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Convolution, average sampling and a Calderon resolution of the identity for shiftinvariant spaces
 J. Fourier Anal. Appl
"... Abstract. In this paper, we study three interconnected inverse problems in shift invariant spaces: 1) the convolution/deconvolution problem; 2) the uniformly sampled convolution and the reconstruction problem; 3) the sampled convolution followed by sampling on irregular grid and the reconstruction p ..."
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Cited by 22 (14 self)
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Abstract. In this paper, we study three interconnected inverse problems in shift invariant spaces: 1) the convolution/deconvolution problem; 2) the uniformly sampled convolution and the reconstruction problem; 3) the sampled convolution followed by sampling on irregular grid and the reconstruction problem. In all three cases, we study both the stable reconstruction as well as illposed reconstruction problems. We characterize the convolutors for stable deconvolution as well as those giving rise to illposed deconvolution. We also characterize the convolutors that allow stable reconstruction as well as those giving rise to illposed reconstruction from uniform sampling. The connection between stable deconvolution, and stable reconstruction from samples after convolution is subtle, as will be demonstrated by several examples and theorems that relate the two problems. 1.
Frames in spaces with finite rate of innovations
 Adv. Comput. Math
"... Abstract. Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space Vq(Φ, Λ) modelling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applic ..."
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Cited by 20 (14 self)
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Abstract. Signals with finite rate of innovation are those signals having finite degrees of freedom per unit of time that specify them. In this paper, we introduce a prototypical space Vq(Φ, Λ) modelling signals with finite rate of innovation, such as stream of (different) pulses found in GPS applications, cellular radio and ultra wideband communication. In particular, the space Vq(Φ, Λ) is generated by a family of welllocalized molecules Φ of similar size located on a relativelyseparated set Λ using ℓ q coefficients, and hence is locally finitelygenerated. Moreover that space Vq(Φ, Λ) includes finitelygenerated shiftinvariant spaces, spaces of nonuniform splines, and the twisted shiftinvariant space in Gabor (Wilson) system as its special cases. Use the welllocalization property of the generator Φ, we show that if the generator Φ is a frame for the space V2(Φ, Λ) and has polynomial (subexponential) decay, then its canonical dual (tight) frame has the same polynomial (subexponential) decay. We apply the above result about the canonical dual frame to the study of the Banach frame property of the generator Φ for the space Vq(Φ, Λ) with q = 2, and of the polynomial (subexponential) decay property of the mask associated with a refinable function that has polynomial (subexponential) decay. Advances in Computational Mathematics, to appear 1.
Convergence of cascade algorithms and smoothness of refinable distributions
 Chinese Ann. Math. Ser. B
"... In this paper, we at first develop a method to study convergence of the cascade algorithm in a Banach space without stable assumption on the initial (Theorem 2.7). Then we apply the previous result on the convergence to characterize compactly supported refinable distributions in fractional Sobolev s ..."
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Cited by 11 (3 self)
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In this paper, we at first develop a method to study convergence of the cascade algorithm in a Banach space without stable assumption on the initial (Theorem 2.7). Then we apply the previous result on the convergence to characterize compactly supported refinable distributions in fractional Sobolev spaces and Hölder continuous spaces (Theorems 3.1, 3.8, and 3.9). Finally we apply the above characterization to choose appropriate initial to guarantee the convergence of the cascade algorithm (Theorem 4.3).
Connection between pframes and pRiesz bases in locally finite SIS of ...
"... Let 1 p 1 and \Phi = (OE 1 ; : : : ; OE r ) T be a vectorvalued compactly supported L p function on R d . Define V p (\Phi) = n P r i=1 P j2Z d d i (j)OE i (\Delta \Gamma j) : (d i (j)) j2Z d 2 ` p ; 1 i r o : In this paper, we consider the pframe property of the space V p (\Phi) ..."
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Cited by 2 (1 self)
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Let 1 p 1 and \Phi = (OE 1 ; : : : ; OE r ) T be a vectorvalued compactly supported L p function on R d . Define V p (\Phi) = n P r i=1 P j2Z d d i (j)OE i (\Delta \Gamma j) : (d i (j)) j2Z d 2 ` p ; 1 i r o : In this paper, we consider the pframe property of the space V p (\Phi) with \Phi being compactly supported function in L p " L p=(p\Gamma1) . Moreover, for the onedimensional case, we show that if fOE i (\Delta \Gamma j) : 1 i r; j 2 Zg is a pframe for V p (\Phi), then there exist a positive integer s r and compactly supported functions / 1 ; : : : ; / s 2 V p (\Phi) such that f/ i (\Delta \Gamma j) : 1 i s; j 2 Zg is a pRiesz basis of V p (\Phi), where \Psi = (/ 1 ; : : : ; / s ) T .
Localization of stability and pframe in the Fourier domain
 In Proceeding of Special Session on Wavelets, Frames and Operator Theory
"... Abstract. In this paper, we introduce and study the localization of stability and pframe properties of a finitely generated shiftinvariant system in the Fourier domain, and then provide more information to that shift invariant system. Especially for a shiftinvariant system generated by finitely man ..."
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Abstract. In this paper, we introduce and study the localization of stability and pframe properties of a finitely generated shiftinvariant system in the Fourier domain, and then provide more information to that shift invariant system. Especially for a shiftinvariant system generated by finitely many compactly supported functions, we show that it has pframe property at almost all frequencies, and that it either has stability property at almost all frequencies or does not have stability property at all frequencies. 1.
Localization of Calderon Convolution in the Fourier Domain
"... Abstract. In this paper, we introduce and study the localization of Calderon convolution for a finitely generated shiftinvariant space in the Fourier domain. We say that a linear space V of functions on R d is shiftinvariant if f ∈ V implies that f( · − k) ∈ V for all k ∈ Z d. For instance, the ..."
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Abstract. In this paper, we introduce and study the localization of Calderon convolution for a finitely generated shiftinvariant space in the Fourier domain. We say that a linear space V of functions on R d is shiftinvariant if f ∈ V implies that f( · − k) ∈ V for all k ∈ Z d. For instance, the space of all polynomials of degree at most N, the space of all pintegrable functions, and the space of all bandlimited functions in L 2 are shiftinvariant spaces.
Connection between pframes and pRiesz bases in locally finite SIS of LP(R)
"... Let 1 p 00 and = (c5i..., q)T be a vectorvalued compactly supported L function on Rd. Define V() = { jEZd d(j)(. — j) : (dj(j))ezd E £, 1 i r}. In this paper, we consider the pframe property of the space V() with 4 being compactly supported function in L fl L/(1). Moreover, for the onedimensio ..."
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Let 1 p 00 and = (c5i..., q)T be a vectorvalued compactly supported L function on Rd. Define V() = { jEZd d(j)(. — j) : (dj(j))ezd E £, 1 i r}. In this paper, we consider the pframe property of the space V() with 4 being compactly supported function in L fl L/(1). Moreover, for the onedimensional case, we show that if {çb(. —j) : 1 < i < r,j Z} is a pframe for V(), then there exist a positive integer s r and compactly supported functions..., V() such that {j5(. — j) : 1 i ç, j e Z} is a pRiesz basis of where 'I ' = ('L,..., b3)T.