Search citation statements
Paper Sections
Citation Types
Year Published
Publication Types
Relationship
Authors
Journals
We show that a real-valued function f in the shift-invariant space generated by a totally positive function of Gaussian type is uniquely determined, up to a sign, by its absolute values $$\{|f(\lambda )|: \lambda \in \Lambda \}$$ { | f ( λ ) | : λ ∈ Λ } on any set $$\Lambda \subseteq {\mathbb {R}}$$ Λ ⊆ R with lower Beurling density $$D^{-}(\Lambda )>2$$ D - ( Λ ) > 2 .We consider a totally positive function of Gaussian type, i.e., a function $$g \in L^2({\mathbb {R}})$$ g ∈ L 2 ( R ) whose Fourier transform factors as $$\begin{aligned} \hat{g}(\xi )= \int _{{\mathbb {R}}} g(x) e^{-2\pi i x \xi } dx = C_0 e^{- \gamma \xi ^2}\prod _{\nu =1}^m (1+2\pi i\delta _\nu \xi )^{-1}, \quad \xi \in {\mathbb {R}}, \end{aligned}$$ g ^ ( ξ ) = ∫ R g ( x ) e - 2 π i x ξ d x = C 0 e - γ ξ 2 ∏ ν = 1 m ( 1 + 2 π i δ ν ξ ) - 1 , ξ ∈ R , with $$\delta _1,\ldots ,\delta _m\in {\mathbb {R}}, C_0, \gamma >0, m \in {\mathbb {N}} \cup \{0\}$$ δ 1 , … , δ m ∈ R , C 0 , γ > 0 , m ∈ N ∪ { 0 } , and the shift-invariant space$$\begin{aligned} V^\infty (g) = \Big \{ f=\sum _{k \in {\mathbb {Z}}} c_k\, g(\cdot -k): c \in \ell ^\infty ({\mathbb {Z}}) \Big \}, \end{aligned}$$ V ∞ ( g ) = { f = ∑ k ∈ Z c k g ( · - k ) : c ∈ ℓ ∞ ( Z ) } , generated by its integer shifts within $$L^\infty ({\mathbb {R}})$$ L ∞ ( R ) . As a consequence of (1), each $$f \in V^\infty (g)$$ f ∈ V ∞ ( g ) is continuous, the defining series converges unconditionally in the weak$$^*$$ ∗ topology of $$L^\infty $$ L ∞ , and the coefficients $$c_k$$ c k are unique [6, Theorem 3.5].
We show that a real-valued function f in the shift-invariant space generated by a totally positive function of Gaussian type is uniquely determined, up to a sign, by its absolute values $$\{|f(\lambda )|: \lambda \in \Lambda \}$$ { | f ( λ ) | : λ ∈ Λ } on any set $$\Lambda \subseteq {\mathbb {R}}$$ Λ ⊆ R with lower Beurling density $$D^{-}(\Lambda )>2$$ D - ( Λ ) > 2 .We consider a totally positive function of Gaussian type, i.e., a function $$g \in L^2({\mathbb {R}})$$ g ∈ L 2 ( R ) whose Fourier transform factors as $$\begin{aligned} \hat{g}(\xi )= \int _{{\mathbb {R}}} g(x) e^{-2\pi i x \xi } dx = C_0 e^{- \gamma \xi ^2}\prod _{\nu =1}^m (1+2\pi i\delta _\nu \xi )^{-1}, \quad \xi \in {\mathbb {R}}, \end{aligned}$$ g ^ ( ξ ) = ∫ R g ( x ) e - 2 π i x ξ d x = C 0 e - γ ξ 2 ∏ ν = 1 m ( 1 + 2 π i δ ν ξ ) - 1 , ξ ∈ R , with $$\delta _1,\ldots ,\delta _m\in {\mathbb {R}}, C_0, \gamma >0, m \in {\mathbb {N}} \cup \{0\}$$ δ 1 , … , δ m ∈ R , C 0 , γ > 0 , m ∈ N ∪ { 0 } , and the shift-invariant space$$\begin{aligned} V^\infty (g) = \Big \{ f=\sum _{k \in {\mathbb {Z}}} c_k\, g(\cdot -k): c \in \ell ^\infty ({\mathbb {Z}}) \Big \}, \end{aligned}$$ V ∞ ( g ) = { f = ∑ k ∈ Z c k g ( · - k ) : c ∈ ℓ ∞ ( Z ) } , generated by its integer shifts within $$L^\infty ({\mathbb {R}})$$ L ∞ ( R ) . As a consequence of (1), each $$f \in V^\infty (g)$$ f ∈ V ∞ ( g ) is continuous, the defining series converges unconditionally in the weak$$^*$$ ∗ topology of $$L^\infty $$ L ∞ , and the coefficients $$c_k$$ c k are unique [6, Theorem 3.5].
We give a full description of complete interpolating sequences for the shiftinvariant space generated by the Gaussian. As a consequence, we rederive the known density conditions for sampling and interpolation. Main resultsConsider the shift-invariant space of functions on R with Gaussian generator g(z) = e −az 2 for a > 0 defined asWe consider the space V 2 as a subspace of L 2 (R) with the usual L 2 -norm.The space V 2 belongs to the general family of shift-invariant spaces. Given a generator g ∈ L 2 (R), such a space is defined asThe primary example is the classical Paley-Wiener space P W = {f ∈ L 2 (R) : supp f ⊆ [−1/2, 1/2]}, which, by the sampling theorem of Shannon-Whittaker-Kotelnikov, can be identified with the shift-invariant space V 2 ( sin πx πx ). In signal processing [1] shift-invariant spaces are often taken as a substitute for the Paley-Wiener space. A unifying feature of both V 2 (Gaussian generator) and P W (sinc-generator) is the fact that both spaces can be viewed as spaces of entire functions by interpreting the variable z to be in C.It is easy to see that the norm equivalence f L 2 (R) ≍ (c n ) ℓ 2 (Z) holds on V 2 with some absolute constants. In what follows it will be convenient for us to work with the second quantity and so we put f V 2 := (c n ) ℓ 2 (Z) , and we often identify V 2 with ℓ 2 (Z) via the mapping (c n ) → f (z) = n∈Z c n e −(z−n) 2 .A sequence Λ = {λ n } n∈Z ⊂ R is said to be sampling for V 2 , if
Sampling and reconstruction of signals in a shift-invariant space are generally studied under the requirement that the generator is in a stronger Wiener amalgam space, and the error estimates are usually given in the sense of $L_{p,{1 / \omega }}$ L p , 1 / ω -norm. Since we often need to reflect the local characteristics of reconstructing error, the asymptotic pointwise error estimates for nonuniform and average sampling in a non-decaying shift-invariant space are discussed under the assumption that the generator is in a hybrid-norm space. Based on Lemma 2.1–Lemma 2.6, we first rewrite the iterative reconstruction algorithms for two kinds of average sampling functionals and prove their convergence. Then, the asymptotic pointwise error estimates are presented for two algorithms under the case that the average samples are corrupted by noise.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
customersupport@researchsolutions.com
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.