2). Finally, we defineฬ‚๐œ‚๐œ‡(๐œ‰)โˆถ=๐œ™๐‘—,๐‘™(๐œ‰)๐‘’๐‘—,๐‘˜๎‚€๐‘…โŠค๐œƒ๐œ‡๐œ‰๎‚,๐œ‡=(๐‘—,๐‘™,๐‘˜),(2.6) where ๐‘…๐œƒ๐œ‡ is rotation by the angle ๐œƒ๐œ‡โˆถ=๐œ‹2โŒŠโˆ’๐‘—/2โŒ‹๐‘™, and as coarse-scale elements we define ฬ‚๐œ‚1,0,๐‘˜(๐œ‰)โˆถ=๐›ฟ0โˆ’1๐œ™1,0(๐œ‰)๐‘’๐‘–๐‘˜โ‹…๐œ‰/๐›ฟ0, where ๐œ™1,0(๐œ‰)โˆถ=๐œ”0(|๐œ‰|) and ๐›ฟ0>0 is sufficiently small. The system {๐œ‚๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is called curvelets, ๐’ฅโˆถ={(๐‘—,๐‘™)โˆฃ๐‘—โ‰ฅ1,๐‘™=0,1,โ€ฆ,2โŒŠ๐‘—/2โŒ‹โˆ’1}. It can be shown that curvelets constitute a tight frame for ๐ฟ2(โ„2) (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 ๐œ…โˆˆโ„•20, we define |๐œ…|โˆถ=๐œ…1+๐œ…2, and for suitably differentiable functions we define ๐‘“(๐œ…)โˆถ=๐œ•|๐œ…|๐‘“/(๐œ•๐œ…1๐œ‰1๐œ•๐œ…2๐œ‰2).

Definition 2.1. A family of functions {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is said to be a family of curvelet molecules with regularity ๐‘…, ๐‘…โˆˆโ„•, if for ๐‘—โ‰ฅ2 they may be expressed as ๐œ“๐œ‡(๐‘ฅ)=23๐‘—/4๐‘Ž๐œ‡๎‚ต๐ท2โˆ’๐‘—๐‘…๐œƒ๐œ‡๎‚ต๐‘˜๐‘ฅโˆ’1๐›ฟ1,๐‘˜2๐›ฟ2๎‚ถ๎‚ถ,(2.7) where ๐ท2โˆ’๐‘—๐‘ฅ=(2๐‘—๐‘ฅ1,2๐‘—/2๐‘ฅ2),๐›ฟ1,๐›ฟ2>0 and all functions ๐‘Ž๐œ‡ satisfy the following.(i)For |๐œ…|โ‰ค๐‘… there exist constants ๐ถ>0 independent of ๐œ‡ such that ||๐‘Ž๐œ‡(๐œ…)||(๐‘ฅ)โ‰ค๐ถ(1+|๐‘ฅ|)โˆ’2๐‘….(2.8)(ii)There exist constants ๐ถ>0 independent of ๐œ‡ such that ||ฬ‚๐‘Ž๐œ‡||๎€ท(๐œ‰)โ‰ค๐ถmin1,2โˆ’๐‘—+||๐œ‰1||+2โˆ’๐‘—/2||๐œ‰2||๎€ธ๐‘….(2.9)The coarse-scale molecules, ๐‘—=1, must take the form ๐œ“๐œ‡(๐‘ฅ)=๐‘Ž๐œ‡(๐‘ฅโˆ’๐‘˜/๐›ฟ0),๐›ฟ0>0, where ๐‘Ž๐œ‡ satisfies (2.8).

It can be shown that curvelets constitute a family of curvelet molecules with regularity ๐‘… for any ๐‘…โˆˆโ„• (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 ๐‘„๐‘—,๐‘™โˆถ=supp(๐œ™๐‘—,๐‘™) for (๐‘—,๐‘™)โˆˆ๐’ฅ, where ๐œ™๐‘—,๐‘™ was defined in (2.3). A bounded admissible partition of unity (BAPU) is a family of functions {ฮจ๐‘—,๐‘™}(๐‘—,๐‘™)โˆˆ๐’ฅโŠ‚๐’ฎโˆถ=๐’ฎ(โ„2) satisfying:(i)supp(ฮจ๐‘—,๐‘™)โŠ†๐‘„๐‘—,๐‘™,(๐‘—,๐‘™)โˆˆ๐’ฅ; (ii)โˆ‘(๐‘—,๐‘™)โˆˆ๐’ฅฮจ๐‘—,๐‘™(๐œ‰)=1,๐œ‰โˆˆโ„2;(iii)sup(๐‘—,๐‘™)โˆˆ๐’ฅ|๐‘„๐‘—,๐‘™|1/๐‘โˆ’1โ€–โ„ฑโˆ’1ฮจ๐‘—,๐‘™โ€–๐ฟ๐‘(โ„2)<โˆž,๐‘โˆˆ(0,1].

An example of a BAPU is {|๐œ™๐‘—,๐‘™|2}(๐‘—,๐‘™)โˆˆ๐’ฅ which follows from the construction of ๐œ™๐‘—,๐‘™ (see (2.1) and (2.2)) and curvelets being curvelet molecules with regularity ๐‘… for any ๐‘…โˆˆโ„•. We are now ready to define curvelet-type spaces.

Definition 2.3. Let {ฮจ๐‘—,๐‘™}(๐‘—,๐‘™)โˆˆ๐’ฅ be a BAPU and ฮจ๐‘—,๐‘™(๐ท)๐‘“โˆถ=โ„ฑโˆ’1(ฮจ๐‘—,๐‘™โ„ฑ๐‘“). For ๐‘ โˆˆโ„,0<๐‘ž<โˆž and 0<๐‘โ‰คโˆž, we define ๐บ๐‘ ๐‘,๐‘žโˆถ=๐บ๐‘ ๐‘,๐‘ž(โ„2) as the set of distributions ๐‘“โˆˆ๐’ฎ๎…žโˆถ=๐’ฎ๎…ž(โ„2) satisfying โ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘ž๎ƒฉ๎“โˆถ=(๐‘—,๐‘™)โˆˆ๐’ฅโ€–โ€–2๐‘—๐‘ ฮจ๐‘—,๐‘™โ€–โ€–(๐ท)๐‘“๐‘ž๐ฟ๐‘๎ƒช1/๐‘ž<โˆž.(2.10)

It can be shown that ๐บ๐‘ ๐‘,๐‘ž is a quasi-Banach space (Banach space for ๐‘,๐‘žโ‰ฅ1), and ๐’ฎ is dense in ๐บ๐‘ ๐‘,๐‘ž (see [7, 8]). Furthermore, ๐บ๐‘ ๐‘,๐‘ž 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 โ€–๐‘“๐‘˜โ€– instead of โ€–{๐‘“๐‘˜}๐‘˜โˆˆ๐พโ€– when the index set is clear from the context.

Definition 2.4. For ๐‘ โˆˆโ„,0<๐‘ž<โˆž and 0<๐‘โ‰คโˆž, we define the sequence space ๐‘”๐‘ ๐‘,๐‘ž as the set of sequences {๐‘ง๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2โŠ‚โ„‚ satisfying โ€–โ€–๐‘ง๐œ‡โ€–โ€–๐‘”๐‘ ๐‘,๐‘žโ€–โ€–โ€–โ€–2โˆถ=๐‘—(๐‘ +(3/2)(1/2โˆ’1/๐‘))๎ƒฉ๎“๐‘˜โˆˆโ„ค2||๐‘ง๐œ‡||๐‘๎ƒช1/๐‘โ€–โ€–โ€–โ€–๐‘™๐‘ž<โˆž,(2.11) where the ๐‘™๐‘-norm is replaced with the ๐‘™โˆž-norm if ๐‘=โˆž.

Notice that the sequence spaces ๐‘™๐‘ž are special cases of ๐‘”๐‘ ๐‘,๐‘ž as we have ๐‘”โˆ’(3/2)(1/2โˆ’1/๐‘ž)๐‘ž,๐‘ž=๐‘™๐‘ž.

Next, we introduce frames for ๐บ๐‘ ๐‘,๐‘ž and use the notation ๐นโ‰๐บ when there exist two constants 0<๐ถ1โ‰ค๐ถ2<โˆž, depending only on โ€œallowableโ€ parameters, such that ๐ถ1๐นโ‰ค๐บโ‰ค๐ถ2๐น.

Definition 2.5. We say that a family of functions {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 in the dual of ๐บ๐‘ ๐‘,๐‘ž is a frame for ๐บ๐‘ ๐‘,๐‘ž if for all ๐‘“โˆˆ๐บ๐‘ ๐‘,๐‘ž we have โ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘žโ‰โ€–โ€–โŸจ๐‘“,๐œ“๐œ‡โŸฉโ€–โ€–๐‘”๐‘ ๐‘,๐‘ž.(2.12) The following is called the frame expansion of {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 when it exists: ๎“๐‘“=๐œ‡โˆˆ๐’ฅร—โ„ค2โŸจ๐‘“,๐‘†โˆ’1๐œ“๐œ‡โŸฉ๐œ“๐œ‡(2.13) in the sense of ๐’ฎ๎…ž, where ๐‘† is the frame operator โˆ‘๐‘†๐‘“=๐œ‡โˆˆ๐’ฅร—โ„ค2โŸจ๐‘“,๐œ“๐œ‡โŸฉ๐œ“๐œ‡, ๐‘“โˆˆ๐บ๐‘ ๐‘,๐‘ž.

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 ๐‘† that is equal to the identity, ๐‘†=๐ผ.

Proposition 2.6. Assume that ๐‘ โˆˆโ„, 0<๐‘ž<โˆž and 0<๐‘โ‰คโˆž. For any finite sequence {๐‘ง๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2โŠ‚โ„‚, one has โ€–โ€–โ€–โ€–๎“๐œ‡โˆˆ๐’ฅร—โ„ค2๐‘ง๐œ‡๐œ‚๐œ‡โ€–โ€–โ€–โ€–๐บ๐‘ ๐‘,๐‘žโ€–โ€–๐‘งโ‰ค๐ถ๐œ‡โ€–โ€–๐‘”๐‘ ๐‘,๐‘ž.(2.14) Furthermore, {๐œ‚๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a frame for ๐บ๐‘ ๐‘,๐‘ž with frame operator ๐‘†=๐ผ, โ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘žโ‰โ€–โ€–โŸจ๐‘“,๐œ‚๐œ‡โŸฉโ€–โ€–๐‘”๐‘ ๐‘,๐‘ž,๐‘“โˆˆ๐บ๐‘ ๐‘,๐‘ž.(2.15)

Notice that frame expansions for two frames {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 and {๐œ‚๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 have the same sparseness when measured in the associated sequence space ๐‘”๐‘ ๐‘,๐‘ž if {๐‘†โˆ’1๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 and {๐‘†โˆ’1๐œ‚๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 also constitute frames for ๐บ๐‘ ๐‘,๐‘ž, โ€–โ€–๎ซ๐‘“,๐‘†โˆ’1๐œ“๐œ‡๎ฌโ€–โ€–๐‘”๐‘ ๐‘,๐‘žโ‰โ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘ž(โ„2)โ‰โ€–โ€–โŸจ๐‘“,๐‘†โˆ’1๐œ‚๐œ‡โŸฉโ€–โ€–๐‘”๐‘ ๐‘,๐‘ž.(2.16) Hence, to get a curvelet-type system {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 with the same sparse representation properties as curvelets {๐œ‚๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2, it suffices to prove that {๐‘†โˆ’1๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 constitutes a frame for ๐บ๐‘ ๐‘,๐‘ž.

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 ๐‘”๐‘ ๐‘,๐‘ž. 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 ๐‘”๐‘ ๐‘,๐‘ž.

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 ๐œ‚๐œ‡ in direct space, ๐‘ฅ๐œ‡โˆถ=๐‘…๐œƒ๐œ‡(๐‘˜12โˆ’๐‘—/๐›ฟ1,๐‘˜22โˆ’๐‘—/2๐›ฟ2), and the โ€œdirectionโ€ of ๐œ‚๐œ‡,๐œŒ๐œ‡โˆถ=(cos๐œƒ๐œ‡,sin๐œƒ๐œ‡).

Definition 3.1. Given a pair of indices ๐œ‡=(๐‘—,๐‘™,๐‘˜) and ๐œ‡๎…ž=(๐‘—๎…ž,๐‘™๎…ž,๐‘˜๎…ž), we define the dyadic-parabolic pseudodistance as ๐œ”๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆถ=2|๐‘—โˆ’๐‘—โ€ฒ|๎‚€๎‚€21+min๐‘—,2๐‘—โ€ฒ๎‚๐‘‘๎€ท๐œ‡,๐œ‡๎…ž๎€ธ๎‚,(3.1) where ๐‘‘๎€ท๐œ‡,๐œ‡๎…ž๎€ธ||๐œƒโˆถ=๐œ‡โˆ’๐œƒ๐œ‡โ€ฒ||2+||๐‘ฅ๐œ‡โˆ’๐‘ฅ๐œ‡โ€ฒ||2+||๎ซ๐œŒ๐œ‡,๐‘ฅ๐œ‡โˆ’๐‘ฅ๐œ‡โ€ฒ๎ฌ||.(3.2)

The dyadic-parabolic distance was studied in detail in [3], and from there we can deduce the following properties.(i)For ๐›ฟ>0 there exists ๐ถ>0 such that ๎“๐‘˜โˆˆโ„ค2๐œ”๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’3/2โˆ’๐›ฟโ‰ค๐ถ.(3.3)(ii)For ๐›ฟ>0 there exists ๐ถ>0 such that ๎“(๐‘—,๐‘™)โˆˆ๐’ฅ๐œ”๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’1/2โˆ’๐›ฟโ‰ค๐ถ.(3.4)(iii)For ๐‘โ‰ฅ2 and ๐›ฟ>0 there exists ๐ถ>0 such that ๎“๐œ‡โ€ฒโ€ฒโˆˆ๐’ฅร—โ„ค2๐œ”๎€ท๐œ‡,๐œ‡๎…ž๎…ž๎€ธโˆ’๐‘โˆ’๐›ฟ๐œ”๎€ท๐œ‡๎…ž๎…ž,๐œ‡๎…ž๎€ธโˆ’๐‘โˆ’๐›ฟ๎€ทโ‰ค๐ถ๐œ”๐œ‡,๐œ‡๎…ž๎€ธโˆ’๐‘โˆ’๐›ฟ/2.(3.5)(iv)Let {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 and {๐œ‚๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 be two families of curvelet molecules with regularity 4๐‘…, ๐‘…โˆˆโ„•. Then there exists ๐ถ>0 such that ||๎ซ๐œ“๐œ‡,๐œ‚๐œ‡โ€ฒ๎ฌ||๎€ทโ‰ค๐ถ๐œ”๐œ‡,๐œ‡๎…ž๎€ธโˆ’๐‘….(3.6)

These properties lead us to the following definition of almost diagonal matrices on ๐‘”๐‘ ๐‘,๐‘ž.

Definition 3.2. Assume that ๐‘ โˆˆโ„, 0<๐‘ž<โˆž and 0<๐‘โ‰คโˆž. Let ๐‘Ÿโˆถ=min(1,๐‘,๐‘ž) and ๐‘กโˆถ=๐‘ +(3/2)(1/2โˆ’1/๐‘). A matrix ๐€={๐‘Ž๐œ‡๐œ‡โ€ฒ}๐œ‡,๐œ‡โ€ฒโˆˆ๐’ฅร—โ„ค2 is called almost diagonal on ๐‘”๐‘ ๐‘,๐‘ž if there exists ๐ถ,๐›ฟ>0 such that ||๐‘Ž๐œ‡๐œ‡โ€ฒ||โ‰ค๐ถ2(๐‘—โ€ฒโˆ’๐‘—)๐‘ก๐œ”๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’2/๐‘Ÿโˆ’๐›ฟ.(3.7)

Remark 3.3. Note that by using (3.5), we get that the composition of two almost diagonal matrices on ๐‘”๐‘ ๐‘,๐‘ž gives a new almost diagonal matrix on ๐‘”๐‘ ๐‘,๐‘ž.

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 ๐€ is almost diagonal on ๐‘”๐‘ ๐‘,๐‘ž, then ๐€ is bounded on ๐‘”๐‘ ๐‘,๐‘ž.

Proof. We only prove the result for ๐‘<โˆž as the result for ๐‘=โˆž follows in a similar way with ๐‘™๐‘ replaced by ๐‘™โˆž. Let ๐œ”0(๐œ‡,๐œ‡๎…ž)โˆถ=๐œ”(๐‘—,๐‘™,0,๐‘—๎…ž,๐‘™๎…ž,0),๐‘งโˆถ={๐‘ง๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2โˆˆ๐‘”๐‘ ๐‘,๐‘ž, and assume for now that ๐‘โ‰ฅ1. We begin with looking at the ๐‘™๐‘-norm of โ€–๐€๐‘งโ€–๐‘”๐‘ ๐‘,๐‘ž. By using Minkowskiโ€™s inequality, Hรถlderโ€™s inequality and (3.3) we get ๎ƒฉ๎“๐‘˜โˆˆโ„ค2||(๐€๐‘ง)๐œ‡||๐‘๎ƒช1/๐‘โŽ›โŽœโŽœโŽ๎“โ‰ค๐ถ๐‘˜โˆˆโ„ค2โŽ›โŽœโŽœโŽ๎“๎€ท๐‘—โ€ฒ,๐‘™โ€ฒ๎€ธโˆˆ๐’ฅ2(๐‘—โ€ฒโˆ’๐‘—)๐‘ก๐œ”0๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’1/2๐‘Ÿโˆ’๐›ฟ/2๎“๐‘˜โ€ฒโˆˆโ„ค2||๐‘ง๐œ‡โ€ฒ||๐œ”๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’3/2๐‘Ÿโˆ’๐›ฟ/2โŽžโŽŸโŽŸโŽ ๐‘โŽžโŽŸโŽŸโŽ 1/๐‘๎“โ‰ค๐ถ๎€ท๐‘—โ€ฒ,๐‘™โ€ฒ๎€ธโˆˆ๐’ฅ2(๐‘—โ€ฒโˆ’๐‘—)๐‘ก๐œ”0๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’1/2๐‘Ÿโˆ’๐›ฟ/2โŽ›โŽœโŽœโŽ๎“๐‘˜โˆˆโ„ค2โŽ›โŽœโŽœโŽ๎“๐‘˜โ€ฒโˆˆโ„ค2||๐‘ง๐œ‡โ€ฒ||๐œ”๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’3/2๐‘Ÿโˆ’๐›ฟ/2โŽžโŽŸโŽŸโŽ ๐‘โŽžโŽŸโŽŸโŽ 1/๐‘๎“โ‰ค๐ถ๎€ท๐‘—โ€ฒ,๐‘™โ€ฒ๎€ธโˆˆ๐’ฅ2(๐‘—โ€ฒโˆ’๐‘—)๐‘ก๐œ”0๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’1/2๐‘Ÿโˆ’๐›ฟ/2ร—โŽ›โŽœโŽœโŽ๎“๐‘˜โˆˆโ„ค2๎“๐‘˜โ€ฒโˆˆโ„ค2||๐‘ง๐œ‡โ€ฒ||๐‘๐œ”๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’3/2๐‘Ÿโˆ’๐›ฟ/2โŽ›โŽœโŽœโŽ๎“๐‘˜โ€ฒโˆˆโ„ค2๐œ”๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’3/2๐‘Ÿโˆ’๐›ฟ/2โŽžโŽŸโŽŸโŽ ๐‘โˆ’1โŽžโŽŸโŽŸโŽ 1/๐‘๎“โ‰ค๐ถ๎€ท๐‘—โ€ฒ,๐‘™โ€ฒ๎€ธโˆˆ๐’ฅ2(๐‘—โ€ฒโˆ’๐‘—)๐‘ก๐œ”0๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’1/2๐‘Ÿโˆ’๐›ฟ/2โŽ›โŽœโŽœโŽ๎“๐‘˜โ€ฒโˆˆโ„ค2||๐‘ง๐œ‡โ€ฒ||๐‘โŽžโŽŸโŽŸโŽ 1/๐‘.(3.8) We then have โ€–๐€๐‘งโ€–๐‘”๐‘ ๐‘,๐‘žโŽ›โŽœโŽœโŽ๎“โ‰ค๐ถ(๐‘—,๐‘™)โˆˆ๐’ฅโŽ›โŽœโŽœโŽ๎“๎€ท๐‘—โ€ฒ,๐‘™โ€ฒ๎€ธโˆˆ๐’ฅ2๐‘—โ€ฒ๐‘ก๐œ”0๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’1/2๐‘Ÿโˆ’๐›ฟ/2โŽ›โŽœโŽœโŽ๎“๐‘˜โ€ฒโˆˆโ„ค2||๐‘ง๐œ‡โ€ฒ||๐‘โŽžโŽŸโŽŸโŽ 1/๐‘โŽžโŽŸโŽŸโŽ ๐‘žโŽžโŽŸโŽŸโŽ 1/๐‘ž.(3.9) For ๐‘žโ‰ฅ1 we use Hรถlderโ€™s inequality and (3.4) to get โ€–๐€๐‘งโ€–๐‘”๐‘ ๐‘,๐‘žโŽ›โŽœโŽœโŽ๎“โ‰ค๐ถ(๐‘—,๐‘™)โˆˆ๐’ฅ๎“๎€ท๐‘—โ€ฒ,๐‘™โ€ฒ๎€ธโˆˆ๐’ฅ2๐‘—โ€ฒ๐‘ž๐‘ก๐œ”0๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’1/2๐‘Ÿโˆ’๐›ฟ/2ร—โŽ›โŽœโŽœโŽ๎“๐‘˜โ€ฒโˆˆโ„ค2||๐‘ง๐œ‡โ€ฒ||๐‘โŽžโŽŸโŽŸโŽ ๐‘ž/๐‘โŽ›โŽœโŽœโŽ๎“๎€ท๐‘—โ€ฒ,๐‘™โ€ฒ๎€ธโˆˆ๐’ฅ๐œ”0๎€ท๐œ‡,๐œ‡๎…ž๎€ธโˆ’1/2๐‘Ÿโˆ’๐›ฟ/2โŽžโŽŸโŽŸโŽ ๐‘žโˆ’1โŽžโŽŸโŽŸโŽ 1/๐‘žโ‰ค๐ถโ€–๐‘งโ€–๐‘”๐‘ ๐‘,๐‘ž.(3.10) For ๐‘ž<1 the result follows by a direct estimate. The case ๐‘<1 remains, and here we first observe that ๎‚๎€ฝ๐€โˆถ=ฬƒ๐‘Ž๐œ‡๐œ‡โ€ฒ๎€พ๐œ‡,๐œ‡โ€ฒโˆˆ๐’ฅร—โ„ค2=๎‚†||๐‘Ž๐œ‡๐œ‡โ€ฒ||๐‘2(๐‘—โ€ฒโˆ’๐‘—)(๐‘กโˆ’๐‘ก๐‘)๎‚‡๐œ‡,๐œ‡โ€ฒโˆˆ๐’ฅร—โ„ค2(3.11) is almost diagonal on ๐‘”๐‘ 1,๐‘ž/๐‘. Furthermore, if we let ๐‘ฃโˆถ={๐‘ฃ๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2โˆถ={|๐‘ง๐œ‡|๐‘2โˆ’๐‘—(๐‘กโˆ’๐‘ก๐‘)}๐œ‡โˆˆ๐’ฅร—โ„ค2 we have โ€–๐‘ฃโ€–๐‘”1/๐‘๐‘ 1,๐‘ž/๐‘=โŽ›โŽœโŽœโŽ๎“(๐‘—,๐‘™)โˆˆ๐’ฅ๎ƒฉ๎“๐‘˜โˆˆโ„ค22๐‘—๐‘ก๐‘||๐‘ง๐œ‡||๐‘๎ƒช๐‘ž/๐‘โŽžโŽŸโŽŸโŽ 1/๐‘ž=โ€–๐‘งโ€–๐‘”๐‘ ๐‘,๐‘ž.(3.12) Before we can put these two observations into use, we need that ||(๐€๐‘ง)๐œ‡||๐‘โ‰ค๎“๎€ท๐‘—โ€ฒ,๐‘™โ€ฒ๎€ธโˆˆ๐’ฅ๎“๐‘˜โ€ฒโˆˆโ„ค2||๐‘Ž๐œ‡๐œ‡โ€ฒ||๐‘||๐‘ง๐œ‡โ€ฒ||๐‘=2๐‘—(๐‘กโˆ’๐‘ก๐‘)๎“(๐‘—๎…ž,๐‘™โ€ฒ)โˆˆ๐’ฅ๎“๐‘˜โ€ฒโˆˆโ„ค2ฬƒ๐‘Ž๐œ‡๐œ‡โ€ฒ๐‘ฃ๐œ‡.(3.13) We then have โ€–๐€๐‘งโ€–๐‘”๐‘ ๐‘,๐‘žโ‰คโ€–โ€–๎‚โ€–โ€–๐€๐‘ฃ๐‘”1/๐‘๐‘ 1,๐‘ž/๐‘โ‰ค๐ถโ€–๐‘ฃโ€–๐‘”1/๐‘๐‘ 1,๐‘ž/๐‘=๐ถโ€–๐‘งโ€–๐‘”๐‘ ๐‘,๐‘ž.(3.14)

4. Curvelet-Type Frames

In this section we study a family of curvelet molecules {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 which is a small perturbation of curvelets {๐œ‚๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2. The goal is first to show that if {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is close enough to {๐œ‚๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2, then {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a frame for ๐บ๐‘ ๐‘,๐‘ž. Next to get a frame expansion, we show that {๐‘†โˆ’1๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is also a frame. The results are inspired by [6] where perturbations of frames were studied in Triebel-Lizorkin and Besov spaces.

Let {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2โŠ‚๐ฟ2(โ„2) be a system that is close to {๐œ‚๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 in the sense that there exists ๐œ€,๐›ฟ>0 such that for ๐‘—โ‰ฅ2๐œ‚๐œ‡(๐‘ฅ)โˆ’๐œ“๐œ‡(๐‘ฅ)=23๐‘—/4๐‘๐œ‡๎‚ต๐ท2โˆ’๐‘—๐‘…๐œƒ๐œ‡๎‚ต๐‘˜๐‘ฅโˆ’1๐›ฟ1,๐‘˜2๐›ฟ2๎‚ถ๎‚ถ,(4.1) where ๐ท2โˆ’๐‘—๐‘ฅ=(2๐‘—๐‘ฅ1,2๐‘—/2๐‘ฅ2), ๐›ฟ1,๐›ฟ2>0 and all functions ๐‘๐œ‡ satisfy the following.(i)For |๐œ…|โ‰ค4โŒˆ|๐‘ก|+2/๐‘ŸโŒ‰+๐›ฟ we need ||๐‘๐œ‡(๐œ…)||(๐‘ฅ)โ‰ค๐œ€(1+|๐‘ฅ|)โˆ’8โŒˆ|๐‘ก|+2/๐‘Ÿ+๐›ฟโŒ‰.(4.2)(ii)Furthermore we need ||ฬ‚๐‘๐œ‡||๎€ท(๐œ‰)โ‰ค๐œ€min1,2โˆ’๐‘—+||๐œ‰1||+2โˆ’๐‘—/2||๐œ‰2||๎€ธ4โŒˆ|๐‘ก|+2/๐‘Ÿ+๐›ฟโŒ‰.(4.3)

We have used the notation from Definition 3.2. The coarse-scale molecules, ๐‘—=1, must take the form ๐œ‚๐œ‡(๐‘ฅ)โˆ’๐œ“๐œ‡(๐‘ฅ)=๐‘๐œ‡(๐‘ฅโˆ’๐‘˜/๐›ฟ0), ๐›ฟ0>0, where ๐‘๐œ‡ satisfies (4.2).

Then {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a family of curvelet molecules with regularity 4โŒˆ|๐‘ก|+2/๐‘Ÿ+๐›ฟโŒ‰ and motivated by {๐œ‚๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 being a tight frame for ๐ฟ2(โ„2), we formally define โŸจ๐‘“,๐œ“๐œ‡๎…žโŸฉ as ๎ซ๐‘“,๐œ“๐œ‡โ€ฒ๎ฌ๎“โˆถ=๐œ‡โˆˆ๐’ฅร—โ„ค2๎ซ๐œ‚๐œ‡,๐œ“๐œ‡โ€ฒ๎ฌ๎ซ๐‘“,๐œ‚๐œ‡๎ฌ,๐‘“โˆˆ๐บ๐‘ ๐‘,๐‘ž.(4.4) It follows from (3.6) and Proposition 3.4 that โŸจโ‹…,๐œ“๐œ‡๎…žโŸฉ is a bounded linear functional on ๐บ๐‘ ๐‘,๐‘ž; in fact we have ๎“๐œ‡โˆˆ๐’ฅร—โ„ค2||๎ซ๐œ‚๐œ‡,๐œ“๐œ‡โ€ฒ๎ฌ||||๎ซ๐‘“,๐œ‚๐œ‡๎ฌ||โ‰คโ€–โ€–โ€–โ€–โ€–โŽงโŽชโŽจโŽชโŽฉ๎“๐œ‡โˆˆ๐’ฅร—โ„ค2||๎ซ๐œ‚๐œ‡,๐œ“๐œ‡โ€ฒ๎ฌ||||๎ซ๐‘“,๐œ‚๐œ‡๎ฌ||โŽซโŽชโŽฌโŽชโŽญ๐œ‡โ€ฒโˆˆ๐’ฅร—โ„ค2โ€–โ€–โ€–โ€–โ€–๐‘”๐‘ ๐‘,๐‘žโ€–โ€–๎ซโ‰ค๐ถ๐‘“,๐œ‚๐œ‡๎ฌโ€–โ€–๐‘”๐‘ ๐‘,๐‘žโ‰ค๐ถโ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘ž.(4.5) Furthermore, {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a norming family for ๐บ๐‘ ๐‘,๐‘ž as it satisfies โ€–โŸจ๐‘“,๐œ“๐œ‡โŸฉโ€–๐‘”๐‘ ๐‘,๐‘žโ‰ค๐ถโ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘ž. This can be used to show that ๐‘† is a bounded operator on ๐บ๐‘ ๐‘,๐‘ž, and for small enough ๐œ€ this will be the key to showing that {๐œ“๐œ‡} is a frame for ๐บ๐‘ ๐‘,๐‘ž.

Theorem 4.1. There exists ๐œ€0,๐ถ1,๐ถ2>0 such that if {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 satisfies (4.1) for some 0<๐œ€โ‰ค๐œ€0, then one has ๐ถ1โ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘žโ‰คโ€–โ€–โŸจ๐‘“,๐œ“๐œ‡โŸฉโ€–โ€–๐‘”๐‘ ๐‘,๐‘žโ‰ค๐ถ2โ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘ž๐‘“โˆˆ๐บ๐‘ ๐‘,๐‘ž.(4.6)

Proof. That {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a norming family gives the upper bound; thus we only need to establish the lower bound. For this we use that {๐œ€โˆ’1(๐œ‚๐œ‡โˆ’๐œ“๐œ‡)}๐œ‡โˆˆ๐’ฅร—โ„ค2 is also a norming family so we have โ€–โ€–โŸจ๐‘“,๐œ‚๐œ‡โˆ’๐œ“๐œ‡โŸฉโ€–โ€–๐‘”๐‘ ๐‘,๐‘žโ‰ค๐ถ๐œ€โ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘ž.(4.7) It then follows that โ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘žโ€–โ€–โ‰ค๐ถโŸจ๐‘“,๐œ‚๐œ‡โŸฉโ€–โ€–๐‘”๐‘ ๐‘,๐‘ž๎‚€โ€–โ€–๎ซโ‰ค๐ถ๐‘“,๐œ“๐œ‡๎ฌโ€–โ€–๐‘”๐‘ ๐œ‡+โ€–โ€–๎ซ๐‘“,๐œ‚๐œ‡โˆ’๐œ“๐œ‡๎ฌโ€–โ€–๐‘”๐‘ ๐‘,๐‘ž๎‚๎‚€โ€–โ€–๎ซโ‰ค๐ถ๐‘“,๐œ“๐œ‡๎ฌโ€–โ€–๐‘”๐‘ ๐‘,๐‘ž+๐œ€โ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘ž๎‚.(4.8) By choosing ๐œ€<1/๐ถ we get the lower bound.

As one might guess from Theorem 4.1, the boundedness of the matrix {โŸจ๐œ‚๐œ‡,๐‘†โˆ’1๐œ“๐œ‡โ€ฒโŸฉ}๐œ‡,๐œ‡โ€ฒโˆˆ๐’ฅร—โ„ค2 on ๐‘”๐‘ ๐‘,๐‘ž is the key to showing that {๐‘†โˆ’1๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is also a frame for ๐บ๐‘ ๐‘,๐‘ž.

Proposition 4.2. There exists ๐œ€0>0 such that if {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 satisfies (4.1) for some 0<๐œ€โ‰ค๐œ€0 and furthermore is a frame for ๐บ022=๐ฟ2(โ„2), then {โŸจ๐œ‚๐œ‡,๐‘†โˆ’1๐œ“๐œ‡๎…žโŸฉ}๐œ‡๐œ‡โ€ฒโˆˆ๐’ฅร—โ„ค2 is bounded on ๐‘”๐‘ ๐‘,๐‘ž.

Proof. The fact that {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a frame for ๐ฟ2(โ„2) ensures that ๐‘†โˆ’1 is a bounded operator on ๐ฟ2(โ„2). We first show that ๐‘†โˆ’1 is bounded on ๐บ๐‘ ๐‘,๐‘ž. This will follow from showing that โ€–โ€–(๐ผโˆ’๐‘†)๐‘“๐บ๐‘ ๐‘,๐‘žโ‰ค๐ถ๐œ€โ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘ž,๐‘“โˆˆ๐บ๐‘ ๐‘,๐‘ž,(4.9) choosing ๐œ€ small enough and using the Neumann series. Assume for a moment that ๐’Ÿโˆถ={๐‘‘๐œ‡โ€ฒ๐œ‡}๐œ‡โ€ฒ,๐œ‡โˆˆ๐’ฅร—โ„ค2โˆถ={โŸจ(๐ผโˆ’๐‘†)๐œ‚๐œ‡,๐œ‚๐œ‡โ€ฒโŸฉ}๐œ‡โ€ฒ,๐œ‡โˆˆ๐’ฅร—โ„ค2 satisfies โ€–๐’Ÿ๐‘งโ€–๐‘”๐‘ ๐‘,๐‘žโ‰ค๐ถ๐œ€โ€–๐‘งโ€–๐‘”๐‘ ๐‘,๐‘ž.(4.10) By using that ๐‘† is self-adjoint, we then have โ€–โ€–(๐ผโˆ’๐‘†)๐‘“๐บ๐‘ ๐‘,๐‘žโ€–โ€–โ‰ค๐ถ๎€ฝ๎ซ(๐ผโˆ’๐‘†)๐‘“,๐œ‚๐œ‡โ€ฒโ€–โ€–๎ฌ๎€พ๐‘”๐‘ ๐‘,๐‘žโ€–โ€–๐’Ÿ=๐ถ๎€ฝ๎ซ๐‘“,๐œ‚๐œ‡โ€–โ€–๎ฌ๎€พ๐‘”๐‘ ๐‘,๐‘žโ€–โ€–โ‰ค๐ถ๐œ€{โŸจ๐‘“,๐œ‚๐œ‡โ€–โ€–โŸฉ}๐‘”๐‘ ๐‘,๐‘žโ‰ค๐ถ๐œ€โ€–๐‘“โ€–๐บ๐‘ ๐‘,๐‘ž.(4.11) So to show (4.9) it suffices to prove (4.10). Note that โŸจ(๐ผโˆ’๐‘†)๐œ‚๐œ‡,๐œ‚๐œ‡โ€ฒ๎“โŸฉ=๐œ‡โ€ฒโ€ฒโˆˆ๐’ฅร—โ„ค2โŸจ๐œ‚๐œ‡,๐œ‚๐œ‡โ€ฒโ€ฒโŸฉโŸจ๐œ‚๐œ‡โ€ฒโ€ฒ,๐œ‚๐œ‡โ€ฒ๎“โŸฉโˆ’๐œ‡โ€ฒโˆˆ๐’ฅร—โ„ค2โŸจ๐œ‚๐œ‡,๐œ“๐œ‡โ€ฒโ€ฒโŸฉโŸจ๐œ“๐œ‡โ€ฒโ€ฒ,๐œ‚๐œ‡โ€ฒโŸฉ=๎“๐œ‡โ€ฒโ€ฒโˆˆ๐’ฅร—โ„ค2โŸจ๐œ‚๐œ‡,๐œ‚๐œ‡โ€ฒโ€ฒโŸฉโŸจ๐œ‚๐œ‡โ€ฒโ€ฒโˆ’๐œ“๐œ‡โ€ฒโ€ฒ,๐œ‚๐œ‡โ€ฒ๎“โŸฉ+๐œ‡โ€ฒโ€ฒโˆˆ๐’ฅร—โ„ค2๎ซ๐œ‚๐œ‡,๐œ‚๐œ‡โ€ฒโ€ฒโˆ’๐œ“๐œ‡โ€ฒโ€ฒ๐œ“๎ฌ๎ซ๐œ‡โ€ฒโ€ฒ,๐œ‚๐œ‡โ€ฒ๎ฌ.(4.12) By setting ๐’Ÿ1๎€ฝ๐‘‘โˆถ=1(๐œ‡โ€ฒ)(๐œ‡โ€ฒโ€ฒ)๎€พ๐œ‚โˆถ=๎€ฝ๎ซ๐œ‡โ€ฒโ€ฒโˆ’๐œ“๐œ‡โ€ฒโ€ฒ,๐œ‚๐œ‡โ€ฒ,๐’Ÿ๎ฌ๎€พ2๎€ฝ๐‘‘โˆถ=2(๐œ‡โ€ฒโ€ฒ)(๐œ‡)๎€พ๐œ‚โˆถ=๎€ฝ๎ซ๐œ‡,๐œ‚๐œ‡โ€ฒโ€ฒ,๐’Ÿ๎ฌ๎€พ3๎€ฝ๐‘‘โˆถ=3(๐œ‡โ€ฒ)(๐œ‡โ€ฒโ€ฒ)๎€พ๐œ“โˆถ=๎€ฝ๎ซ๐œ‡โ€ฒโ€ฒ,๐œ‚๐œ‡โ€ฒ,๐’Ÿ๎ฌ๎€พ4๎€ฝ๐‘‘โˆถ=4(๐œ‡โ€ฒโ€ฒ)(๐œ‡)๎€พ๐œ‚โˆถ=๎€ฝ๎ซ๐œ‡,๐œ‚๐œ‡โ€ฒโ€ฒโˆ’๐œ“๐œ‡โ€ฒโ€ฒ,๎ฌ๎€พ(4.13) we have the decomposition ๐’Ÿ=๐’Ÿ1๐’Ÿ2+๐’Ÿ3๐’Ÿ4.(4.14) Since {๐œ€โˆ’1(๐œ‚๐œ‡โˆ’๐œ“๐œ‡)}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a family of curvelet molecules with regularity 4โŒˆ|๐‘ก|+2/๐‘Ÿ+๐›ฟโŒ‰, we have from (3.6) that ๐œ€โˆ’1๐’Ÿ1,๐’Ÿ2,๐’Ÿ3,๐œ€โˆ’1๐’Ÿ4 are almost diagonal on ๐‘”๐‘ ๐‘,๐‘ž. Next, we use Remark 3.3, and by Proposition 3.4, โ€–๐’Ÿ๐‘งโ€–๐‘”๐‘ ๐‘,๐‘žโ‰ค๐ถ๐œ€โ€–๐‘งโ€–๐‘”๐‘ ๐‘,๐‘ž.(4.15) Consequently, (4.9) holds, and for sufficiently small ๐œ€ the operator ๐‘†โˆ’1 is bounded on ๐บ๐‘ ๐‘,๐‘ž. Finally, let ๐‘งโˆถ={๐‘ง๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2โˆˆ๐‘”๐‘ ๐‘,๐‘ž and โˆ‘โ„Ž=โˆถ๐œ‡๐‘ง๐œ‡๐œ‚๐œ‡. By using (2.9) we have that โ„Žโˆˆ๐บ๐‘ ๐‘,๐‘ž, and as {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a frame for ๐ฟ2(โ„2), we have that ๐‘†โˆ’1 is self-adjoint which gives ๎“๐œ‡โˆˆ๐’ฅร—โ„ค2โŸจ๐œ‚๐œ‡,๐‘†โˆ’1๐œ“๐œ‡๎…žโŸฉ๐‘ง๐œ‡=๎“๐œ‡โˆˆ๐’ฅร—โ„ค2โŸจ๐‘†โˆ’1๐œ‚๐œ‡,๐œ“๐œ‡๎…žโŸฉ๐‘ง๐œ‡=โŸจ๐‘†โˆ’1โ„Ž,๐œ“๐œ‡๎…žโŸฉ.(4.16) If we combine this with {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 being a norming family (4.5), we get โ€–โ€–โ€–โ€–๎“๐œ‡โˆˆ๐’ฅร—โ„ค2๎ซ๐œ‚๐œ‡,๐‘†โˆ’1๐œ“๐œ‡โ€ฒ๎ฌ๐‘ง๐œ‡โ€–โ€–โ€–โ€–๐‘”๐‘ ๐‘,๐‘ž=โ€–โ€–โŸจ๐‘†โˆ’1โ„Ž,๐œ“๐œ‡โ€ฒโŸฉโ€–โ€–๐‘”๐‘ ๐‘,๐‘žโ€–โ€–๐‘†โ‰ค๐ถโˆ’1โ„Žโ€–โ€–๐บ๐‘ ๐‘,๐‘žโ‰ค๐ถโ€–โ„Žโ€–๐บ๐‘ ๐‘,๐‘žโ‰ค๐ถโ€–๐‘งโ€–๐‘”๐‘ ๐‘,๐‘ž(4.17) which proves that {โŸจ๐œ‚๐œ‡,๐‘†โˆ’1๐œ“๐œ‡๎…žโŸฉ}๐œ‡,๐œ‡๎…žโˆˆ๐’ฅร—โ„ค2 is bounded on ๐‘”๐‘ ๐‘,๐‘ž.

That {๐‘†โˆ’1๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a frame for ๐บ๐‘ ๐‘,๐‘ž now follows as a consequence of {โŸจ๐œ‚๐œ‡,๐‘†โˆ’1๐œ“๐œ‡๎…žโŸฉ}๐œ‡,๐œ‡๎…žโˆˆ๐’ฅร—โ„ค2 being bounded on ๐‘”๐‘ ๐‘,๐‘ž. 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 {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a family of curvelet molecules with regularity 4โŒˆ|๐‘ก|+2/๐‘Ÿ+๐›ฟโŒ‰ and a frame for ๐ฟ2(โ„2). If {โŸจ๐œ‚๐œ‡,๐‘†โˆ’1๐œ“๐œ‡๎…žโŸฉ}๐œ‡,๐œ‡๎…žโˆˆ๐’ฅร—โ„ค2 is bounded on ๐‘”๐‘ ๐‘,๐‘ž, then for ๐‘“โˆˆ๐บ๐‘ ๐‘,๐‘ž one has ๎“๐‘“=๐œ‡โˆˆ๐’ฅร—โ„ค2โŸจ๐‘“,๐‘†โˆ’1๐œ“๐œ‡โŸฉ๐œ“๐œ‡(4.18) in the sense of ๐’ฎ๎…ž.

Next, we have that {๐‘†โˆ’1๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a frame for ๐บ๐‘ ๐‘,๐‘ž.

Theorem 4.4. Assume that {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a family of curvelet molecules with regularity 4โŒˆ|๐‘ก|+2/๐‘Ÿ+๐›ฟโŒ‰ and a frame for ๐ฟ2(โ„2). Then {๐‘†โˆ’1๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a frame for ๐บ๐‘ ๐‘,๐‘ž if and only if {โŸจ๐œ‚๐œ‡,๐‘†โˆ’1๐œ“๐œ‡๎…žโŸฉ}๐œ‡,๐œ‡๎…žโˆˆ๐’ฅร—โ„ค2 is bounded on ๐‘”๐‘ ๐‘,๐‘ž.

It follows from Proposition 4.2, Lemma 4.3, and Theorem 4.4 that if {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a family of curvelet molecules which is close enough to curvelets, then the representation โˆ‘๐œ‡โˆˆ๐’ฅร—โ„ค2โŸจ๐‘“,๐‘†โˆ’1๐œ“๐œ‡โŸฉ๐œ“๐œ‡,๐‘“โˆˆ๐บ๐‘ ๐‘,๐‘ž, has the same sparse representation properties as curvelets when measured in ๐‘”๐‘ ๐‘,๐‘ž. As a final result we also have a frame expansion with {๐‘†โˆ’1๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2.

Lemma 4.5. Assume that {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 is a family of curvelet molecules with regularity 4โŒˆ|๐‘ก|+2/๐‘Ÿ+๐›ฟโŒ‰ and a frame for ๐ฟ2(โ„2). If the transpose of {โŸจ๐œ‚๐œ‡,๐‘†โˆ’1๐œ“๐œ‡๎…žโŸฉ}๐œ‡,๐œ‡๎…žโˆˆ๐’ฅร—โ„ค2 is bounded on ๐‘”๐‘ ๐‘,๐‘ž, then for ๐‘“โˆˆ๐บ๐‘ ๐‘,๐‘ž one has ๎“๐‘“=๐œ‡โˆˆ๐’ฅร—โ„ค2โŸจ๐‘“,๐œ“๐œ‡โŸฉ๐‘†โˆ’1๐œ“๐œ‡(4.19) in the sense of ๐’ฎ๎…ž.

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 ๐‘” with sufficient smoothness and decay can be used to construct a system {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 that satisfies (4.1). From the previous section, we then have that the representation โˆ‘๐œ‡โˆˆ๐’ฅร—โ„ค2โŸจ๐‘“,๐‘†โˆ’1๐œ“๐œ‡โŸฉ๐œ“๐œ‡, ๐‘“โˆˆ๐บ๐‘ ๐‘,๐‘ž, has the same sparse representation properties as curvelets when measured in ๐‘”๐‘ ๐‘,๐‘ž.

First we take ๐‘”โˆˆ๐ถ๐‘€+1(โ„2), ฬ‚๐‘”(0)โ‰ 0, which for fixed ๐‘๎…ž>2,๐‘€>0 satisfies||๐‘”(๐œ…)||(๐‘ฅ)โ‰ค๐ถ(1+|๐‘ฅ|)โˆ’๐‘โ€ฒ,|๐œ…|โ‰ค๐‘€+1.(5.1) Next, for ๐‘šโ‰ฅ1 we define ๐‘”๐‘š(๐‘ฅ)โˆถ=๐ถ๐‘”๐‘š2๐‘”(๐‘š๐‘ฅ), where ๐ถ๐‘”=โˆถฬ‚๐‘”(0)โˆ’1. It then follows that||g๐‘š(๐œ…)||(๐‘ฅ)โ‰ค๐ถ๐‘š2+|๐œ…|(1+๐‘š|๐‘ฅ|)โˆ’๐‘โ€ฒ,๎€œ|๐œ…|โ‰ค๐‘€+1,โ„2๐‘”๐‘š(๐‘ฅ)d๐‘ฅ=1.(5.2) We recall that curvelets (2.6) are a family of curvelet molecules for any regularity ๐‘…โˆˆโ„•. From the definition of a family of curvelet molecules (Definition 2.1), we have that for ๐‘—โ‰ฅ2 curvelet molecules can be expressed as ๐œ‚๐œ‡(๐‘ฅ)=23๐‘—/4๐‘Ž๐œ‡๎‚ต๐ท2โˆ’๐‘—๐‘…๐œƒ๐œ‡๎‚ต๐‘˜๐‘ฅโˆ’1๐›ฟ1,๐‘˜2๐›ฟ2๎‚ถ๎‚ถ,(5.3) where ๐‘Ž๐œ‡ must satisfy (2.8) and (2.9). So to construct a family of curvelet molecules {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 which satisfy (4.1), we need to construct a family of functions {๐‘๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 such that ๐‘Ž๐œ‡โˆ’๐‘๐œ‡ satisfy (4.2) and (4.3). We define {๐œ“๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 as ๐œ“๐œ‡(๐‘ฅ)โˆถ=23๐‘—/4๐‘๐œ‡๎‚ต๐ท2โˆ’๐‘—๐‘…๐œƒ๐œ‡๎‚ต๐‘˜๐‘ฅโˆ’1๐›ฟ1,๐‘˜2๐›ฟ2๎‚ถ๎‚ถ(5.4) for ๐‘—โ‰ฅ2 and to construct {๐‘๐œ‡}๐œ‡โˆˆ๐’ฅร—โ„ค2 we also need the following set of finite linear combinations:ฮ˜๐พ,๐‘š๎ƒฏ๐‘โˆถ=๐œ‡โˆถ๐‘๐œ‡(โ‹…)=๐พ๎“๐‘–=1๐‘๐‘–๐‘”๐‘š๎€ทโ‹…+๐‘‘๐‘–๎€ธ,๐‘๐‘–โˆˆโ„‚,๐‘‘๐‘–โˆˆโ„2๎ƒฐ.(5.5) We have omitted the construction of ๐œ“๐œ‡ for ๐‘—=1 as it follow in a similar way.

Proposition 5.1. Let ๐‘โ€ฒ>๐‘>2,๐‘€>0 and ๐‘—>0. If ๐‘”โˆˆ๐ถ๐‘€+1(โ„2),ฬ‚๐‘”(0)โ‰ 0, fulfills (5.1) and ๐‘Ž๐œ‡โˆˆ๐ฟ2(โ„2)โˆฉ๐ถ๐‘€+1(โ„2) fulfills ||๐‘Ž๐œ‡(๐œ…)||(๐‘ฅ)โ‰ค๐ถ(1+|๐‘ฅ|)โˆ’๐‘โ€ฒ,|||๐œ…|โ‰ค๐‘€+1,ฬ‚๐‘Ž๐œ‡||๎€ท(๐œ‰)โ‰ค๐ถmin1,2โˆ’๐‘—+||๐œ‰1||+2โˆ’๐‘—/2||๐œ‰2||๎€ธ๐‘€+1,(5.6) then for any ๐œ€>0 there exist ๐พ,๐‘šโ‰ฅ1 (๐‘š independent of ๐‘—) and ๐‘๐œ‡โˆˆฮ˜๐พ,๐‘š such that ||๐‘Ž๐œ‡(๐œ…)(๐‘ฅ)โˆ’๐‘๐œ‡(๐œ…)||(๐‘ฅ)โ‰ค๐œ€(1+|๐‘ฅ|)โˆ’๐‘||,|๐œ…|โ‰ค๐‘€,ฬ‚๐‘Ž๐œ‡ฬ‚๐‘(๐œ‰)โˆ’๐œ‡||๎€ท(๐œ‰)โ‰ค๐œ€min1,2โˆ’๐‘—+||๐œ‰1||+2โˆ’๐‘—/2||๐œ‰2||๎€ธ๐‘€.(5.7)

Proof. Let ๐œ€>0 and ๐œ…,|๐œ…|โ‰ค๐‘€, be given. We construct the approximation of ๐‘Ž๐œ‡ in direct space in three steps. First by a convolution operator ๐œ”๐‘š=๐‘Ž๐œ‡โˆ—๐‘”๐‘š, then by ๐œƒ๐‘ž,๐‘š which is the integral in ๐œ”๐‘š taken over a dyadic cube ๐‘„, and finally by a discretization over smaller dyadic cubes ๐‘๐‘™,๐‘ž,๐‘š. From (5.2) we have ๐‘Ž๐œ‡(๐œ…)(๐‘ฅ)โˆ’๐œ”๐‘š(๐œ…)(๎€œ๐‘ฅ)=โ„2๎‚€๐‘Ž๐œ‡(๐œ…)(๐‘ฅ)โˆ’๐‘Ž๐œ‡(๐œ…)(๎‚๐‘”๐‘ฅโˆ’๐‘ฆ)๐‘š(๐‘ฆ)d๐‘ฆ.(5.8) Define ๐‘ˆโˆถ=๐‘š๐œ†/2๐‘, where ๐œ†โˆถ=min(1,๐‘โ€ฒโˆ’๐‘). For |๐‘ฅ|โ‰ค๐‘ˆ, we use the mean value theorem to get ||๐‘Ž๐œ‡(๐œ…)(๐‘ฅ)โˆ’๐‘Ž๐œ‡(๐œ…)||๎€ท||๐‘ฆ||๎€ธ.(๐‘ฅโˆ’๐‘ฆ)โ‰ค๐ถmin1,(5.9) Inserting this in (5.8) we have ||๐‘Ž๐œ‡(๐œ…)(๐‘ฅ)โˆ’๐œ”๐‘š(๐œ…)||๎€œ(๐‘ฅ)โ‰ค๐ถโ„2๎€ท||๐‘ฆ||๎€ธ๐‘šmin1,2๎€ท||๐‘ฆ||๎€ธ1+๐‘š๐‘โ€ฒd๐‘ฆโ‰ค๐ถ๐‘šโˆ’๐œ†โ‰ค๐ถ๐‘šโˆ’๐œ†/2๐‘ˆ๐‘โ‰ค๐ถ๐‘šโˆ’๐œ†/2(1+|๐‘ฅ|)๐‘.(5.10) For |๐‘ฅ|>๐‘ˆ, we split the integral over ฮฉโˆถ={๐‘ฆโˆถ|๐‘ฆ|โ‰ค|๐‘ฅ|/2} and ฮฉ๐‘. If ๐‘ฆโˆˆฮฉ, then |๐‘ฅโˆ’๐‘ฆ|โ‰ฅ|๐‘ฅ|/2, and we have ๎€œฮฉ||๐‘Ž๐œ‡(๐œ…)(๐‘ฅ)โˆ’๐‘Ž๐œ‡(๐œ…)(||||๐‘”๐‘ฅโˆ’๐‘ฆ)๐‘š(||๐‘ฆ)d๐‘ฆโ‰ค๐ถ(1+|๐‘ฅ|)โˆ’๐‘โ€ฒโ‰ค๐ถ(1+๐‘ˆ)๐œ†(1+|๐‘ฅ|)๐‘โ‰ค๐ถ๐‘šโˆ’๐œ†2/2๐‘(1+|๐‘ฅ|)๐‘.(5.11) Integrating over ฮฉ๐‘ with |๐‘ฅ|>๐‘ˆ gives ๎€œฮฉ๐‘||๐‘Ž๐œ‡(๐œ…)(๐‘ฅ)โˆ’๐‘Ž๐œ‡(๐œ…)(||||๐‘”๐‘ฅโˆ’๐‘ฆ)๐‘š(||โ‰ค๐ถ๐‘ฆ)d๐‘ฆ(1+|๐‘ฅ|)๐‘โ€ฒ+๎€œฮฉ๐‘๐ถ๐‘š2๎€ท||||๎€ธ1+๐‘ฅโˆ’๐‘ฆ๐‘โ€ฒ๎€ท||๐‘ฆ||๎€ธ1+๐‘š๐‘โ€ฒโ‰ค๐ถd๐‘ฆ(1+|๐‘ฅ|)๐‘โ€ฒ+๐ถ๐‘šโˆ’๐œ†(1+|๐‘ฅ|)๐‘โ‰ค๐ถ๎‚€๐‘šโˆ’๐œ†2/2๐‘+๐‘šโˆ’๐œ†๎‚(1+|๐‘ฅ|)๐‘.(5.12) So by choosing ๐‘š sufficiently large in (5.10)โ€“(5.12), we get ||๐‘Ž๐œ‡(๐œ…)(๐‘ฅ)โˆ’๐œ”๐‘š(๐œ…)||โ‰ค๐œ€(๐‘ฅ)3(1+|๐‘ฅ|)โˆ’๐‘.(5.13) For the next step we fix ๐‘š and choose ๐‘žโˆˆโ„•. Let ๐‘„ denote the dyadic cube with sidelength 2๐‘ž+1, sides parallel with the axes and centered at the origin. We then approximate ๐œ”๐‘š with ๐œƒ๐‘ž,๐‘š defined as ๐œƒ๐‘ž,๐‘š(๎€œโ‹…)=๐‘„๐‘Ž๐œ‡(๐‘ฆ)๐‘”๐‘š(โ‹…โˆ’๐‘ฆ)d๐‘ฆ.(5.14) In which case we have ๐œ”๐‘š(๐œ…)(๐‘ฅ)โˆ’๐œƒ(๐œ…)๐‘ž,๐‘š(๎€œ๐‘ฅ)=๐‘„๐‘๐‘Ž๐œ‡(๐‘ฆ)๐‘”๐‘š(๐œ…)(๐‘ฅโˆ’๐‘ฆ)d๐‘ฆ,(5.15) and it follows that ||๐œ”๐‘š(๐œ…)(๐‘ฅ)โˆ’๐œƒ(๐œ…)๐‘ž,๐‘š||โ‰ค๎€œ(๐‘ฅ)|๐‘ฆ|โ‰ฅ2๐‘ž๐ถ๐‘š2+|๐œ…|๎€ท||๐‘ฆ||๎€ธ1+๐‘โ€ฒ๎€ท||||1+๐‘š๐‘ฅโˆ’๐‘ฆ๐ต๎€ธ๐‘โ€ฒd๐‘ฆโˆถ=๐ฟ.(5.16) We first estimate the integral for |๐‘ฅ|โ‰ค2๐‘žโˆ’1 which gives |๐‘ฆ|>|๐‘ฅ| and |๐‘ฅโˆ’๐‘ฆ|โ‰ฅ2๐‘žโˆ’1. Hence we obtain ๐ฟโ‰ค๐ถ๐‘š2+|๐œ…|(1+|๐‘ฅ|)๐‘โ€ฒ๎€œ|๐‘ข|โ‰ฅ2๐‘žโˆ’11(1+๐‘š|๐‘ข|)๐‘๎…žd๐‘ขโ‰ค๐ถ๐‘š|๐œ…|โˆ’๐œ†2โˆ’๐œ†๐‘ž(1+|๐‘ฅ|)๐‘โ€ฒ.(5.17) For |๐‘ฅ|>2๐‘žโˆ’1, we split the integral over ฮฉโˆถ={๐‘ฆโˆถ|๐‘ฆ|โ‰ฅ2๐‘ž}โˆฉ{๐‘ฆโˆถ|๐‘ฆ|โ‰ค|๐‘ฅ|/2} and ฮฉ๎…žโˆถ={๐‘ฆโˆถ|๐‘ฆ|โ‰ฅ2๐‘ž}โˆฉ{๐‘ฆโˆถ|๐‘ฆ|>|๐‘ฅ|/2}. If ๐‘ฆโˆˆฮฉ, then |๐‘ฅโˆ’๐‘ฆ|โ‰ฅ|๐‘ฅ|/2, and we get ๎€œฮฉ๐‘š2+|๐œ…|๎€ท||๐‘ฆ||๎€ธ1+๐‘โ€ฒ๎€ท||||๎€ธ1+๐‘š๐‘ฅโˆ’๐‘ฆ๐‘โ€ฒd๐‘ฆโ‰ค๐ถ๐‘š2+|๐œ…|(1+๐‘š|๐‘ฅ|)๐‘โ€ฒ๎€œ|๐‘ฆ|โ‰ฅ2๐‘ž1๎€ท||๐‘ฆ||๎€ธ1+๐‘โ€ฒโ‰คd๐‘ฆ๐ถ๐‘š|๐œ…|โˆ’๐œ†2โˆ’๐œ†๐‘ž(1+|๐‘ฅ|)๐‘.(5.18) Similar for ฮฉโ€ฒ we have ๎€œฮฉโ€ฒ๐‘š2+|๐œ…|๎€ท||๐‘ฆ||๎€ธ1+๐‘๎…ž๎€ท||||๎€ธ1+๐‘š๐‘ฅโˆ’๐‘ฆ๐‘โ€ฒ๐ถd๐‘ฆโ‰ค(1+|๐‘ฅ|)๐‘โ€ฒ๎€œโ„2๐‘š2+|๐œ…|๎€ท||||๎€ธ1+๐‘š๐‘ฅโˆ’๐‘ฆ๐‘โ€ฒโ‰คd๐‘ฆ๐ถ๐‘š|๐œ…|(1+|๐‘ฅ|)๐‘โ€ฒโ‰ค๐‘š|๐œ…|2โˆ’๐œ†๐‘ž(1+|๐‘ฅ|)๐‘.(5.19) By choosing ๐‘ž sufficiently large in (5.17)โ€“(5.19), we obtain ||๐œ”๐‘š(๐œ…)(๐‘ฅ)โˆ’๐œƒ(๐œ…)๐‘ž,๐‘š||โ‰ค๐œ€(๐‘ฅ)3(1+|๐‘ฅ|)โˆ’๐‘.(5.20) For the final step we fix ๐‘ž, choose ๐‘™โˆˆโ„•, and approximate ๐œƒ๐‘ž,๐‘š by a discretization ๐‘๐‘™,๐‘ž,๐‘š(๎“โ‹…)=๐ผโˆˆ๐ป๐‘™,๐‘ž||๐ผ||๐‘Ž๐œ‡๎€ท๐‘ฅ๐ผ๎€ธ๐‘”๐‘š๎€ทโ‹…โˆ’๐‘ฅ๐ผ๎€ธ,(5.21) where ๐‘ฅ๐ผ is the center of the dyadic cube ๐ผ and ๐ป๐‘™,๐‘ž is the set of dyadic cubes with sidelength 2โˆ’๐‘™ which together give ๐‘„. Note that ๐‘๐‘™,๐‘ž,๐‘šโˆˆฮ˜๐พ,๐‘š, ๐พ=2๐‘ž+๐‘™+1. We introduce ๐น(โ‹…)โˆถ=๐‘Ž๐œ‡(โ‹…)๐‘”๐‘š(๐œ…)(๐‘ฅโˆ’โ‹…) which gives |||๐œƒ(๐œ…)๐‘ž,๐‘š(๐‘ฅ)โˆ’๐‘(๐œ…)๐‘™,๐‘ž,๐‘š(|||โ‰ค๎“๐‘ฅ)๐ผโˆˆ๐ป๐‘™,๐‘ž๎€œ๐ผ||๐‘Ž๐œ‡(๐‘ฆ)๐‘”๐‘š(๐œ…)(๐‘ฅโˆ’๐‘ฆ)โˆ’๐‘Ž๐œ‡๎€ท๐‘ฅ๐ผ๎€ธ๐‘”๐‘š(๐œ…)๎€ท๐‘ฅโˆ’๐‘ฅ๐ผ๎€ธ||โ‰ค๎“d๐‘ฆ๐ผโˆˆ๐ป๐‘™,๐‘ž๎€œ๐ผ||๎€ท๐‘ฅ๐น(๐‘ฆ)โˆ’๐น๐ผ๎€ธ||d๐‘ฆ.(5.22) By using the mean value theorem, we then get |||๐œƒ(๐œ…)๐‘ž,๐‘š(๐‘ฅ)โˆ’๐‘(๐œ…)๐‘™,๐‘ž,๐‘š(|||โ‰ค๎“๐‘ฅ)๐ผโˆˆ๐ป๐‘™,๐‘ž๎€œ๐ผ||๐‘ฆโˆ’๐‘ฅ๐ผ||max๐ผ|||๐œ…โ€ฒ|||๐‘งโˆˆ๐‘™(๐‘ฅ,๐‘ฆ)โ‰ค1|||๐น(๐œ…โ€ฒ)(|||๐‘ง)d๐‘ฆโ‰ค๐ถ22๐‘žโˆ’๐‘™maxโ€ฒ|๐‘ง|โ‰ค2๐‘ž+1|๐œ…|โ‰ค|๐œ…|+1|||๐‘”(๐œ…โ€ฒ)๐‘š(|||,๐‘ฅโˆ’๐‘ง)(5.23) where ๐‘™(๐‘ฅ๐ผ,๐‘ฆ) is the line segment between ๐‘ฅ๐ผ and ๐‘ฆ. If |๐‘ฅ|โ‰ค2๐‘ž+2 and |๐œ…๎…ž|โ‰ค|๐œ…|+1, then we have |||๐‘”(๐œ…โ€ฒ)๐‘š|||(๐‘ฅโˆ’๐‘ง)โ‰ค๐ถ๐‘š3+|๐œ…|โ‰ค๐ถ๐‘š3+|๐œ…|2๐‘ž๐‘(1+|๐‘ฅ|)๐‘.(5.24) For |๐‘ฅ|>2๐‘ž+2 and |๐‘ง|โ‰ค2๐‘ž+1, we have |๐‘ฅโˆ’๐‘ง|โ‰ฅ|๐‘ฅ|/2, and hence for |๐œ…โ€ฒ|โ‰ค|๐œ…|+1, it follows that |||๐‘”(๐œ…โ€ฒ)๐‘š|||โ‰ค(๐‘ฅโˆ’๐‘ง)๐ถ๐‘š3+|๐œ…|(1+๐‘š|๐‘ฅ|)๐‘โ€ฒโ‰ค๐ถ๐‘š3+|๐œ…|(1+|๐‘ฅ|)๐‘โ€ฒ.(5.25) By choosing ๐‘™ sufficiently large, we obtain by combining (5.23)โ€“(5.25) that |||๐œƒ(๐œ…)๐‘ž,๐‘š(๐‘ฅ)โˆ’๐‘(๐œ…)๐‘™,๐‘ž,๐‘š|||โ‰ค๐œ€(๐‘ฅ)3(1+|๐‘ฅ|)โˆ’๐‘.(5.26) Finally by combining (5.13), (5.20) and (5.26), we get |||๐‘Ž๐œ‡(๐œ…)(๐‘ฅ)โˆ’๐‘(๐œ…)๐‘™,๐‘ž,๐‘š|||(๐‘ฅ)โ‰ค๐œ€(1+|๐‘ฅ|)โˆ’๐‘.(5.27) To approximate ๐‘Ž๐œ‡ in frequency space we use three steps similar to the approximation in direct space. Note that ๐‘๐‘™,๐‘ž,๐‘š still fulfills (5.27) if we choose ๐‘™,๐‘ž,๐‘š even larger. First we use ๎๐œ”๐‘š to approximate ฬ‚๐‘Ž๐œ‡ in which case we have ||ฬ‚๐‘Ž๐œ‡(๐œ‰)โˆ’๎๐œ”๐‘š||=||||(๐œ‰)ฬ‚๐‘Ž๐œ‡(๐œ‰)๐‘€/(1+๐‘€)ฬ‚๐‘Ž๐œ‡(๐œ‰)1/(1+๐‘€)๎‚ต1โˆ’๐ถ๐‘”๎‚ต๐œ‰ฬ‚๐‘”๐‘š||||๎€ท๎‚ถ๎‚ถโ‰ค๐ถmin1,2โˆ’๐‘—+||๐œ‰1||+2โˆ’๐‘—/2||๐œ‰2||๎€ธ๐‘€๎€ท||๐œ‰||๎€ธ1+โˆ’1||||1โˆ’๐ถ๐‘”๎‚ต๐œ‰ฬ‚๐‘”๐‘š๎‚ถ||||.(5.28) By choosing ๐œ‰๐‘”>0 such that ๐ถ(1+๐œ‰๐‘”)โˆ’1|1โˆ’๐ถ๐‘”ฬ‚๐‘”(๐œ‰/๐‘š)|โ‰ค๐œ€/3 and ๐‘š such that ๐ถ|1โˆ’๐ถ๐‘”ฬ‚๐‘”(๐œ‰/๐‘š)|โ‰ค๐œ€/3 for |๐œ‰|<๐œ‰๐‘”, we get ||ฬ‚๐‘Ž๐œ‡(๐œ‰)โˆ’๎๐œ”๐‘š||โ‰ค๐œ€(๐œ‰)3๎€ทmin1,2โˆ’๐‘—+||๐œ‰1||+2โˆ’๐‘—/2||๐œ‰2||๎€ธ๐‘€.(5.29) Next, we fix ๐‘š, choose ๐‘ž and limit the Fourier integral of ๐‘Ž๐œ‡ to ๐‘„ from the approximation in direct space, ๐œƒ๎…ž๐‘ž,๐‘š(๐œ‰)=ฬ‚๐‘”๐‘š(๎€œ๐œ‰)๐‘„๐‘Ž๐œ‡(๐‘ฅ)๐‘’๐‘–๐‘ฅโ‹…๐œ‰d๐‘ฅ.(5.30) This gives |||๎๐œ”๐‘š(๐œ‰)โˆ’๐œƒโ€ฒ๐‘ž,๐‘š(|||โ‰ค||๐œ‰)ฬ‚๐‘”๐‘š(||๎€œ๐œ‰)|๐‘ฅ|>2๐‘ž||๐‘Ž๐œ‡(๐‘ฅ)๐‘’๐‘–๐‘ฅโ‹…๐œ‰||d๐‘ฅโ‰ค๐ถ2โˆ’๐œ†๐‘ž.(5.31) In the last step, we fix ๐‘ž and approximate ๐œƒโ€ฒ๐‘ž,๐‘š by ฬ‚๐‘๐‘™,๐‘ž,๐‘š. We introduce ๐บ(๐‘ฅ)โˆถ=๐‘Ž๐œ‡(๐‘ฅ)๐‘’๐‘–๐‘ฅโ‹…๐œ‰ which gives ||๐œƒ๎…ž๐‘ž,๐‘šฬ‚๐‘(๐œ‰)โˆ’๐‘™,๐‘ž,๐‘š||โ‰ค||(๐œ‰)ฬ‚๐‘”๐‘š|||||||๎€œ(๐œ‰)๐‘„๐‘Ž๐œ‡(๐‘ฅ)๐‘’๐‘–๐‘ฅโ‹…๐œ‰๎“d๐‘ฅโˆ’๐ผโˆˆ๐ป๐‘™,๐‘ž|||||๐ผ||๐‘Ž๐œ‡๎€ท๐‘ฅ๐ผ๎€ธ๐‘’๐‘–๐‘ฅ๐ผโ‹…๐œ‰||โ‰ค||ฬ‚๐‘”๐‘š(||๎“๐œ‰)๐ผโˆˆ๐ป๐‘™,๐‘ž๎€œ๐ผ||๎€ท๐‘ฅ๐บ(๐‘ฅ)โˆ’๐บ๐ผ๎€ธ||โ‰คd๐‘ฅ๐ถ22๐‘žโˆ’๐‘™||||1+๐œ‰/๐‘šmax2|||๐œ…โ€ฒ|||๐‘ฅโˆˆโ„โ‰ค1|||๐บ(๐œ…โ€ฒ)|||(๐‘ฅ)โ‰ค๐ถ๐‘š22๐‘žโˆ’๐‘™.(5.32) By combining (5.29)โ€“(5.32) for sufficiently large l,๐‘ž,๐‘š, we get ||ฬ‚๐‘Ž๐œ‡ฬ‚๐‘(๐œ‰)โˆ’๐‘™,๐‘ž,๐‘š||๎€ท(๐œ‰)โ‰ค๐œ€min1,2โˆ’๐‘—+||๐œ‰1||+2โˆ’๐‘—/2||๐œ‰2||๎€ธ๐‘€.(5.33) It follows that by choosing ๐‘™,๐‘ž,๐‘š large enough ๐‘๐‘™,๐‘ž,๐‘š fulfills both (5.7) and (5.4). Furthermore, we have ๐‘๐‘™,๐‘ž,๐‘šโˆˆฮ˜๐พ,๐‘š, ๐พ=2๐‘ž+๐‘™+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 ๐‘”๐‘ ๐‘,๐‘ž. 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 ๐‘’โˆ’|โ‹…|2 and the rational functions (1+|โ‹…|2)โˆ’๐‘ with ๐‘ 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.

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Copyright © 2012 Kenneth N. Rasmussen and Morten Nielsen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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