Hindawi Publishing Corporation Journal of Function Spaces and Applications Volume 2013, Article ID 610917, 6 pages http://dx.doi.org/10.1155/2013/610917
Research Article An Interplay between Gabor and Wilson Frames S. K. Kaushik1 and Suman Panwar2 1 2
Department of Mathematics, Kirori Mal College, University of Delhi, Delhi 110 007, India Department of Mathematics, University of Delhi, Delhi 110 007, India
Correspondence should be addressed to S. K. Kaushik;
[email protected] Received 6 May 2013; Accepted 12 September 2013 Academic Editor: Wilfredo Urbina Copyright Š 2013 S. K. Kaushik and S. Panwar. 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. Wilson frames {đđđ : đ¤0 , đ¤â1 â đż2 (R)} đâZ,đâN0 as a generalization of Wilson bases have been defined and studied. We give necessary condition for a Wilson system to be a Wilson frame. Also, sufficient conditions for a Wilson system to be a Wilson Bessel sequence are obtained. Under the assumption that the window functions đ¤0 and đ¤â1 for odd and even indices of đ are the same, we obtain sufficient conditions for a Wilson system to be a Wilson frame (Wilson Bessel sequence). Finally, under the same conditions, a characterization of Wilson frame in terms of Zak transform is given.
1. Introduction In 1946, Gabor [1] proposed a decomposition of a signal in terms of elementary signals, which displays simultaneously the local time and frequency content of the signal, as opposed to the classical Fourier transform which displays only the global frequency content for the entire signal. On the basis of this development, in 1952, Duffin and Schaeffer [2] introduced frames for Hilbert spaces to study some deep problems in nonharmonic Fourier series. In fact, they abstracted the fundamental notion of Gabor for studying signal processing. Janssen [3] showed that while being complete in đż2 (R), the set suggested by Gabor is not a Riesz basis. This apparent failure of Gabor system was then rectified by resorting to the concept of frames. Since then, the theory of Gabor systems has been intimately related to the theory of frames, and many problems in frame theory find their origins in Gabor analysis. For example, the localized frames were first considered in the realm of Gabor frames [4â7]. Gabor frames have found wide applications in signal and image processing. In view of Balian-Low theorem [8], Gabor frame for đż2 (R) (which is a Riesz basis) has bad localization properties in time or frequency. Thus, a system to replace Gabor systems which does not have bad localization properties in time and frequency was required. For more literature on Gabor frames one may refer to [8â12]. Wilson et al. [13, 14] suggested a system of functions which are localized around the positive and
negative frequency of the same order. The idea of Wilson was used by Daubechies et al. [15] to construct orthonormal âWilson basesâ which consist of functions given by đ { if đ is even, đđ cos (2đđđĽ) đ¤ (đĽ â ) , { { 2 đđđ (đĽ) = { đ+1 { { ) , if đ is odd, 2 sin (2 (đ + 1) đđĽ) đ¤ (đĽ â 2 { â2, đđ = { 2,
if đ = 0, if đ â N, (1)
with a smooth well-localized window function đ¤. For such bases the disadvantage described in the Balian-Low theorem is completely removed. Independently from the work of Daubechies, Jaffard, and Journe, orthonormal local trigonometric bases consisting of the functions đ¤đ cos(đ + (1/2))đ(â
â đ), đ â Z, đ â N0 were introduced by Malvar [16]. Some generalizations of Malvar bases exist in [17, 18]. A drawback of Malvarâs construction is the restriction on the support of the window functions. But the restriction on orthonormal bases allows only a small class of window functions. In [19], it has been proved that Wilson bases of exponential decay are not unconditional bases for all modulation spaces on R including the classical Bessel potential space and the Schwartz spaces. Also, Wilson bases
2
Journal of Function Spaces and Applications
are not unconditional bases for the ordinary đżđ spaces for đ ≠ 2, shown in [19]. Approximation properties of Wilson bases are studied in [20]. Wilson bases for general timefrequency lattices are studied in [21]. Generalizations of Wilson bases to nonrectangular lattices are discussed in [13] with motivation from wireless communication and cosines modulated filter banks. Modified Wilson bases are studied in [22]. Bittner [23] considered the Wilson bases introduced by Daubechies et al. with nonsymmetrical window functions for odd and even indices of đ. In this paper, we generalize the concept of Wilson bases and define Wilson frames. We give necessary condition for a Wilson system to be a Wilson frame. Also, sufficient conditions for a Wilson system to be a Wilson Bessel sequence are obtained. Under the assumption that the window functions for odd and even indices of đ are the same, we obtain sufficient conditions for a Wilson system to be a Wilson frame (Wilson Bessel sequence). Finally, under the same conditions, a characterization of Wilson frame in terms of Zak transform is given.
2. Preliminaries We assume that the reader is familiar with the theory of Gabor frames, refer [8, 9] for further details. Definition 1. Let H denote a Hilbert space. Let đź denote a countable index set. A family of vectors {đđ }đâđź is called a frame for H if there exist constants đ´ and đľ with 0 < đ´ ⤠đľ < â such that óľ¨ óľ¨2 óľŠ óľŠ2 óľŠ óľŠ2 đ´óľŠóľŠóľŠđóľŠóľŠóľŠ ⤠âóľ¨óľ¨óľ¨â¨đ, đđ âŠóľ¨óľ¨óľ¨ ⤠đľóľŠóľŠóľŠđóľŠóľŠóľŠ , đâđź
âđ â H.
(2)
The positive constants đ´ and đľ are called lower frame bound and upper frame bound for the family {đđ }đâđź , respectively. The inequality (2) is called the frame inequality. If in (2) only the upper inequality holds, then {đđ }đâđź is called a Bessel sequence. Definition 2 (see [9]). Let đ â đż2 (R) and đ, đ are positive constants. The sequence {đ¸đđ đđđ đ}đ,đâZ is called a Gabor system for đż2 (R). Further, (i) if {đ¸đđ đđđ đ}đ,đâZ is a frame for đż2 (R), then it is called a Gabor frame; (ii) if {đ¸đđ đđđ đ}đ,đâZ is a Bessel sequence for đż2 (R), then it is called a Gabor Bessel sequence. Definition 3 (see [23]). The Wilson system associated with đ¤0 , đ¤â1 â đż2 (R) is defined as a sequence of functions {đđđ : đ¤0 , đ¤â1 â đż2 (R)} đâZ,đâN0 in đż2 (R) given by đđđ
đ { if đ is even, {đđ cos (2đđđĽ) đ¤0 (đĽ â ), 2 (đĽ) = { đ + 1 {2 sin (2 (đ + 1) đđĽ) đ¤ (đĽ â ), if đ is odd, â1 { 2 (3)
where N0 = N ⪠{0} and â2, đđ = { 2,
if đ = 0, if đ â N.
(4)
If đ¤0 = đ¤â1 = đ, then the Wilson system is given as {đđđ : đ â đż2 (R)}đâZ,đâN0 . Definition 4 (see [9]). The Zak transform of đ â đż2 (R) is defined as a function of two variables given by (đđ)(đĄ, V) = âđâZ đ(đĄ â đ) exp(2đđđV), đĄ, V â R.
3. Main Results We begin this section with the definition of a Wilson frame. Definition 5. The Wilson System {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 for đż2 (R) associated with đ¤0 , đ¤â1 â đż2 (R) is called a Wilson frame if there exist constants đ´ and đľ with 0 < đ´ ⤠đľ < â such that óľ¨ óľ¨2 óľŠ óľŠ2 óľŠ óľŠ2 đ´óľŠóľŠóľŠđóľŠóľŠóľŠ ⤠â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ ⤠đľóľŠóľŠóľŠđóľŠóľŠóľŠ ,
âđ â đż2 (R) .
đâZ đâN0
(5)
The constants đ´ and đľ are called lower frame bound and upper frame bound, respectively, for the Wilson frame {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 . Definition 6. In (5), if only the upper inequality holds for all đ â đż2 (R), then the Wilson system {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 , associated with đ¤0 , đ¤â1 â đż2 (R), is called a Wilson Bessel sequence with Bessel bound đľ. Example 7. (a) Let đ = đ¤0 = đ¤â1 = đ[0,1) . Then {đđđ : đ â đż2 (R)}đâZ,đâN0 is a Wilson frame for đż2 (R). (b) Let đ¤0 ≠ đ¤â1 such that |đ¤â1 (đĽ)| ⤠đś(1 + |đĽ|)â1âđ , |đ¤0 (đĽ)| ⤠đś(1 + |đĽ|)â1âđ for some constant đś and đ > 0. Let đ+ = (0, 1/2) Ă [â1/2, 1/2]. Consider the matrix đđ¤0 (đĽ, đ) đđ¤0 (âđĽ, đ) ). đ (đĽ, đ) = ( âđđ¤â1 (đĽ, đ) đđ¤â1 (âđĽ, đ)
(6)
Let đ´ 0 = ess inf (đĽ,đ)âđ+ â đâ1 (đĽ, đ)ââ2 2 and đľ0 = ess sup(đĽ,đ)âđ+ â đ(đĽ, đ)â22 . If 0 < đ´ 0 ⤠đľ0 < â, then the Wilson system {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 is a Wilson frame for đż2 (R) with bounds đ´ 0 and đľ0 . (c) If we choose đ = đ¤0 = đ¤â1 = đ[0,1/2) , then {đđđ : đ â 2 đż (R)}đâZ,đâN0 is not a Wilson frame for đż2 (R). (d) Let { sin đđĽ , if đĽ ≠ 0, đ¤0 (đĽ) = { đđĽ otherwise. {1,
(7)
Journal of Function Spaces and Applications
3 óľ¨óľ¨ + 4 â óľ¨óľ¨óľ¨óľ¨âŤ đ (đĽ) sin (2đđđĽ) đâZ óľ¨
Then {đđđ : đ¤0 â đż2 (R)}đâZ,đâN0 is a tight Wilson frame for đż2 (R) with frame bound 2. (e) Let đ(đĽ) = đ¤0 (đĽ) = đ¤â1 (đĽ) = 21/2 đâđĽ đ[0,â) (đĽ). Then {đđđ : đ â đż2 (R)}đâZ,đâN0 is a Wilson frame for đż2 (R).
đâN
óľ¨óľ¨2 Ă đ¤â1 (đĽ â đ)đđĽóľ¨óľ¨óľ¨óľ¨ . óľ¨
1/2
{đđđ
(f) Let đ(đĽ) = đ¤0 (đĽ) = đ¤â1 (đĽ) = 2 /(1 + 2đđđĽ). Then : đ â đż2 (R)}đâZ,đâN0 is a Wilson frame for đż2 (R). 2
(11)
2
(g) If đ¤0 (đĽ) = đâđ(đĽâ1/4) and đ¤â1 (đĽ) = đâđ(đĽ+1/4) , where đ > 0, then {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 is a Wilson frame for đż2 (R). (h) Let 1 + đĽ, if đĽ â [0, 1) , { { { {đĽ đ (đĽ) = { , if đĽ â [1, 2) , {2 { { otherwise. {0,
Thus, óľ¨óľ¨ óľ¨óľ¨2 óľ¨ óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ = 2 â óľ¨óľ¨óľ¨óľ¨âŤ đ (đĽ) đđ đ¤0 (đĽ)đđĽóľ¨óľ¨óľ¨óľ¨ óľ¨ đâZ đâZóľ¨
đâN0
óľ¨ óľ¨2 + 4 â óľ¨óľ¨óľ¨óľ¨â¨đ, cos (2đđâ
) đđ đ¤0 (â
)âŠóľ¨óľ¨óľ¨óľ¨
(8)
đâZ đâN
óľ¨2 óľ¨ + 4 â óľ¨óľ¨óľ¨óľ¨â¨đ, sin (2đđâ
) đđ đ¤â1 (â
)âŠóľ¨óľ¨óľ¨óľ¨
Then {đđđ : đ â đż2 (R)}đâZ,đâN0 is a Wilson frame for đż2 (R). Next, we give two Lemmas which will be used in the subsequent results. Lemma 8 is also proved in [24], but for the sake of completeness, we give the proof.
đâZ đâN
óľ¨ óľ¨2 = 2 â (óľ¨óľ¨óľ¨óľ¨â¨đ, cos (2đđâ
) đđ đ¤0 (â
)âŠóľ¨óľ¨óľ¨óľ¨ đ,đâZ
óľ¨ óľ¨2 + óľ¨óľ¨óľ¨óľ¨â¨đ, sin (2đđâ
) đđ đ¤â1 (â
)âŠóľ¨óľ¨óľ¨óľ¨ ) . (12)
Lemma 8. Let {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 be the Wilson system associated with đ¤0 , đ¤â1 â đż2 (R). Then, for đ â đż2 (R), óľ¨ óľ¨2 óľ¨ óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ = 2 â (óľ¨óľ¨óľ¨óľ¨â¨đ, cos (2đđâ
) đđ đ¤0 (â
)âŠóľ¨óľ¨óľ¨óľ¨
đâZ đâN0
đ,đâZ
(9) óľ¨2 óľ¨ + óľ¨óľ¨óľ¨óľ¨â¨đ, sin (2đđâ
) đđ đ¤â1 (â
)âŠóľ¨óľ¨óľ¨óľ¨ ) .
Lemma 9. Let {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 be the Wilson system associated with đ¤0 , đ¤â1 â đż2 (R). Then, for đ â đż2 (R), óľ¨ óľ¨ óľ¨ óľ¨2 óľ¨2 óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ = â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤0 âŠóľ¨óľ¨óľ¨óľ¨ + â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤â1 âŠóľ¨óľ¨óľ¨óľ¨
Proof. Let đ â đż2 (R). Then
đâZ đâN0
óľ¨ óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨
đ,đâZ
đ,đâZ
óľ¨2 óľ¨ + â (óľ¨óľ¨óľ¨óľ¨â¨đ, cos (2đđâ
) đđ đ¤0 (â
)âŠóľ¨óľ¨óľ¨óľ¨
đâZ đâN0
đ,đâZ
óľ¨óľ¨ óľ¨óľ¨2 đ óľ¨ óľ¨ = â óľ¨óľ¨óľ¨óľ¨âŤ đ (đĽ) đđ cos (2đđđĽ) đ¤0 (đĽ â )đđĽóľ¨óľ¨óľ¨óľ¨ 2 óľ¨ óľ¨óľ¨ đ:evenóľ¨
óľ¨2 óľ¨ + óľ¨óľ¨óľ¨óľ¨â¨đ, sin (2đđâ
) đđ đ¤â1 (â
)âŠóľ¨óľ¨óľ¨óľ¨ ) óľ¨2 óľ¨ â â (óľ¨óľ¨óľ¨óľ¨â¨đ, cos (2đđâ
) đđ đ¤â1 (â
)âŠóľ¨óľ¨óľ¨óľ¨
đâN0
đ,đâZ
óľ¨óľ¨ óľ¨óľ¨2 đ+1 óľ¨ óľ¨ + â óľ¨óľ¨óľ¨óľ¨âŤ 2đ (đĽ) sin (2 (đ + 1) đđĽ) đ¤â1 (đĽ â )đđĽóľ¨óľ¨óľ¨óľ¨ . 2 óľ¨ óľ¨óľ¨ đ:oddóľ¨
óľ¨2 óľ¨ + óľ¨óľ¨óľ¨óľ¨â¨đ, sin (2đđâ
) đđ đ¤0 (â
)âŠóľ¨óľ¨óľ¨óľ¨ ) .
đâN0
(10) Proof. We have
This gives óľ¨óľ¨2 óľ¨óľ¨ óľ¨ óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ = 2 â óľ¨óľ¨óľ¨óľ¨âŤ đ (đĽ) đ¤0 (đĽ â đ)đđĽóľ¨óľ¨óľ¨óľ¨ óľ¨ đâZ đâZóľ¨
đâN0
óľ¨óľ¨ + 4 â óľ¨óľ¨óľ¨óľ¨âŤ đ (đĽ) cos (2đđđĽ) đâZ óľ¨ đâN
óľ¨óľ¨2 Ă đ¤0 (đĽ â đ)đđĽóľ¨óľ¨óľ¨óľ¨ óľ¨
óľ¨ óľ¨ óľ¨ óľ¨2 óľ¨2 óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ = â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ + â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨
đâZ đâN0
đ:even đâN0
đ:odd đâN0
óľ¨ óľ¨2 = 2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđ đ¤0 âŠóľ¨óľ¨óľ¨óľ¨ đâZ
óľ¨ óľ¨2 + â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤0 + đ¸âđ đđ đ¤0 âŠóľ¨óľ¨óľ¨óľ¨ đâZ đâN
(13)
4
Journal of Function Spaces and Applications óľ¨ óľ¨2 + â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤â1 â đ¸âđ đđ đ¤â1 âŠóľ¨óľ¨óľ¨óľ¨
its lower frame bound. Then
đâZ đâN
đ´ óľ¨ óľ¨2 óľ¨ óľ¨2 ⤠â (óľ¨óľ¨đ¤ (đĽ â đ)óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨đ¤â1 (đĽ â đ)óľ¨óľ¨óľ¨ ) . 2 đâZ óľ¨ 0
óľ¨ óľ¨ = 2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđ đ¤0 âŠóľ¨óľ¨óľ¨ + â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤0 âŠóľ¨óľ¨óľ¨ óľ¨óľ¨2
đâZ
óľ¨óľ¨2
đ,đâZ đ ≠ 0
(17)
More precisely, if the inequality (17) is not satisfied, then the given Wilson system does not satisfy the lower frame condition.
óľ¨ óľ¨2 + â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤â1 âŠóľ¨óľ¨óľ¨óľ¨
Proof. Assume that condition (17) is violated. Then there exists a measurable set Î â R having positive measure such that
đ,đâZ đ ≠ 0
+ 2 Re â â¨đ, đ¸đ đđ đ¤0 ⊠â¨đ, đ¸âđ đđ đ¤0 âŠ
đ´ óľ¨ óľ¨2 óľ¨ óľ¨2 đ (đĽ) = â (óľ¨óľ¨óľ¨đ¤0 (đĽ â đ)óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨đ¤â1 (đĽ â đ)óľ¨óľ¨óľ¨ ) < 2 đâZ
đ,đâZ đ ≠ 0
on Î. (18)
â 2 Re â â¨đ, đ¸đ đđ đ¤â1 ⊠â¨đ, đ¸âđ đđ đ¤â1 âŠ. đ,đâZ đ ≠ 0
(14)
We can assume that this Î is contained in an interval of length 1. Let Î 0 = {đĽ â Î : đ (đĽ) â¤
Therefore, using
đ´ 1 đ´ 1 Î đ = {đĽ â Î : â < đ (đĽ) < â }. 2 đ 2 đ+1
óľ¨ óľ¨2 Re â¨đ, đ¸đ đđ đ⊠â¨đ, đ¸âđ đđ đ⊠= (óľ¨óľ¨óľ¨óľ¨â¨đ, cos (2đđâ
) đđ đ (â
)âŠóľ¨óľ¨óľ¨óľ¨ óľ¨ óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, sin (2đđâ
) đđ đ (â
)âŠóľ¨óľ¨óľ¨óľ¨ ) , (15) we finally get the result.
óľ¨ óľ¨ óľ¨2 óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤0 âŠóľ¨óľ¨óľ¨óľ¨ + â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤â1 âŠóľ¨óľ¨óľ¨óľ¨
Then Î is partitioned into disjoint measurable sets such that at least one of these measurable sets will have a positive measure. Let this set be Î đó¸ . Choose đ = đÎ đó¸ . Then âđâ = |đÎ đó¸ |, where |đÎ đó¸ | = measure of đÎ đó¸ .
óľ¨ óľ¨ óľ¨2 óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤0 âŠóľ¨óľ¨óľ¨óľ¨ = â óľ¨óľ¨óľ¨óľ¨â¨đđđ đ¤0 , đ¸đ âŠóľ¨óľ¨óľ¨óľ¨
đ,đâZ
óľ¨2 óľ¨ = â (óľ¨óľ¨óľ¨óľ¨â¨đ, cos (2đđâ
) đđ đ¤â1 (â
)âŠóľ¨óľ¨óľ¨óľ¨
đâZ
đâZ
óľ¨2 óľ¨ óľ¨ óľ¨2 = ⍠óľ¨óľ¨óľ¨đ (đĽ)óľ¨óľ¨óľ¨ óľ¨óľ¨óľ¨đ¤0 (đĽ â đ)óľ¨óľ¨óľ¨ đđĽ.
đ,đâZ
óľ¨2 óľ¨ + óľ¨óľ¨óľ¨óľ¨â¨đ, sin (2đđâ
) đđ đ¤0 (â
)âŠóľ¨óľ¨óľ¨óľ¨ )
(19)
Since for đ â Z, the functions đđđ đ¤0 and đđđ đ¤â1 have support in Î đó¸ , {đ¸đ }đâZ constitute an orthonormal basis for đż2 (đź), for every interval đź of length 1 and Î đó¸ is contained in an interval of length 1, we have
Remark 10. Combining Lemmas 8 and 9, we get
đ,đâZ
đ´ â 1} , 2
R
(16)
óľ¨2 óľ¨ + â (óľ¨óľ¨óľ¨óľ¨â¨đ, cos (2đđâ
) đđ đ¤0 (â
)âŠóľ¨óľ¨óľ¨óľ¨ đ,đâZ
óľ¨ óľ¨2 + óľ¨óľ¨óľ¨óľ¨â¨đ, sin (2đđâ
) đđ đ¤â1 (â
)âŠóľ¨óľ¨óľ¨óľ¨ ) . Remark 11. In view of Lemma 8 and Remark 10, the Wilson system obtained by interchanging đ¤0 and đ¤â1 is also a Wilson Bessel sequence if both the Gabor systems {đ¸đ đđ đ¤0 }đ,đâZ and {đ¸đ đđ đ¤â1 }đ,đâZ are Bessel sequences. The following result gives a necessary condition for a Wilson system {đđđ : đ¤0 , đ¤â1 }đâZ,đâN0 associated with đ¤0 , đ¤â1 â đż2 (R) to be a Wilson frame. The following result is motivated by Proposition 9.1.2 in [9]. Theorem 12. Let {đđđ : đ¤0 , đ¤â1 â đż2 (R)đâZ,đâN0 be a Wilson frame for đż2 (R) associated with đ¤0 , đ¤â1 â đż2 (R). Let đ´ denote
(20)
Also, since đ = đÎ đó¸ , we have óľ¨ óľ¨2 óľ¨ óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤0 âŠóľ¨óľ¨óľ¨óľ¨ = â ⍠óľ¨óľ¨óľ¨đ¤0 (đĽ â đ)óľ¨óľ¨óľ¨ đđĽ. Î ó¸ đâZ
đ,đâZ
(21)
đ
Similarly, we obtain óľ¨ óľ¨2 óľ¨ óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤â1 âŠóľ¨óľ¨óľ¨óľ¨ = â ⍠óľ¨óľ¨óľ¨đ¤â1 (đĽ â đ)óľ¨óľ¨óľ¨ đđĽ. Î ó¸ đâZ
đ,đâZ
(22)
đ
Therefore, óľ¨ óľ¨2 óľ¨ óľ¨2 â (óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤0 âŠóľ¨óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤â1 âŠóľ¨óľ¨óľ¨óľ¨ )
đ,đâZ
= ââŤ
đâZ Î đó¸
=âŤ
Î đó¸
óľ¨ óľ¨2 óľ¨ óľ¨2 (óľ¨óľ¨óľ¨đ¤0 (đĽ â đ)óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨đ¤â1 (đĽ â đ)óľ¨óľ¨óľ¨ ) đđĽ
đ (đĽ) đđĽ.
(23)
Journal of Function Spaces and Applications
5
Further, since óľ¨ óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ đâZ,đâN0
óľ¨ óľ¨2 óľ¨ óľ¨2 ⤠2 â (óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤0 âŠóľ¨óľ¨óľ¨óľ¨ + óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤â1 âŠóľ¨óľ¨óľ¨óľ¨ ) ,
(24)
Proof. In view of Proposition 9.7.3 in [9], the Gabor systems {đ¸đ đđ đ¤0 }đ,đâZ and {đ¸đ đđ đ¤â1 }đ,đâZ are Gabor Bessel sequences with Bessel bounds đľ1 and đľ2 , respectively. Therefore, using Lemma 8 and Remark 10, the Wilson system {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 is a Wilson Bessel sequence with Bessel bound 2(đľ1 + đľ2 ).
đ,đâZ
we get óľ¨ óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ ⤠2 âŤ
Î đó¸
đâZ đâN0
⤠2(
đ (đĽ) đđĽ
đ´ 1 â ó¸ ) ⍠đđĽ 2 đ + 1 Î đó¸
= (đ´ â
(25)
2 óľŠ óľŠ2 ) óľŠóľŠđóľŠóľŠ . đó¸ + 1 óľŠ óľŠ
(26)
đ,đâZ
Next, we give sufficient conditions for a Wilson system {đđđ : đ¤0 â đż2 (R)}đâZ,đâN0 to be a Wilson frame. Theorem 16. Let đ¤0 â đż2 (R) and
Hence, óľ¨ óľ¨2 óľŠ óľŠ2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ < đ´óľŠóľŠóľŠđóľŠóľŠóľŠ .
Theorem 15. Let đ¤0 , đ¤â1 â đż2 (R), and let there exist đľ1 > 0, đľ2 > 0 such that |đđ¤0 |2 ⤠đľ1 and |đđ¤â1 |2 ⤠đľ2 . Then, the Wilson system {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 is a Wilson Bessel sequence with Bessel bound 2(đľ1 + đľ2 ).
óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ đľ = sup â óľ¨óľ¨óľ¨ â đ¤0 (đĽ â đ) đ¤0 (đĽ â đ â đ)óľ¨óľ¨óľ¨ < â. óľ¨óľ¨ óľ¨ đĽâ[0,1]đâZ óľ¨óľ¨đâZ óľ¨
(28)
Then the Wilson system {đđđ : đ¤0 â đż2 (R}đâZ,đâZ0 is a Wilson Bessel sequence. Further, if
This is a contradiction. Next, we give a sufficient condition for a Wilson system {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 to be a Wilson Bessel sequence.
óľ¨2 óľ¨ đ´ = inf [ â óľ¨óľ¨óľ¨đ¤0 (đĽ â đ)óľ¨óľ¨óľ¨ đĽâ[0,1]
Theorem 13. Let đ¤0 , đ¤â1 â đż (R), óľ¨óľ¨ óľ¨óľ¨óľ¨ óľ¨óľ¨ óľ¨ đľ1 = sup â óľ¨óľ¨óľ¨ â đ¤0 (đĽ â đ) đ¤0 (đĽ â đ â đ)óľ¨óľ¨óľ¨ < â, óľ¨ óľ¨óľ¨ đĽâ[0,1]đâZ óľ¨óľ¨đâZ óľ¨ (27) óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ đľ2 = sup â óľ¨óľ¨óľ¨ â đ¤â1 (đĽ â đ) đ¤â1 (đĽ â đ â đ)óľ¨óľ¨óľ¨ < â. óľ¨óľ¨ đĽâ[0,1]đâZ óľ¨óľ¨óľ¨đâZ óľ¨ {đđđ
2
Then, the Wilson system : đ¤0 , đ¤â1 â đż (R)}đâZ,đâN0 is a Wilson Bessel sequence with Bessel bound 2(đľ1 + đľ2 ). Proof. In view of Theorem 9.1.5 in [9], the Gabor systems {đ¸đ đđ đ¤0 }đ,đâZ and {đ¸đ đđ đ¤â1 }đ,đâZ are Gabor Bessel sequences. Therefore, using Lemma 8 and Remark 10, the Wilson system {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 is a Wilson Bessel sequence with Bessel bound 2(đľ1 + đľ2 ). Corollary 14. Let đ¤0 , đ¤â1 â đż2 (R) be bounded and compactly supported. Then the Wilson system {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 is a Wilson Bessel sequence. Proof. Since, đ¤0 , đ¤â1 â đż2 (R) are bounded and compactly supported, đľ1 and đľ2 as defined in Theorem 13 are both finite, and hence, the Wilson system {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 is a Wilson Bessel sequence. In the following results, we give a sufficient condition for the Wilson system {đđđ : đ¤0 , đ¤â1 â đż2 (R)}đâZ,đâN0 to be a Wilson Bessel sequence in terms of Zak transforms of đ¤0 and đ¤â1 .
đâZ
óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ óľ¨óľ¨ â â óľ¨óľ¨óľ¨ â đ¤0 (đĽ â đ) đ¤0 (đĽ â đ â đ)óľ¨óľ¨óľ¨] > 0, óľ¨óľ¨ óľ¨ đ ≠ 0 óľ¨óľ¨đâZ óľ¨
2
(29)
then the Wilson system {đđđ : đ¤0 â đż2 (R)}đâZ,đâN0 is a Wilson frame with frame bounds đ´/2 and đľ/2. Proof. In view of Theorem 9.1.5 in [9], the Gabor system {đ¸đ đđ đ¤0 }đ,đâZ is a Gabor frame for đż2 (R). If we choose đ¤0 = đ¤â1 in Lemma 9, then óľ¨ óľ¨ óľ¨2 óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ = 2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤0 âŠóľ¨óľ¨óľ¨óľ¨
đâZ đâN0
đ,đâZ
âđ â đż2 (R) .
(30)
Hence, the Wilson system {đđđ : đ¤0 â đż2 (R)}đâZ,đâN0 is a Wilson frame with frame bounds đ´/2 and đľ/2. Corollary 17. Suppose đ¤0 â đż2 (R) has support in an interval of length 1; then the Wilson system {đđđ : đ¤0 â đż2 (R)}đâZ,đâN0 is a Wilson frame for đż2 (R). Proof. Since đ¤0 â đż2 (R) has support in an interval of length 1, we have âđâZ đ¤0 (đĽâđ)đ¤0 (đĽ â đ â đ) = 0, for all đ ≠ 0. Thus, đľ < â, đ´ > 0. Hence, in view of Theorem 16, the Wilson system {đđđ : đ¤0 â đż2 (R)}đâZ,đâN0 is a frame for đż2 (R). The following result gives a class of functions đ¤0 â đż2 (R) for which the associated Wilson system is a Bessel sequence but not a frame.
6
Journal of Function Spaces and Applications
Theorem 18. Suppose đ¤0 â đż2 (R) is a continuous function with compact support. Then the Wilson system {đđđ : đ¤0 â đż2 (R)}đâZ,đâN0 is a Wilson Bessel sequence for đż2 (R) but not a frame. Proof. Since đ¤0 â đż2 (R) is a bounded function with compact support, đľ defined in Theorem 16 is finite, and hence, the Wilson system {đđđ : đ¤0 â đż2 (R}đâZ,đâN0 is a Wilson Bessel sequence for đż2 (R). Moreover, since đ¤0 â đż2 (R) is a continuous function with compact support, in view of corollary 9.7.4 in [9], the Gabor system {đ¸đ đđ đ¤0 }đ,đâZ can never become a frame, and hence, the Wilson system {đđđ : đ¤0 â đż2 (R}đâZ,đâN0 is not a Wilson frame. Finally, we give a necessary and sufficient condition for a Wilson system {đđđ : đ¤0 â đż2 (R}đâZ,đâN0 to be a Wilson frame in terms of the Zak transform of đ¤0 . Theorem 19. The Wilson system {đđđ : đ¤0 â đż2 (R}đâZ,đâN0 is a Wilson frame for đż2 (R) with bounds 2đ´ and 2đľ if and only if đ´ ⤠|đđ¤0 |2 ⤠đľ. Proof. If we choose đ¤0 = đ¤â1 in Lemma 9, then óľ¨ óľ¨ óľ¨2 óľ¨2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đđđ âŠóľ¨óľ¨óľ¨óľ¨ = 2 â óľ¨óľ¨óľ¨óľ¨â¨đ, đ¸đ đđ đ¤0 âŠóľ¨óľ¨óľ¨óľ¨
đâZ đâN0
đ,đâZ
âđ â đż2 (R) .
(31)
Therefore, the Wilson system {đđđ : đ¤0 â đż2 (R}đâZ,đâN0 is a Wilson frame for đż2 (R) with bounds 2đ´ and 2đľ if and only if the Gabor system {đ¸đ đđ đ¤0 }đ,đâZ is a frame for đż2 (R) with bounds đ´ and đľ. Since, the Gabor system {đ¸đ đđ đ¤0 }đ,đâZ is a frame for đż2 (R) with bounds đ´ and đľ if and only if đ´ ⤠|đđ¤0 |2 ⤠đľ, the result follows.
Acknowledgments The authors thank the referee(s) for their useful suggestions and comments towards the improvement of the paper. The research of Suman Panwar is supported by CSIR vide letter no. 09\045(1140)\2011-EMR I dated 16\11\2011.
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[5] R. Balan, P. G. Casazza, C. Heil, and Z. Landau, âDensity, overcompleteness, and localization of frames. II. Gabor systems,â The Journal of Fourier Analysis and Applications, vol. 12, no. 3, pp. 309â344, 2006. [6] P. G. Casazza and G. E. Pfander, âInfinite dimensional restricted invertibility,â Journal of Functional Analysis, vol. 263, no. 12, pp. 3784â3803, 2012. [7] K. Gr¨ochenig, âLocalization of frames, Banach frames, and the invertibility of the frame operator,â The Journal of Fourier Analysis and Applications, vol. 10, no. 2, pp. 105â132, 2004. [8] K. Gr¨ochenig, Foundations of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkh¨auser, Boston, Mass, USA, 2001. [9] O. Christensen, Frames and Bases: An Introductory Course, Applied and Numerical Harmonic Analysis, Birkh¨auser, Boston, Mass, USA, 2008. [10] H. G. Feichtinger and T. Strohmer, Eds., Gabor Analysis and Algorithms: Theory and Applications, Applied and Numerical Harmonic Analysis, Birkh¨auser, Boston, Mass, USA, 1998. [11] H. G. Feichtinger and T. Strohmer, Eds., Advances in Gabor Analysis, Applied and Numerical Harmonic Analysis, Birkh¨auser, Boston, Mass, USA, 2003. [12] T. Strohmer, âApproximation of dual Gabor frames, window decay, and wireless communications,â Applied and Computational Harmonic Analysis, vol. 11, no. 2, pp. 243â262, 2001. [13] D. J. Sullivan, J. J. Rehr, J. W. Wilkins, and K. G. Wilson, Phase Space Warnier Functions in Electronic Structure Calculations, Cornell University, 1987. [14] K. G. Wilson, Generalised Warnier Functions, Cornell university, 1987. [15] I. Daubechies, S. Jaffard, and J.-L. Journ´e, âA simple Wilson orthonormal basis with exponential decay,â SIAM Journal on Mathematical Analysis, vol. 22, no. 2, pp. 554â573, 1991. [16] H. S. Malvar, âLapped transforms for efficient transform/subband coding,â IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 38, no. 6, pp. 969â978, 1990. [17] P. Auscher, G. Weiss, and M. V. Wickerhauser, âLocal sine and cosine bases of Coifman and Meyer and the construction of smooth wavelets,â in Wavelets: A Tutorial in Theory and Applications, C. K. Chui, Ed., vol. 2 of Wavelet Analysis and Its Applications, pp. 237â256, Academic Press, Boston, Mass, USA, 1992. [18] R. R. Coifman and Y. Meyer, âRemarques sur lâanalyse de Fourier a` fenËetre,â Comptes Rendus de lâAcad´emie des Sciences, vol. 312, no. 3, pp. 259â261, 1991. [19] H. G. Feichtinger, K. Gr¨ochenig, and D. Walnut, âWilson bases and modulation spaces,â Mathematische Nachrichten, vol. 155, pp. 7â17, 1992. [20] K. Bittner, âLinear approximation and reproduction of polynomials by Wilson bases,â The Journal of Fourier Analysis and Applications, vol. 8, no. 1, pp. 85â108, 2002. [21] G. Kutyniok and T. Strohmer, âWilson bases for general timefrequency lattices,â SIAM Journal on Mathematical Analysis, vol. 37, no. 3, pp. 685â711, 2005. [22] P. WojdyĹĹo, âModified Wilson orthonormal bases,â Sampling Theory in Signal and Image Processing, vol. 6, no. 2, pp. 223â235, 2007. [23] K. Bittner, âBiorthogonal Wilson bases,â in Wavelet Applications in Signal and Image Processing VII, vol. 3813 of Proceedings of SPIE, pp. 410â421, Denver, Colo, USA, July 1999. [24] S. K. Kaushik and S. Panwar, âA note on Gabor frames,â Communicated.
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