Multi-window dilation-and-modulation frames on the half real line
aa r X i v : . [ m a t h . F A ] A ug Multi-window dilation-and-modulation frames on the half real line ∗ Yun-Zhang Li Wei Zhang College of Applied Sciences,Beijing University of Technology, Beijing 100124, P. R. ChinaE-mail: 1. [email protected]. [email protected]
Abstract
Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulationoperators, respectively. They have been extensively studied. However, dilation-and-modulation systemshave not, and they cannot be derived from wavelet or Gabor systems. In this paper, we investigate aclass of dilation-and-modulation systems in the causal signal space L ( R + ). L ( R + ) can be identified asubspace of L ( R ) consisting of all L ( R )-functions supported on R + , and is unclosed under the Fouriertransform. So the Fourier transform method does not work in L ( R + ). In this paper, we introduce thenotion of Θ a -transform in L ( R + ), using Θ a -transform we characterize dilation-and-modulation framesand dual frames in L ( R + ); and present an explicit expression of all duals with the same structure fora general dilation-and-modulation frame for L ( R + ). Interestingly, we prove that an arbitrary frame ofthis form is always nonredundant whenever the number of the generators is 1, and is always redundantwhenever it is greater than 1. Some examples are also provided to illustrate the generality of our results. Key Words : frame; wavelet frame; Gabor frame; dilation-and-modulation frame; multi-window dilation-and-modulation frame : 42C40, 42C15
It is well known that translation, modulation and dilation are fundamental operations in wavelet analysis.The translation operator T x , the modulation operator M x with x ∈ R and the dilation operator D c with0 < c = 1 are defined by T x f ( · ) = f ( · − x ) , M x f ( · ) = e πix · f ( · ) and D c f ( · ) = √ cf ( c · )for f ∈ L ( R ), respectively. Given a finite subset Ψ of L ( R ), Gabor frames of the form { M mb T na ψ : m, n ∈ Z , ψ ∈ Ψ } (1.1)and wavelet frames of the form { D a j T bk ψ : j, k ∈ Z , ψ ∈ Ψ } (1.2) ∗ Supported by the National Natural Science Foundation of China (Grant No. 11271037). Y.-Z. Li & W. Zhang with a , b > { M mb D a j ψ : m, j ∈ Z , ψ ∈ Ψ } with a, b > { T mb D a j ˆ ψ : m, j ∈ Z , ψ ∈ Ψ } (1.4)It does not fall into the framework of the above wavelet and Gabor systems. Our focus in this paper will beon a class of dilation-and-modulation frames for L ( R + ) with R + = (0 , ∞ ). L ( R + ) can be considered asa closed subspace of L ( R ) consisting of all functions in L ( R ) which vanish outside R + . And it can modelcausal signal space. In practice, time variable cannot be negative.For subspace Gabor and wavelet frames of the forms (1.1) and (1.2) respectively, we refer to [2, 6-8,15-19, 23, 24, 31, 32, 36, 38-40, 44, 48, 50, 51] and references therein for details. It is easy to check thatthere exists no nonzero function ψ such that T nc ψ ( · ) = 0 on ( −∞ , c > n ∈ Z . This implies that L ( R + ) admits no frame of the form (1.1), (1.2) or(1.4). So it is natural to ask how we construct frames for L ( R + ) with good structures. Two methods areknown to us for this purpose. One is to construct frames for L ( R + ) consisting of a subsystem of (1.2) andsome inhomogeneous refinable function-based “boundary wavelets”. For details, we refer to [3, 5, 29, 30,35, 41, 46, 47] and references therein. The other is to use the Cantor group operation and Walsh seriestheory to introduce the notion of (frame) multiresolution analysis in L ( R + ), and then derive wavelet framessimilarly to the case of L ( R ). For details, we refer to [1, 10-12, 33, 34, 42, 43, 45] and references therein.The references [20] and [21] also have something to do with this problem. In [20], numerical experimentswere made to establish that the nonnegative integer shifts of the Gaussian function form a Riesz sequencein L ( R + ). And in [21], a sufficient condition was obtained to determine whether or not the nonnegativetranslates of a given function form a Riesz sequence on L ( R + ).Given a >
1, a measurable function h defined on R + is said to be a - dilation periodic if h ( a · ) = h ( · ) a.e.on R + . Throughout this paper, we denote by { Λ m } m ∈ Z the sequence of a -dilation periodic functions definedby Λ m ( · ) = 1 √ a − e πim · a − on [1 , a ) for each m ∈ Z . (1.5)Motivated by the above works, we in this paper investigate the dilation-and-modulation systems in L ( R + )of the form: MD (Ψ , a ) = { Λ m D a j ψ l : m, j ∈ Z , ≤ l ≤ L } (1.6) ilation-and-modulation frames General setup: (i) a is a fixed positive number greater than 1.(ii) Ψ = { ψ , ψ , · · · , ψ L } is a finite subset of L ( R + ) with cardinality L .For Φ = { ϕ , ϕ , · · · , ϕ L } , we define MD (Φ , a ) similarly to (1.6). The system MD (Ψ , a ) is slightlylike but differs from (1.3). The modulation factor e πimb · in (1.3) is Z -periodic according to addition, whileΛ m in (1.6) is a -dilation periodic. Let a and Ψ be as in the general setup. The system MD (Ψ , a ) is calleda frame for L ( R + ) if there exist 0 < C ≤ C < ∞ such that C k f k L ( R + ) ≤ L X l =1 X m,j ∈ Z (cid:12)(cid:12) h f, Λ m D a j ψ l i L ( R + ) (cid:12)(cid:12) ≤ C k f k L ( R + ) for f ∈ L ( R + ) , (1.7)where C , C are called frame bounds ; it is called a Bessel sequence in L ( R + ) if the right-hand side inequalityin (1.7) holds, where C is called a Bessel bound . In particular, it is called a
Parseval frame if C = C = 1in (1.7). Given a frame MD (Ψ , a ) for L ( R + ), a sequence MD (Φ , a ) is called a dual (or an MD - dual ) of MD (Ψ , a ) if it is a frame such that f = L X l =1 X m,j ∈ Z h f, Λ m D a j ϕ l i L ( R + ) Λ m D a j ψ l for f ∈ L ( R + ) . (1.8)It is easy to check that MD (Ψ , a ) is also a dual of MD (Φ , a ) if MD (Φ , a ) is a dual of MD (Ψ , a ). So, inthis case, we say MD (Ψ , a ) and MD (Φ , a ) form a pair of dual frames for L ( R + ). By the knowledge offrame theory, MD (Ψ , a ) and MD (Φ , a ) form a pair of dual frames for L ( R + ) if they are Bessel sequencesand satisfy (1.8). The fundamentals of frames can be found in [4, 9, 26, 49]. Observe that L ( R + ) is theFourier transform of the Hardy space H ( R ) which is a reducing subspace of L ( R ) defined by H ( R ) = { f ∈ L ( R ) : ˆ f ( · ) = 0 a.e. on( −∞ , } . Wavelet frames in H ( R ) of the form (1.2) were studied in [28, 44, 48]. By the Plancherel theorem, an H ( R )-frame { D a j T bk ψ : j, k ∈ Z , ψ ∈ Ψ } leads to a L ( R + )-frame { e − πia j k · ˆ ψ ( a j · ) : j, k ∈ Z , ψ ∈ Ψ } . (1.9)In (1.9), e − πia j k · is a − j Z -periodic with respect to addition, and the period varies with j . However, Λ m in(1.6) is a -dilation periodic, and unrelated to j . Therefore, the system (1.6) differs from (1.9) for L ( R + ),and is of independent interest. Actually, it is slightly related to a kind of function-valued frames in [25].This paper focuses on the theory of L ( R + )-frames of the form (1.6). It cannot be derived from the wellknown wavelet and Gabor systems, and its operation is more intuitive when compared with the Cantor group Y.-Z. Li & W. Zhang and Walsh series-based systems in [1, 10-12, 33, 34, 42, 43, 45]. Also L ( R + ) is unclosed under the Fouriertransform. In particular, the Fourier transform of a compactly supported nonzero function in L ( R + ) liesoutside this space. Therefore, the Fourier transform cannot be used in our setting, and we need to find anew method.The rest of this paper is organized follows. In Section 2, we introduce the notion of Θ a -transform, and givea Θ a -transform domain characterization of a dilation-and-modulation system MD (Ψ , a ) being complete, aBessel sequence and a frame in L ( R + ), respectively. In Section 3, using Θ a -transform we characterize dualframe pairs of the form ( MD (Ψ , a ) , MD (Φ , a )), and obtain an explicit expression of all MD -duals of ageneral frame MD (Ψ , a ) for L ( R + ). We also prove that an arbitrary frame MD (Ψ , a ) is a Riesz basis ifand only if L = 1. It means that the frame MD (Ψ , a ) is always nonredundant whenever L = 1, and it isalways redundant whenever L >
1. In Section 4, we give some examples of MD -dual frame pairs for L ( R + )to illustrate the generality of our results. They show that the achieved results in this paper provide us withan easy method to construct MD -dual frames for L ( R + ) with the window functions having good propertiessuch as having bounded supports and certain smoothness. Θ a -transform domain frame characterization Let a and Ψ be as in the general setup. In this section, by introducing Θ a -transform we give the conditionsof completeness, Bessel sequence and frame of MD (Ψ , a ) in L ( R + ), respectively. Definition 2.1.
Let a be as in the general setup. For f ∈ L ( R + ) , we define Θ a f ( x, ξ ) = X l ∈ Z a l f ( a l x ) e − πilξ (2.1) for a.e. ( x, ξ ) ∈ R + × R . Remark 2.1.
Observe that, given f ∈ L ( R + ) , Z a j +1 a j X l ∈ Z a l | f ( a l x ) | dx = k f k L ( R + ) < ∞ for j ∈ Z . This implies that P l ∈ Z a l | f ( a l · ) | < ∞ a.e. on R + by the arbitrariness of j . Therefore, (2.1) is well-defined. Lemma 2.1.
Let a be as in the general setup. For m , j ∈ Z , define Λ m as in (1.5), and e m,j by e m,j ( x, ξ ) = Λ m ( x ) e πijξ for ( x, ξ ) ∈ R + × R . Then ( i ) { Λ m : m ∈ Z } and { e m,j : m, j ∈ Z } are orthonormal bases for L ([1 , a )) and L ([1 , a ) × [0 , ,respectively; ilation-and-modulation frames ii ) Given f ∈ L ( R + ) , we have Θ a f ( a j x, ξ + m ) = e πijξ a − j Θ a f ( x, ξ ) for j , m ∈ Z and a.e. ( x, ξ ) ∈ R + × R ; ( iii ) For j, m ∈ Z , f ∈ L ( R + ) , Θ a (Λ m D a j f )( x, ξ ) = e m,j ( x, ξ )Θ a f ( x, ξ ) for a.e. ( x, ξ ) ∈ R + × R ;( iv ) Θ a -transform is a unitary operator from L ( R + ) onto L ([1 , a ) × [0 , ; ( v ) Z [1 , a ) × [0 , | f ( x, ξ ) | dxdξ = X m,j ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [1 , a ) × [0 , f ( x, ξ ) e m,j ( x, ξ ) dxdξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (2.2) for f ∈ L ([1 , a ) × [0 , .Proof. By a standard argument, we have (i)-(iii). Next we prove (iv) and (v).(iv) It is easy to check that Θ a -transform is a linear and bijective mapping from L ( R + ) onto L ([1 , a ) × [0 , f ∈ L ( R + ), k Θ a f k L ((1 ,a ) × (0 , = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X l ∈ Z a l f ( a l x ) e − πilξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ((1 ,a ) × (0 , = Z a dx Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X l ∈ Z a l f ( a l x ) e − πilξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dξ = Z a X l ∈ Z a l | f ( a l x ) | dx = k f k L ( R + ) . This implies that Θ a -transform is norm-preserving.(v) By (i), (2.2) holds if f ∈ L ([1 , a ) × [0 , f ∈ L ([1 , a ) × [0 , \ L ([1 , a ) × [0 , g = X m,j ∈ Z Z [1 , a ) × [0 , f ( x, ξ ) e m,j ( x, ξ ) dxdξ ! e m,j belongs to L ([1 , a ) × [0 , L ([1 , a ) × [0 , f . So f = g by the uniqueness of Fourier coefficients, and thus f ∈ L ([1 , a ) × [0 , Y.-Z. Li & W. Zhang
Remark 2.2.
We call the property (ii) the quasi-periodicity of Θ a -transform. By (iv), an arbitrary function F ∈ L ([1 , a ) × [0 , determines a unique f ∈ L ( R + ) in the following way. Observe that there exists aunique { c m,j } m,j ∈ Z ∈ l ( Z ) such that F ( x, ξ ) = X m,j ∈ Z c m,j e m,j ( x, ξ ) = X j ∈ Z X m ∈ Z c m, j Λ m ( x ) ! e πijξ for a.e. ( x, ξ ) ∈ [1 , a ) × [0 , by (i). Define f on R + by f ( a j x ) = a − j X m ∈ Z c m, − j Λ m ( x ) for j ∈ Z and a.e. x ∈ [1 , a ) . Then Θ a f ( x, ξ ) = F ( x, ξ ) for a.e. ( x, ξ ) ∈ [1 , a ) × [0 , . Therefore, we can define L ( R + ) -functions in Θ a -transform domain. Lemma 2.2.
Let a and Ψ be as in the general setup. Then L X l =1 X m,j ∈ Z (cid:12)(cid:12) h f, Λ m D a j ψ l i L ( R + ) (cid:12)(cid:12) = Z [1 , a ) × [0 , L X l =1 | Θ a ψ l ( x, ξ ) | ! | Θ a f ( x, ξ ) | dxdξ for f ∈ L ( R + ) . Proof.
Fix f ∈ L ( R + ). By Lemma 2.1 (iii) and (iv), we have L X l =1 X m,j ∈ Z (cid:12)(cid:12) h f, Λ m D a j ψ l i L ( R + ) (cid:12)(cid:12) = L X l =1 X m,j ∈ Z (cid:12)(cid:12) h Θ a f, Θ a Λ m D a j ψ l i L ([1 , a ) × [0 , (cid:12)(cid:12) = L X l =1 X m,j ∈ Z (cid:12)(cid:12) h Θ a f, e m,j Θ a ψ l i L ([1 , a ) × [0 , (cid:12)(cid:12) = L X l =1 X m,j ∈ Z (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [1 , a ) × [0 , Θ a ψ l ( x, ξ )Θ a f ( x, ξ ) e m,j ( x, ξ ) dxdξ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Again applying Lemma 2.1 (v) to Θ a ψ l ( x, ξ )Θ a f ( x, ξ ) leads to L X l =1 X m,j ∈ Z (cid:12)(cid:12) h f, Λ m D a j ψ l i L ( R + ) (cid:12)(cid:12) = L X l =1 Z [1 , a ) × [0 , (cid:12)(cid:12)(cid:12) Θ a ψ l ( x, ξ )Θ a f ( x, ξ ) (cid:12)(cid:12)(cid:12) dxdξ = Z [1 , a ) × [0 , L X l =1 | Θ a ψ l ( x, ξ ) | ! | Θ a f ( x, ξ ) | dxdξ. This finishes the proof.
Theorem 2.1.
Let a and Ψ be as in the general setup. Then MD (Ψ , a ) is complete in L ( R + ) if and onlyif L X l =1 | Θ a ψ l ( x, ξ ) | = 0 for a.e. ( x, ξ ) ∈ [1 , a ) × [0 , . (2.3) ilation-and-modulation frames Proof.
By Lemma 2.2, for f ∈ L ( R + ), L X l =1 X m,j ∈ Z (cid:12)(cid:12) h f, Λ m D a j ψ l i L ( R + ) (cid:12)(cid:12) = 0 a.e. on R + (2.4)if and only if L X l =1 | Θ a ψ l ( x, ξ ) | ! | Θ a f ( x, ξ ) | = 0 for a.e. ( x, ξ ) ∈ [1 , a ) × [0 , . (2.5)Observe that MD (Ψ , a ) is complete in L ( R + ) if and only if f = 0 is a unique solution to (2.4) in L ( R + ).It follows that the completeness of MD (Ψ , a ) in L ( R + ) is equivalent to f = 0 being a unique solutionto (2.5) in L ( R + ). This is in turn equivalent to the fact that Θ a f = 0 is a unique solution to (2.5) in L ([1 , a ) × [0 , Theorem 2.2.
Let a and Ψ be as in the general setup. Then MD (Ψ , a ) is a Bessel sequence in L ( R + ) with the Bessel bound B if and only if L X l =1 | Θ a ψ l ( x, ξ ) | ≤ B for a.e. ( x, ξ ) ∈ [1 , a ) × [0 , . (2.6) Proof.
Applying Lemma 2.2, we have L X l =1 X m,j ∈ Z (cid:12)(cid:12) h f, Λ m D a j ψ l i L ( R + ) (cid:12)(cid:12) = Z [1 , a ) × [0 , L X l =1 | Θ a ψ l ( x, ξ ) | ! | Θ a f ( x, ξ ) | dxdξ for f ∈ L ( R + ) . (2.7)So (2.6) implies that L X l =1 X m,j ∈ Z (cid:12)(cid:12) h f, Λ m D a j ψ l i L ( R + ) (cid:12)(cid:12) ≤ B Z [1 , a ) × [0 , | Θ a f ( x, ξ ) | dxdξ = B k f k L R +) (2.8)for f ∈ L ( R + ) by Lemma 2.1 (iv). Thus MD (Ψ , a ) is a Bessel sequence in L ( R + ) with the Bessel bound B . Now we prove the converse implication by contradiction. Suppose MD (Ψ , a ) is a Bessel sequence in L ( R + ) with the Bessel bound B , and L P l =1 | Θ a ψ ( · , · ) | > B on some E ⊂ [1 , a ) × [0 ,
1) with | E | >
0. Take f by Θ a f ( · , · ) = χ E ( · , · ) on [1 , a ) × [0 , χ E denotes the characteristic function of E . Then f is well-defined, k f k L ( R + ) = Z [1 , a ) × [0 , | Θ a f ( x, ξ ) | dxdξ = | E | by Lemma 2.1 (iv), and L X l =1 X m,j ∈ Z (cid:12)(cid:12) h f, Λ m D a j ψ l i L ( R + ) (cid:12)(cid:12) > B | E | = B k f k L R +) . Y.-Z. Li & W. Zhang
It contradicts the fact that MD (Ψ , a ) is a Bessel sequence in L ( R + ) with the Bessel bound B . The proofis completed.By an argument similar to Theorem 2.2, we have Theorem 2.3.
Let a and Ψ be as in the general setup. Then MD (Ψ , a ) is a frame in L ( R + ) with framebounds A and B if and only if A ≤ L P l =1 | Θ a ψ l ( x, ξ ) | ≤ B for a.e. ( x, ξ ) ∈ [1 , a ) × [0 , . Θ a -transform domain expression of duals In this section, we characterize and express MD -duals of a general frame MD (Ψ , a ) for L ( R + ). Andwe also study the redundancy of a general frame MD (Ψ , a ) for L ( R + ). Interestingly, we prove that anarbitrary frame MD (Ψ , a ) for L ( R + ) is always nonredundant if L = 1, and is always redundant if L > D = { f ∈ L ( R + ) : Θ a f ∈ L ∞ ([1 , a ) × [0 , } . (3.1)Then D is dense in L ( R + ) by following Lemma 2.1 (iv) and the fact that L ∞ ([1 , a ) × [0 , L ([1 , a ) × [0 , a and Ψ be as in the general setup, and MD (Ψ , a ) be a Bessel sequence in L ( R + ). We denote by S its frame operator, i.e., Sf = L X l =1 X m,j ∈ Z h f, Λ m D a j ψ l i L ( R + ) Λ m D a j ψ l for f ∈ L ( R + ) . By a standard argument, we have the following lemma which shows that S commutes with the modulationand dilation operators. Lemma 3.1.
Let a and Ψ be as in the general setup. Assume that MD (Ψ , a ) is a Bessel sequence in L ( R + ) , and that S is its frame operator. Then S Λ m f = Λ m Sf, SD a j f = D a j Sf, and thus S Λ m D a j f = Λ m D a j Sf for f ∈ L ( R + ) and m, j ∈ Z . Lemma 3.2.
Let a and Ψ be as in the general setup, and Φ = { ϕ , ϕ , · · · , ϕ L } ⊂ L ( R + ) . Then L X l =1 X m,j ∈ Z h f, Λ m D a j ψ l i L ( R + ) h Λ m D a j ϕ l , g i L ( R + ) = Z [1 , a ) × [0 , Ω( x, ξ )Θ a f ( x, ξ )Θ a g ( x, ξ ) dxdξ (3.2) for f, g ∈ D , where Ω( x, ξ ) = L X l =1 Θ a ϕ l ( x, ξ )Θ a ψ l ( x, ξ ) . ilation-and-modulation frames Proof.
Fix f, g ∈ D . Then L X l =1 X m,j ∈ Z |h f, Λ m D a j ψ l i| < ∞ , and L X l =1 X m,j ∈ Z |h g, Λ m D a j ϕ l i| < ∞ by Lemma 2.2, and thus the series L X l =1 X m,j ∈ Z h f, Λ m D a j ψ l i L ( R + ) h Λ m D a j ϕ l , g i L ( R + ) converges absolutely and is well-defined. By Lemma 2.1 (i), (iii) and (iv), we see that L X l =1 X m,j ∈ Z h f, Λ m D a j ψ l i L ( R + ) h Λ m D a j ϕ l , g i L ( R + ) = L X l =1 X m,j ∈ Z h Θ a f, Θ a Λ m D a j ψ l i L ([1 , a ) × [0 , h Θ a Λ m D a j ϕ l , Θ a g i L ([1 , a ) × [0 , = L X l =1 X m,j ∈ Z h Θ a ψ l Θ a f, e m,j i L ([1 , a ) × [0 , h e m,j , Θ a ϕ l Θ a g i L ([1 , a ) × [0 , = L X l =1 h Θ a f Θ a ψ l , Θ a g Θ a ϕ l i L ([1 , a ) × [0 , = Z [1 , a ) × [0 , Ω( x, ξ )Θ a f ( x, ξ )Θ a g ( x, ξ ) dxdξ. The proof is completed.
Lemma 3.3.
Let a and Ψ be as in the general setup. Assume that MD (Ψ , a ) is a Bessel sequence in L ( R + ) , and that S is its frame operator. Then, for f ∈ L ( R + ) , Θ a Sf ( · , · ) = L X l =1 | Θ a ψ l ( · , · ) | ! Θ a f ( · , · ) (3.3) a.e. on [1 , a ) × [0 , .Proof. By Lemma 3.2, we have h Sf, g i L ( R + ) = Z [1 , a ) × [0 , L X l =1 | Θ a ψ l ( x, ξ ) | ! Θ a f ( x, ξ )Θ a g ( x, ξ ) dxdξ for f, g ∈ D . Since D is dense in L ( R + ) and MD (Ψ , a ) is a Bessel sequence, by Theorem 2.2 and a standardargument, it follows that h Sf, g i L ( R + ) = * L X l =1 | Θ a ψ l ( x, ξ ) | ! Θ a f, Θ a g + L ([1 , a ) × [0 , for f, g ∈ L ( R + ). Also observing that h Sf, g i L ( R + ) = h Θ a Sf, Θ a g i L ([1 , a ) × [0 , by Lemma 2.1 (iv), wesee that h Θ a Sf, Θ a g i L ([1 , a ) × [0 , = * L X l =1 | Θ a ψ l | ! Θ a f, Θ a g + L ([1 , a ) × [0 , for f, g ∈ L ( R + ). This implies (3.3) by Lemma 2.1 (iv). The proof is completed.0 Y.-Z. Li & W. Zhang
Lemma 3.4.
Let a and Ψ be as in the general setup. Then there exists no Riesz sequence MD (Ψ , a ) in L ( R + ) whenever L > .Proof. By contradiction. Suppose
L > MD (Ψ , a ) is a Riesz sequence in L ( R + ). Let S be its frameoperator. Then it commutes Λ m D a j for all m, j ∈ Z by Lemma 3.1. Since S is self-adjoint, invertible andbounded, it follows that S − Λ m D a j ψ l = Λ m D a j S − ψ l for m, j ∈ Z and 1 ≤ l ≤ L. Therefore, MD ( S − (Ψ) , a ) is an orthonormal system in L ( R + ). Write S − ψ l = ϕ l for 1 ≤ l ≤ L . Then h Λ m D a j ϕ l , Λ m D a j ϕ l i L ( R + ) = δ m ,m δ j ,j δ l ,l for m , m , j , j ∈ Z and 1 ≤ l , l ≤ L , where the Kronecker delta is defined by δ n, m = (cid:26) n = m ;0 if n = m. By Lemma 2.1 (iii) and (iv), it is equivalent to h e m ,j Θ a ϕ l , e m ,j Θ a ϕ l i L ([1 , a ) × [0 , = δ m ,m δ j ,j δ l ,l for m , m , j , j ∈ Z and 1 ≤ l , l ≤ L , equivalently,1 √ a − Z [1 , a ) × [0 , Θ a ϕ l ( x, ξ )Θ a ϕ l ( x, ξ ) e m,j ( x, ξ ) dxdξ = δ m, δ j, δ l ,l for m, j ∈ Z and 1 ≤ l , l ≤ L . This is in turn equivalent toΘ a ϕ l ( · , · )Θ a ϕ l ( · , · ) = δ l ,l a.e. on [1 , a ) × [0 , ≤ l , l ≤ L by the uniqueness of Fourier coefficients. In particular, it implies that | Θ a ϕ ( · , · ) | = | Θ a ϕ ( · , · ) | = 1and Θ a ϕ ( · , · )Θ a ϕ ( · , · ) = 0a.e. on [1 , a ) × [0 , Lemma 3.5.
Let a and Ψ be as in the general setup, and L = 1 . Then MD (Ψ , a ) is a Parseval frame for L ( R + ) if and only if it is an orthonormal basis for L ( R + ) . Theorem 3.1.
Let a and Ψ be as in the general setup, and Φ = { ϕ , ϕ , · · · , ϕ L } ⊂ L ( R + ) . Assume that MD (Ψ , a ) and MD (Φ , a ) are Bessel sequences in L ( R + ) . Then MD (Ψ , a ) and MD (Φ , a ) form a pair ofdual frames for L ( R + ) if and only if L X l =1 Θ a ϕ l ( x, ξ )Θ a ψ l ( x, ξ ) = 1 for a.e. ( x, ξ ) ∈ [1 , a ) × [0 , . (3.4) ilation-and-modulation frames Proof.
Since MD (Ψ , a ) and MD (Φ , a ) are Bessel sequences in L ( R + ), and D is dense in L ( R + ), we seethat MD (Ψ , a ) and MD (Φ , a ) form a pair of dual frames for L ( R + ) if and only if L X l =1 X m,j ∈ Z h f, Λ m D a j ψ l i L ( R + ) h Λ m D a j ϕ l , g i L ( R + ) = h f, g i L ( R + ) (3.5)for f, g ∈ D . By Lemma 3.2 and Lemma 2.1 (iv), (3.5) is equivalent to Z [1 , a ) × [0 , L X l =1 Θ a ϕ l ( x, ξ )Θ a ψ l ( x, ξ ) ! Θ a f ( x, ξ )Θ a g ( x, ξ ) dxdξ = Z [1 , a ) × [0 , Θ a f ( x, ξ )Θ a g ( x, ξ ) dxdξ (3.6)for f, g ∈ D . Obviously, (3.4) implies (3.6). Now we prove the converse implication to finish the proof. Sup-pose (3.6) holds. By Theorem 2.2 and the cauchy-Schwarz inequality, we have L P l =1 Θ a ϕ l Θ a ψ l ∈ L ∞ ([1 , a ) × (0 , , a ) × (0 ,
1) is a Lebesgue point of L P l =1 Θ a ϕ l Θ a ψ l . Arbitrarilyfix such a point ( x , ξ ) ∈ (1 , a ) × (0 , f, g ∈ D in (3.6) such thatΘ a f = Θ a g = 1 p | B (( x , ξ ) , ε ) | χ B (( x , ξ ) , ε ) . on [1 , a ) × [0 ,
1) with B (( x , ξ ) , ε ) ⊂ (1 , a ) × (0 ,
1) and ε >
0, where B (( x , ξ ) , ε ) denotes the ε -neighborhood of ( x , ξ ). Then f and g are well-defined by Lemma 2.1 (iv), and we obtain that1 | B (( x , ξ ) , ε ) | Z B (( x , ξ ) , ε ) L X l =1 Θ a ϕ l ( x, ξ )Θ a ψ l ( x, ξ ) dxdξ = 1 . (3.7)Letting ε → L X l =1 Θ a ϕ l ( x , ξ )Θ a ψ l ( x , ξ ) = 1 . This implies (3.4) by the arbitrariness of ( x , ξ ). The proof is completed.Now we turn to the expression of MD -duals. Let a and Ψ be as in the general setup, MD (Ψ , a )be a frame for L ( R + ) and S be its frame operator. By Lemma 3.1, S Λ m D a j = Λ m D a j S , and thus S − Λ m D a j = Λ m D a j S − for m, j ∈ Z . So MD (Ψ , a ) and its canonical dual S − ( MD (Ψ , a )) share thesame dilation-and-modulation structure, that is, S − ( MD (Ψ , a )) = MD ( S − (Ψ) , a ) . The following theorem gives its canonical dual window and all MD -dual windows in Θ a transform domain. Theorem 3.2.
Let a and Ψ be as in the general setup, and MD (Ψ , a ) be a frame for L ( R + ) . Then ( i ) its canonical dual MD ( S − (Ψ) , a ) is given by Θ a S − ψ l ( · , · ) = Θ a ψ l ( · , · ) L P l =1 | Θ a ψ l ( · , · ) | a.e. on [1 , a ) × [0 , for ≤ l ≤ L ;2 Y.-Z. Li & W. Zhang ( ii ) a dilation-and-modulation system MD (Φ , a ) with Φ = { ϕ , ϕ , · · · , ϕ L } is a dual frame of MD (Ψ , a ) if and only if Φ is defined by Θ a ϕ l ( · , · ) = Θ a ψ l ( · , · ) (cid:18) − L P l =1 Θ a ψ l ( · , · ) X l ( · , · ) (cid:19) L P l =1 | Θ a ψ l ( · , · ) | + X l ( · , · ) a.e. on [1 , a ) × [0 , , (3.8) where X l ∈ L ∞ ([1 , a ) × [0 , with ≤ l ≤ L .Proof. (i) Since S is an invertible and bounded operator on L ( R + ), we haveΘ a f ( · , · ) = L X l =1 | Θ a ψ l ( · , · ) | ! Θ a S − f ( · , · ) for f ∈ L ( R + ) (3.9)by Lemma 3.3. Replacing f by ψ l in (3.9) with 1 ≤ l ≤ L , we have (i).(ii) Sufficiency. Suppose Φ is given by (3.8). Then MD (Φ , a ) is a Bessel sequence in L ( R + ) by Theorem2.2. By a simple computation, we have L X l =1 Θ a ϕ l ( · , · )Θ a ψ l ( · , · ) = 1 a.e. on [1 , a ) × [0 , . It follows that MD (Φ , a ) is a dual frame of MD (Ψ , a ) by Theorem 3.1.Necessity. Suppose MD (Φ , a ) is a dual frame of MD (Ψ , a ). Then L X l =1 Θ a ψ l ( · , · )Θ a ϕ l ( · , · ) = 1 a.e. on [1 , a ) × [0 , a ϕ l ∈ L ∞ ([1 , a ) × [0 , X l = Θ a ϕ l , ≤ i ≤ L . The proof iscompleted.The following theorem shows that the cardinality L of Ψ determines whether or not a frame MD (Ψ , a )is redundant. If L = 1, there exists no redundant frame MD (Ψ , a ) for L ( R + ). If L >
1, there exists nononredundant frame MD (Ψ , a ) for L ( R + ). Theorem 3.3.
Let a and Ψ be as in the general setup, and MD (Ψ , a ) be a frame for L ( R + ) . Then MD (Ψ , a ) is a Riesz basis for L ( R + ) if and only if L = 1 .Proof. The necessity is an immediate consequence of Lemma 3.4. Now we show the sufficiency. Suppose L = 1. From the proof of Lemma 3.4, S − ( MD (Ψ , a )) = MD ( S − (Ψ) , a ) . So MD ( S − (Ψ) , a ) is a Parseval frame for L ( R + ) since MD (Ψ , a ) is a frame for L ( R + ). It leads to that MD ( S − (Ψ) , a ) is an orthonormal basis by Lemma 3.5. This is equivalent to the fact that MD (Ψ , a ) is aRiesz basis for L ( R + ). The proof is completed. ilation-and-modulation frames Theorems 2.2, 2.3 and 3.2 provide us with an easy method to construct MD -dual frame pairs for L ( R + ).This section focus on some examples. They show that we can construct MD -dual frame pairs for L ( R + )with good properties such as dual windows having bounded supports and certain smoothness. Example 4.1.
Let c be a finitely supported sequence defined on Z , and its Fourier transform ˆ c ( ξ ) = X l ∈ Z c l e − πilξ have no zero on [0 , . Define ψ ∈ L ( R + ) by Θ a ψ ( x, ξ ) = ˆ c ( ξ ) for ( x, ξ ) ∈ [1 , a ) × [0 , . Then ψ is a step function and of bounded support by the definition of Θ a , and MD ( ψ, a ) is a frame for L ( R + ) by Theorem 2.3 since | ˆ c ( ξ ) | have positive lower and upper bounds due to its continuity and havingno zeros on [0 , . It follows that MD ( ψ, a ) has a unique MD -dual window S − ψ defined by Θ a S − ψ ( x, ξ ) = 1 P l ∈ Z c l e πilξ for ( x, ξ ) ∈ [1 , a ) × [0 , by Theorems 3.2 and 3.3. Observe that, if at least two c l are nonzero, we have P l ∈ Z c l e πilξ = X l ∈ Z d l e − πilξ with d being infinitely supported. It follows that the dual window S − ψ is of unbounded support by thedefinition of Θ a , although ψ is of bounded support. The following example shows that it is possible for us to obtain multi-window MD -dual frame pairs for L ( R + ) with each window being of bounded support. Example 4.2.
Let
L > , m , m , · · · , m L be trigonometric polynomials satisfying | m ( ξ ) | + | m ( ξ ) | + · · · + | m L ( ξ ) | = 1 for ξ ∈ [0 , . Define
Ψ = { ψ , ψ , · · · , ψ L } by Θ a ψ l ( x, ξ ) = m l ( ξ ) for ( x, ξ ) ∈ [1 , a ) × [0 , . Then MD (Ψ , a ) is a frame for L ( R + ) and every ψ l is of bounded support by an argument similar to Example4.1. Define Φ = { ϕ , ϕ , · · · , ϕ L } by Θ a φ ( x, ξ ) = m l ( ξ ) − L X l =1 m l ( ξ ) X l ( x, ξ ) ! + X l ( x, ξ ) for a.e. ( x, ξ ) ∈ [1 , a ) × [0 ,
1) (4.1)4
Y.-Z. Li & W. Zhang with X l ∈ L ∞ ([1 , a ) × [0 , . Then MD (Ψ , a ) and MD (Φ , a ) form a pair of dual frames for L ( R + ) byTheorem 3.2. If, in addition, we require X l ( x, ξ ) = X j ∈ Z d l,j ( x ) e − πijξ for a.e. ( x, ξ ) ∈ [1 , a ) × [0 , . (4.2) with { d l,j ( · ) } j ∈ Z is a finitely supported sequence of functions on [1 , a ) for each ≤ l ≤ L , then each ϕ l with ≤ l ≤ L is of bounded support by (4.1) and the definition of Θ a . Example 4.3.
Let L ≥ , Ψ = { ψ , ψ , · · · , ψ L } be a finite subset of L ( R + ) , and supp ( ψ l ) ⊂ [1 , a ) .Assume that L X l =1 | ψ l ( x ) | = 1 for a.e. x ∈ [1 , a ) . Define
Φ = { ϕ , ϕ , · · · , ϕ L } by Θ a ϕ l ( x, ξ ) = ψ l ( x ) − L X l =1 ψ l ( x ) X l ( x, ξ ) ! + X l ( x, ξ ) for a.e. ( x, ξ ) ∈ [1 , a ) × [0 ,
1) (4.3) with X l ∈ L ∞ ([1 , a ) × [0 , . Then MD (Ψ , a ) and MD (Φ , a ) form a pair of dual frames for L ( R + ) byTheorem 3.2. In particular, if X l , ≤ l ≤ L are required as in (4.2), in addition, each ϕ l with ≤ l ≤ L isof bounded support. In Examples 4.2 and 4.3, Θ a ψ l , ≤ l ≤ L , are defined by univariate functions. Next we give a relativelymore general example. Example 4.4.
Assume that c ( x ) and c ( x ) are two real-valued measurable functions defined on [1 , a ] , andthat there exist two positive constants A and B such that A ≤ | c ( x ) | + | c ( x ) | ≤ B for x ∈ [1 , a ] . Define
Ψ = { ψ , ψ } ⊂ L ( R + ) by Θ a ψ ( x, ξ ) = c ( x ) + c ( x ) e − πiξ Θ a ψ ( x, ξ ) = i p c ( x ) c ( x ) sin 2 πξ if c ( x ) c ( x ) ≥ p − c ( x ) c ( x ) cos 2 πξ if c ( x ) c ( x ) < for a.e. ( x, ξ ) ∈ [1 , a ] × [0 , . Then ψ ( x ) = c ( x ) if ≤ x ≤ a ; a − c ( a − x ) if a ≤ x ≤ a ;0 otherwise , ilation-and-modulation frames and ψ ( x ) = a p c ( ax ) c ( ax ) if a − ≤ x ≤ − a − p c ( a − x ) c ( a − x ) if a ≤ x ≤ a and c ( a − x ) c ( a − x ) ≥ a − p − c ( a − x ) c ( a − x ) if a ≤ x ≤ a and c ( a − x ) c ( a − x ) < otherwise , and | Θ a ψ ( x, ξ ) | + | Θ a ψ ( x, ξ ) | = ( | c ( x ) | + | c ( x ) | ) by a simple computation and the definition of Θ a . It follows that A ≤ | Θ a ψ ( x, ξ ) | + | Θ a ψ ( x, ξ ) | ≤ B (4.4) for a.e. ( x, ξ ) ∈ [1 , a ] × [0 , , and thus MD (Ψ , a ) is a frame for L ( R + ) by Theorem 2.3. Obviously, ψ and ψ are real-valued and of bounded support.Now we check the MD -duals of MD (Ψ , a ) . Define Φ = { ϕ , ϕ } by Θ a ϕ l ( x, ξ ) = Θ a ψ l ( x, ξ ) (cid:16) − Θ a ψ ( x, ξ ) X ( x, ξ ) − Θ a ψ ( x, ξ ) X ( x, ξ ) (cid:17) ( | c ( x ) | + | c ( x ) | ) + X l ( x, ξ ) for ≤ l ≤ and a.e. ( x, ξ ) ∈ [1 , a ] × [0 , with X , X ∈ L ∞ ([1 , a ] × [0 , . Then MD (Ψ , a ) and MD (Φ , a ) form a pair of dual frames for L ( R + ) by Theorem 3.2. If X l ( x, ξ ) = X j ∈ Z d l,j ( x ) e − πijξ for a.e. ( x, ξ ) ∈ [1 , a ] × [0 , . with { d l,j ( · ) } j ∈ Z being a finitely supported sequence of real-valued function on [1 , a ] for each ≤ l ≤ , then ϕ and ϕ are also real-valued and of bounded support. Also we can obtain Φ with good smoothness by choosinggood X and X . For example, if we make further assumption that c ( x ) , c ( x ) and p | c ( x ) c ( x ) | are k -thcontinuously differentiable on (1 , a ) , that c (1) c (1) = c ( a ) c ( a ) = 0 , and c ( x ) c ( x ) > for x ∈ (1 , a ) .Then ψ and ψ are continuous functions on R + and k -th continuously differentiable on (1 , a ) ∪ ( a , a ) and ( a − , ∪ ( a, a ) , respectively. In this case, if we further require that | c ( x ) | + | c ( x ) | is a constant on [1 , a ] ,and X ( x, ξ ) and X ( x, ξ ) satisfy X ( x, ξ ) = X j ∈ Z d ,j e − πijξ and X ( x, ξ ) = X j ∈ Z d ,j e − πijξ for ξ ∈ [0 , with { d ,j } j ∈ Z and { d ,j } j ∈ Z being two finitely supported real number sequences. Then ϕ and ϕ are real-valued, of bounded support, and have the same continuity and differentiability as ψ and ψ . Y.-Z. Li & W. Zhang
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