Phase-Retrieval in Shift-Invariant Spaces with Gaussian Generator

Gröchenig, Karlheinz

doi:10.1007/s00041-020-09755-5

Cited by 21 publications

(16 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…See [2] for motivation for sign retrieval in shift-invariant spaces. When g is a Gaussian function-corresponding to m = 0 in (1)-Theorem 1 was recently obtained in [3,Theorem 1]. Here, that result is extended to all totally positive functions of Gaussian type.…”

Section: Theorem 1 (Sign Retrieval) Let G Be a Totally Positive Functmentioning

confidence: 87%

“…For example, 1 := ∩ (−∞, 0] and 2 := ∩ (0, ∞) have always zero lower Beurling density. The proof of the sign retrieval theorem for Gaussian generators in [3] resorts instead to a special property of the Gaussian function, namely that…”

Section: Theorem 1 (Sign Retrieval) Let G Be a Totally Positive Functmentioning

confidence: 99%

“…In contrast to[3, Theorem 1], in Theorem 1 it is not assumed that is separated, or even relatively separated. See[3] for an example of a separated set with D − ( ) < 2 for which sign retrieval fails in a shift-invariant space with Gaussian generator.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Sign Retrieval in Shift-Invariant Spaces with Totally Positive Generator

Romero

2021

J Fourier Anal Appl

View full text Add to dashboard Cite

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].

show abstract

Section: Theorem 1 (Sign Retrieval) Let G Be a Totally Positive Functmentioning

confidence: 87%

Section: Theorem 1 (Sign Retrieval) Let G Be a Totally Positive Functmentioning

confidence: 99%

See 1 more Smart Citation

Sign Retrieval in Shift-Invariant Spaces with Totally Positive Generator

Romero

2021

J Fourier Anal Appl

View full text Add to dashboard Cite

show abstract

“…Analogously, replacing w by w −1 and using the estimate (7) for G − , we conclude that H(w) → 0 as w → 0. Consequently the singularity of H at 0 is removable, and H is thus entire and bounded.…”

Section: Proof Of Theorem 13: Sufficiencymentioning

confidence: 56%

Complete interpolating sequences for the Gaussian shift-invariant space

Baranov¹,

Belov²,

Gröchenig³

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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

show abstract

“…Since the sinc function has infinite support and slow decay, the space of bandlimited functions is often unsuitable for numerical implementations. Retaining some of the simplicity and structure of bandlimited models, sampling in non-bandlimited shift-invariant spaces is more amenable and realistic for many applications [2][3][4][5][6][7][8][9][10][11][12][13]. Sampling and reconstruction of signals in a shift-invariant space…”

Section: Introductionmentioning

confidence: 99%

Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space

Fan

Tang

2021

J Inequal Appl

View full text Add to dashboard Cite

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.

show abstract

Phase-Retrieval in Shift-Invariant Spaces with Gaussian Generator

Cited by 21 publications

References 20 publications

Sign Retrieval in Shift-Invariant Spaces with Totally Positive Generator

Sign Retrieval in Shift-Invariant Spaces with Totally Positive Generator

Complete interpolating sequences for the Gaussian shift-invariant space

Asymptotic pointwise error estimates for reconstructing shift-invariant signals with generators in a hybrid-norm space

Contact Info

Product

Resources

About