In this paper, we introduce the notion of generalized multiresolution structure (GMS) in the set-ting of reducing subspaces of L2(Rd). For a general expansive matrix, we obtain a necessary and sufficient condition for...In this paper, we introduce the notion of generalized multiresolution structure (GMS) in the set-ting of reducing subspaces of L2(Rd). For a general expansive matrix, we obtain a necessary and sufficient condition for GMS, and prove the existence of GMS in a reducing subspace. Using GMS, we obtain a pyramid decomposition and a frame-like expansion for signals in reducing subspaces.展开更多
Given L, N, M ∈ N and an NZ-periodic set S in Z, let l2(S) be the closed subspace of l2(Z) consisting of sequences vanishing outside S. For f = { fl : 0≤l≤L-1 }l2(Z), we denote by G(f, N, M) the Gabor system genera...Given L, N, M ∈ N and an NZ-periodic set S in Z, let l2(S) be the closed subspace of l2(Z) consisting of sequences vanishing outside S. For f = { fl : 0≤l≤L-1 }l2(Z), we denote by G(f, N, M) the Gabor system generated by f, and by L(f, N, M) the closed linear subspace generated by G(f, N, M). This paper addresses density results, frame conditions for a Gabor system G(g, N, M) in l2(S), and Gabor duals of the form G(a, N, M) in some L(h, N, M) for a frame G(g, N, M) in l2(S) (so-called oblique duals). We obtain a density theorem for a Gabor system G(g, N, M) in l2(S), and show that such condition is suficient for theexistence of g={XE1:0≤l≤L-1} with G(g,N,m) We characterize g with G(g,N,m) being respectively a frame for L(g,N,m) being a tight frame for l2(S).and G(h, N, M ) in L(h, N, M ), we establish a criterion for the existence of an oblique Gabor dual of g in L(h, N, M), study the uniqueness of oblique Gabor dual, and derive an explicit expression of a class of oblique Gabor duals (among which the one with the smallest norm is obtained). Some examples are also provided.展开更多
Since a frame for a Hilbert space must be a Bessel sequence, many results on(quasi-)affine bi-frame are established under the premise that the corresponding(quasi-)affine systems are Bessel sequences. However,it is ve...Since a frame for a Hilbert space must be a Bessel sequence, many results on(quasi-)affine bi-frame are established under the premise that the corresponding(quasi-)affine systems are Bessel sequences. However,it is very technical to construct a(quasi-)affine Bessel sequence. Motivated by this observation, in this paper we introduce the notion of weak(quasi-)affine bi-frame(W(Q)ABF) in a general reducing subspace for which the Bessel sequence hypothesis is not needed. We obtain a characterization of WABF, and prove the equivalence between WABF and WQABF under a mild condition. This characterization is used to recover some related known results in the literature.展开更多
Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This paper addresses Gabor systems on discrete periodic sets, which can model signals to appear peri...Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This paper addresses Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. Complete Gabor systems and Gabor frames on discrete periodic sets are characterized; a sufficient and necessary condition on what periodic sets admit complete Gabor systems is obtained; this condition is also proved to be sufficient and necessary for the existence of sets E such that the Gabor systems generated by χE are tight frames on these periodic sets; our proof is constructive, and all tight frames of the above form with a special frame bound can be obtained by our method; periodic sets admitting Gabor Riesz bases are characterized; some examples are also provided to illustrate the general theory.展开更多
基金supported by Beijing Natural Science Foundation (Grant No. 1122008)the Scientific Research Common Program of Beijing Municipal Commission of Education (Grant No.KM201110005030)
文摘In this paper, we introduce the notion of generalized multiresolution structure (GMS) in the set-ting of reducing subspaces of L2(Rd). For a general expansive matrix, we obtain a necessary and sufficient condition for GMS, and prove the existence of GMS in a reducing subspace. Using GMS, we obtain a pyramid decomposition and a frame-like expansion for signals in reducing subspaces.
基金supported by National Natural Science Foundation of China (Grant Nos. 10901013, 10671008)Beijing Natural Science Foundation (Grant No. 1092001)+1 种基金the Scientific Research Common Program of Beijing Municipal Commission of Education (Grant No. KM201110005030)the Project Sponsored by SRF for ROCS, SEM of China
文摘Given L, N, M ∈ N and an NZ-periodic set S in Z, let l2(S) be the closed subspace of l2(Z) consisting of sequences vanishing outside S. For f = { fl : 0≤l≤L-1 }l2(Z), we denote by G(f, N, M) the Gabor system generated by f, and by L(f, N, M) the closed linear subspace generated by G(f, N, M). This paper addresses density results, frame conditions for a Gabor system G(g, N, M) in l2(S), and Gabor duals of the form G(a, N, M) in some L(h, N, M) for a frame G(g, N, M) in l2(S) (so-called oblique duals). We obtain a density theorem for a Gabor system G(g, N, M) in l2(S), and show that such condition is suficient for theexistence of g={XE1:0≤l≤L-1} with G(g,N,m) We characterize g with G(g,N,m) being respectively a frame for L(g,N,m) being a tight frame for l2(S).and G(h, N, M ) in L(h, N, M ), we establish a criterion for the existence of an oblique Gabor dual of g in L(h, N, M), study the uniqueness of oblique Gabor dual, and derive an explicit expression of a class of oblique Gabor duals (among which the one with the smallest norm is obtained). Some examples are also provided.
基金supported by National Natural Science Foundation of China(Grant No.11271037)Beijing Natural Science Foundation(Grant No.1122008)
文摘Since a frame for a Hilbert space must be a Bessel sequence, many results on(quasi-)affine bi-frame are established under the premise that the corresponding(quasi-)affine systems are Bessel sequences. However,it is very technical to construct a(quasi-)affine Bessel sequence. Motivated by this observation, in this paper we introduce the notion of weak(quasi-)affine bi-frame(W(Q)ABF) in a general reducing subspace for which the Bessel sequence hypothesis is not needed. We obtain a characterization of WABF, and prove the equivalence between WABF and WQABF under a mild condition. This characterization is used to recover some related known results in the literature.
基金supported by National Natural Science Foundation of China (Grant No. 10671008)Beijing Natural Science Foundation (Grant No. 1092001)PHR (IHLB) and the project sponsored by SRF for ROCS,SEM of China
文摘Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This paper addresses Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. Complete Gabor systems and Gabor frames on discrete periodic sets are characterized; a sufficient and necessary condition on what periodic sets admit complete Gabor systems is obtained; this condition is also proved to be sufficient and necessary for the existence of sets E such that the Gabor systems generated by χE are tight frames on these periodic sets; our proof is constructive, and all tight frames of the above form with a special frame bound can be obtained by our method; periodic sets admitting Gabor Riesz bases are characterized; some examples are also provided to illustrate the general theory.
