Tight Gabor Sets on Discrete Periodic Sets
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Tight Gabor Sets on Discrete Periodic Sets Yun-Zhang Li · Qiao-Fang Lian
Received: 30 September 2008 / Accepted: 15 December 2008 / Published online: 6 January 2009 © Springer Science+Business Media B.V. 2008
Abstract This paper addresses Gabor analysis on a discrete periodic set. Such a scenario can potentially find its applications in signal processing where signals may present on a union of disconnected discrete index sets. We focus on the Gabor systems generated by characteristic functions. A sufficient and necessary condition for a set to be a tight Gabor set in discrete periodic sets is obtained; discrete periodic sets admitting a tight Gabor set are also characterized; the perturbation of tight Gabor sets is investigated; an algorithm to determine whether a set is a tight Gabor set is presented. Furthermore, we prove that an arbitrary Gabor frame set can be represented as the union of a tight Gabor set and a Gabor Bessel set. Keywords Discrete periodic set · Gabor system · Gabor Bessel set · Gabor frame set · Tight Gabor set Mathematics Subject Classification (2000) 42C40
1 Introduction Before proceeding, we first introduce some notations and notions. We denote by Z, N and l 2 (Z) the set of integers, the set of positive integers and the Hilbert space of squaresummable sequences on Z, respectively. Given a set E ⊂ Z, χE denotes the characteristic
Supported by the National Natural Science Foundation of China (Grant No. 10671008), PHR (IHLB), the Project-sponsored by SRF for ROCS, SEM of China. Y.-Z. Li College of Applied Sciences, Beijing University of Technology, Beijing 100124, China e-mail: [email protected] Q.-F. Lian () Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China e-mail: [email protected]
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function of E. Given M, N ∈ N, and g, γ ∈ l 2 (Z). Write NM = {0, 1, . . . , M − 1}, and define the modulation operator E Mm with m ∈ NM and translation operator TnN with n ∈ Z by m
E Mm f (·) := e2π i M · f (·),
TnN f (·) := f (· − nN )
for f ∈ l 2 (Z), respectively. We denote by G (g, N, M) the Gabor system generated by g:
G (g, N, M) := E Mm TnN g : m ∈ NM , n ∈ Z .
(1)
When G (g, N, M) and G (γ , N, M) are both Bessel sequences in l 2 (Z), define the operator Sγ , g by Sγ , g f :=
M−1 f, E Mm TnN γ E Mm TnN g n∈Z m=0
for f ∈ l (Z). For a nonempty subset S of Z, we denote by l 2 (S) the closed subspace of l 2 (Z): 2
/ S}. l 2 (S) := {f ∈ l 2 (Z) : f (j ) = 0 for j ∈
(2)
Clearly, it is a Hilbert space with the inner product in l 2 (Z). A nonempty subset S of Z is said to be N Z-periodic if j + nN ∈ S for j ∈ S and n ∈ Z. Two subsets S1 and S2 of Z are called N Z-congruent if there exist partitions {S1, k }k∈Z of S1 and {S2, k }k∈Z of S2 such that S2, k = S1, k + kN for all k ∈ Z. In this paper, we focus on Gabor analysis in l 2 (S), where S is an N Z-periodic set in Z. The following proposition shows that only periodic S can be suitable for discrete Gabor analysis. Proposition 1 Given N , M ∈ N,and a nonempty set S in Z. Let g ∈ l
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