Three Problems on Exponential Bases

Carli, Laura De; Mizrahi, Alberto; Tepper, Alexander

doi:10.4153/cmb-2018-015-6

Cited by 3 publications

(4 citation statements)

References 25 publications

(25 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…To prove the orthogonality, one must first show the existence of a matrix C such that for all n ∈ Z d As mentioned earlier, the exponential completeness of a set does not imply the completeness of the set in general. For a recent developments in study of the completeness, frame and Riesz bases properties of exponentials we refer the reader to the paper by De Carli and her co-authors [2].…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

Non-separable Lattices, Gabor Orthonormal Bases and Tilings

Lai

Mayeli

2019

J Fourier Anal Appl

View full text Add to dashboard Cite

Let K ⊂ R d be a bounded set with positive Lebesgue measure. Let Λ = M (Z 2d ) be a lattice in R 2d with density dens(Λ) = 1. It is well-known that if M is a diagonal block matrix with diagonal matrices A and B, then G(|K| −1/2 χK , Λ) is an orthonormal basis for L 2 (R d ) if and only if K tiles both by A(Z d ) and B −t (Z d ). However, there has not been any intensive study when M is not a diagonal matrix. We investigate this problem for a large class of important cases of M . In particular, if M is any lower block triangular matrix with diagonal matrices A and B, we prove that if G(|K| −1/2 χK , Λ) is an orthonormal basis, then K can be written as a finite union of fundamental domains of A(Z d ) and at the same time, as a finite union of fundamental domains of B −t (Z d ). If A t B is an integer matrix, then there is only one common fundamental domain, which means K tiles by a lattice and is spectral. However, surprisingly, we will also illustrate by an example that a union of more than one fundamental domain is also possible. We also provide a constructive way for forming a Gabor window function for a given upper triangular lattice. Our study is related to a Fuglede's type problem in Gabor setting and we give a partial answer to this problem in the case of lattices.

show abstract

Section: Discussion and Open Problemsmentioning

confidence: 99%

Non-separable Lattices, Gabor Orthonormal Bases and Tilings

Lai

Mayeli

2019

J Fourier Anal Appl

View full text Add to dashboard Cite

show abstract

“…It has been recently proved in [15] that convex sets tile by translations if and only if they have an exponential basis. In [6], it is proved that the set is an exponential basis on a domain of measure if and only if D tiles . Furthermore, is orthonormal for .…”

Section: Introductionmentioning

confidence: 99%

“…The aforementioned results in [6] are related to Theorem 1 in [9], where it is proved that if a set is an orthonormal basis on a domain , then is periodic, i.e., for some .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Examples of exponential bases on union of intervals

Oleg¹,

Drezels²

2023

Can. Math. Bull.

View full text Add to dashboard Cite

In this paper, we construct explicit exponential bases of unions of segments of total measure one. Our construction applies to finite or infinite unions of segments, with some conditions on the gaps between them. We also construct exponential bases on finite or infinite unions of cubes in $\mathbb {R}^d$ and prove a stability result for unions of segments that generalize Kadec’s $\frac 14$ -theorem.

show abstract

Recombination of stable sampling sets and stable interpolation sets in functional quasinormed spaces

López Nicolás

2024

B Soc Paran Mat

View full text Add to dashboard Cite

This contribution is aimed in obtaining new results in combining stable sampling sets (respectively, stable interpolation sets) of a given quasinormed space in order to obtain other new ones. We apply these results to Paley-Wiener spaces and. In addition, we study the problem of obtaining a generator system of a given quasinormed space, and obtain conditions for a finite product of subsets of a given quasinormed space to be a generator system, using the interpolation and sampling theory for quasinormed spaces of functions.

show abstract

Three Problems on Exponential Bases

Cited by 3 publications

References 25 publications

Non-separable Lattices, Gabor Orthonormal Bases and Tilings

Non-separable Lattices, Gabor Orthonormal Bases and Tilings

Examples of exponential bases on union of intervals

Recombination of stable sampling sets and stable interpolation sets in functional quasinormed spaces

Contact Info

Product

Resources

About