arXiv subjects

Anupam Gumber

Publications and source records attributed to Anupam Gumber.

9 recordsLinked to original sources

arXiv2026

Characterization and Construction of Pairwise Orthogonal Parseval Frames with Applications to Sampling

In this paper, we provide a characterization of pairwise orthogonal frames with generalized translation-invariant (GTI) structures, based on the unconditional convergence property (UCP). These GTI systems are generated by translating functions over a countable family of closed, co-compact subgroups of a locally compact abelian (LCA) group $G$, where the families of subgroups associated with each system may differ. As an application of this characterization, we establish necessary and sufficient criteria for the orthogonality of various structured systems, including Gabor, wavelet, and shearlet systems on LCA groups. Furthermore, we derive a characterization of GTI Parseval (tight) frames and present explicit constructions of pairs of GTI systems using filters. Each constructed system satisfies the $\infty$-UCP and admits a Calder\'on sum equal to one. As a consequence of these results, the constructed systems form Parseval frames and are pairwise orthogonal. The proposed construction improves upon the technique in \cite{RGS} by relaxing the stationary assumption on the families of subgroups. Finally, we illustrate the results with examples using $B$-splines as generating functions and discuss applications of pairwise orthogonal frames in sampling theory.

math.FA↗

arXiv2023

Pseudo-differential calculi and entropy estimates with Orlicz modulation spaces

We deduce continuity properties for pseudo-differential operators with symbols in Orlicz modulation spaces when acting on other Orlicz modulation spaces. In particular we extend well-known results in the literature. We also show that the entropy functional is continuous on a suitable Orlicz modulation space between $M^p$ and $M^2$ when $p<2$, though the functional is discontinuous on $M^2=L^2$.

math.FA↗

arXiv2022

The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators

We study the spectral bounds of self-adjoint operators on the Hilbert space of square-integrable functions, arising from the representation theory of the Heisenberg group. Interestingly, starting either with the von Neumann lattice or the hexagonal lattice of density 2, the spectral bounds obey well-known arithmetic-geometric mean iterations. This follows from connections to Jacobi theta functions and Ramanujan's corresponding theories. As a consequence we re-discover that these operators resemble the identity operator as the density of the lattice grows. We also prove that the conjectural value of Landau's constant is obtained as the cubic arithmetic-geometric mean of $\sqrt[3]{2}$ and 1, which we believe to be a new result.

math.CA↗

arXiv2022

Completeness of Sets of Shifts in Invariant Banach Spaces of Tempered Distributions via Tauberian conditions

The main result of this paper is a far reaching generalization of the completeness result given by V.~Katsnelson in a recent paper [35]. Instead of just using a collection of dilated Gaussians it is shown that the key steps of an earlier paper [27] by the authors, combined with the use of Tauberian conditions (i.e. the non-vanishing of the Fourier transform) allow us to show that the linear span of the translates of a single function $g \in {\boldsymbol{\mathcal {S}({\mathbb{R}}^d)}}$ is a dense subspace of any Banach space satisfying certain double invariance properties. In fact, a much stronger statement is presented: for a given compact subset $M$ in such a Banach space $({\boldsymbol B}, \, \|\,\cdot\,\|_{\boldsymbol B})$ one can construct a finite rank operator, whose range is contained in the linear span of finitely many translates of $g$, and which approximates the identity operator over $M$ up to a given level of precision. The setting of tempered distributions allows to reduce the technical arguments to methods which are widely used in Fourier Analysis. The extension to non-quasi-analytic weights respectively locally compact Abelian groups is left to a forthcoming paper, which will be technically much more involved and uses different ingredients.

math.FA↗

arXiv2022

Fourier characterizations of Pilipovic spaces

Let $f$ be a function or distribution on $\rr d$. We show that $f$ belongs to a certain Pilipovi{ć} space, if and only if $f$ and suitable partial fractional Fourier transforms of $f$ satisfy certain types of estimates.

math.FA↗

arXiv2020

Completeness of shifted dilates in invariant Banach spaces of tempered distributions

We show that well-established methods from the theory of Banach modules and time-frequency analysis allow to derive completeness results for the collection of shifted and dilated version of a given (test) function in a quite general setting. While the basic ideas show strong similarity to the arguments used in a recent paper by V.~Katsnelson we extend his results in several directions, both relaxing the assumptions and widening the range of applications. There is no need for the Banach spaces considered to be embedded into $(L^2(\mathbb{R}), ||\cdot{}||_2)$, nor is the Hilbert space structure relevant. We choose to present the results in the setting of the Euclidean spaces, because then the Schwartz space $\mathcal{S}^{\prime}(\mathbb{R}^d)$ ($d \geq 1$) of tempered distributions provides a well-established environment for mathematical analysis. We also establish connections to modulation spaces and Shubin classes $({Q}_{s}(\mathbb{R}^d), ||\cdot{}||_{Q_s})$, showing that they are special cases of Katsnelson's setting (only) for $s \geq 0$.

math.FA↗

arXiv2020

Translation and modulation invariant Hilbert spaces

We show that for any Hilbert space of distributions on $\textbf{R}^d$ which is translation and modulation invariant, is equal to $L^2(\textbf{R}^d)$, with the same norm apart from a multiplicative constant.

math.FA↗

arXiv2017

Orthogonality and duality of frames over locally compact abelian groups

Motivated by the recent work of Bownik and Ross \cite{BR}, and Jakobsen and Lemvig \cite{JL}, this article generalizes latest results on reproducing formulas for generalized translation invariant (GTI) systems to the setting of super-spaces over a second countable locally compact abelian (LCA) group $G$. To do so, we introduce the notion of a super-GTI system with finite sequences as generators from a super-space $L^2(G) \oplus \cdots \oplus L^2(G) $ ($N$ summands). We characterize the generators of two super-GTI systems in the super-space such that they form a super-dual frame pair. For this, we first give necessary and sufficient conditions for two Bessel families to be orthogonal frames (we call as GTI-orthogonal frame systems) when the Bessel families have the form of GTI systems in $L^2(G)$. As a consequence, we deduce similar results for several function systems including the case of TI systems, and GTI systems on compact abelian groups. As an application, we apply our duality result for super-GTI systems to the Bessel families with a wave-packet structure (combination of wavelet as well as Gabor structure), and hence a characterization for dual super wave-packet systems on LCA groups is obtained. In addition, we relate the well established theory from literature with our results by observing several deductions in context of wavelet and Gabor systems over LCA groups with $G=\mathbb{R}^d,\mathbb{Z}^d$, etc.

math.FA↗

arXiv2015

Orthonormal Wavelet System in $\ell^2 ({\mathbb{Z}}^2_N)$

Using the group theoretic approach based on the set of digits, we first investigate a finite collection of functions in $\ell^2 ({\mathbb{Z}}^2_N)$ that satisfies some localization properties in a region of the time-frequency plane. The digits are associated with an invertible (expansive/non-expansive) matrix having integer entries. Next, we study and characterize an orthonormal wavelet system for $\ell^2 ({\mathbb{Z}}^2_N)$. In addition, some results connecting the uncertainty principle with functions that generate the orthonormal wavelet system having time-frequency localization properties are obtained.

math.FA↗