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Anirudha Poria

Publications and source records attributed to Anirudha Poria.

At least 19 recordsLinked to original sources
arXiv2026

Sampling and Density Theorems for the Fractional Opdam-Cherednik Transform

In this paper, we establish sampling and density results for the fractional Opdam-Cherednik transform. Using the Sturm-Liouville structure of Jacobi functions, we construct a Riesz basis associated with the Opdam-Cherednik kernel and derive an explicit sampling formula. The sampling nodes are determined by the zeros of a shifted Jacobi function. We further develop a concentration-operator approach and obtain Landau-type necessary density conditions for sampling and interpolation in fractional Opdam-Cherednik bandlimited spaces. The non-fractional Opdam-Cherednik sampling formula is recovered when $θ=π/2$.

math.FA↗
arXiv2026

Localization operators on discrete Orlicz modulation spaces

In this paper, we introduce Orlicz spaces on $ \mathbb Z^n \times \mathbb T^n $ and Orlicz modulation spaces on $\mathbb Z^n$, and study inclusion relations, convolution relations, and duality of these spaces. We show that the Orlicz modulation space $M^Φ(\mathbb Z^n)$ is close to the modulation space $M^{2}(\mathbb Z^n)$ for some particular Young function $Φ$. Then, we study localization operators on $\mathbb Z^n$. In particular, using appropriate classes for symbols, we prove that these operators are bounded on Orlicz modulation spaces on $\mathbb Z^n$, compact and in the Schatten--von Neumann classes.

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arXiv2025

The heat semigroup associated with the Jacobi--Cherednik operator and its applications

In this paper, we study the heat equation associated with the Jacobi--Cherednik operator on the real line. We establish some basic properties of the Jacobi--Cherednik heat kernel and heat semigroup. We also provide a solution to the Cauchy problem for the Jacobi--Cherednik heat operator and prove that the heat kernel is strictly positive. Then, we characterize the image of the space $L^2(\mathbb R, A_{α, β})$ under the Jacobi--Cherednik heat semigroup as a reproducing kernel Hilbert space. As an application, we solve the modified Poisson equation and present the Jacobi--Cherednik--Markov processes.

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arXiv2023

Uncertainty principles for the short-time Fourier transform on the lattice

In this paper, we study a few versions of the uncertainty principle for the short-time Fourier transform on the lattice $\mathbb Z^n \times \mathbb T^n$. In particular, we establish the uncertainty principle for orthonormal sequences, Donoho--Stark's uncertainty principle, Benedicks-type uncertainty principle, Heisenberg-type uncertainty principle and local uncertainty inequality for this transform on $\mathbb Z^n \times \mathbb T^n$. Also, we obtain the Heisenberg-type uncertainty inequality using the $k$-entropy of the short-time Fourier transform on $\mathbb Z^n \times \mathbb T^n$.

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arXiv2023

Uncertainty principles for the windowed Opdam--Cherednik transform

In this paper, we study a few versions of the uncertainty principle for the windowed Opdam--Cherednik transform. In particular, we establish the uncertainty principle for orthonormal sequences, Donoho--Stark's uncertainty principle, Benedicks-type uncertainty principle, Heisenberg-type uncertainty principle and local uncertainty inequality for this transform. We also obtain the Heisenberg-type uncertainty inequality using the $k$-entropy of the windowed Opdam--Cherednik transform.

math.FA↗
arXiv2023

Localization Operators On Discrete Modulation Spaces

In this paper, we study a class of pseudo-differential operators known as time-frequency localization operators on $\mathbb Z^n$, which depend on a symbol $ς$ and two windows functions $g_1$ and $g_2$. We define the short-time Fourier transform on $ \mathbb Z^n \times \mathbb T^n $ and modulation spaces on $\mathbb Z^n$, and present some basic properties. Then, we use modulation spaces on $\mathbb Z^n \times \mathbb T^n$ as appropriate classes for symbols, and study the boundedness and compactness of the localization operators on modulation spaces on $\mathbb Z^n$. Then, we show that these operators are in the Schatten--von Neumann class. Also, we obtain the relation between the Landau--Pollak--Slepian type operator and the localization operator on $\mathbb Z^n$. Finally, under suitable conditions on the symbols, we prove that the localization operators are paracommutators, paraproducts and Fourier multipliers.

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arXiv2023

Localization operators associated with the windowed Opdam-Cherednik transform on modulation spaces

In this paper, we study a class of pseudodifferential operators known as time-frequency localization operators, which depend on a symbol $ς$ and two windows functions $g_1$ and $g_2$. We first present some basic properties of the windowed Opdam-Cherednik transform. Then, we use modulation spaces associated with the Opdam-Cherednik transform as appropriate classes for symbols and windows, and study the boundedness and compactness of the localization operators associated with the windowed Opdam-Cherednik transform on modulation spaces. Finally, we show that these operators are in the Schatten-von Neumann class.

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arXiv2022

Weighted Norm Inequalities for the Opdam--Cherednik Transform

In this paper, we study several weighted norm inequalities for the Opdam--Cherednik transform. We establish different versions of the Heisenberg--Pauli--Weyl inequality for this transform. In particular, we give an extension of this inequality using weights with different exponents and present a variation of the inequality that incorporates $L^p$-norms for the Opdam--Cherednik transform. Also, we prove a version of the Hardy--Littlewood inequality for this transform. Finally, we give other variations of the Heisenberg--Pauli--Weyl inequality such as the Nash-type and Clarkson-type inequalities for the Opdam--Cherednik transform.

math.FA↗
arXiv2022

Qualitative uncertainty principles for the windowed Opdam--Cherednik transform on weighted modulation spaces

The aim of this paper is to establish a few qualitative uncertainty principles for the windowed Opdam-Cherednik transform on weighted modulation spaces associated with this transform. In particular, we obtain the Cowling-Price's, Hardy's and Morgan's uncertainty principles for this transform on weighted modulation spaces. The proofs of the results are based on versions of the Phragm{é}n-Lindl{ö}f type result for several complex variables on weighted modulation spaces and the properties of the Gaussian kernel associated with the Jacobi-Cherednik operator.

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arXiv2022

Hausdorff operators associated with the Opdam--Cherednik transform in Lebesgue spaces

In this paper, we introduce the Hausdorff operator associated with the Opdam--Cherednik transform and study the boundedness of this operator in various Lebesgue spaces. In particular, we prove the boundedness of the Hausdorff operator in Lebesgue spaces, in grand Lebesgue spaces, and in quasi-Banach spaces that are associated with the Opdam--Cherednik transform. Also, we give necessary and sufficient conditions for the boundedness of the Hausdorff operator in these spaces.

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arXiv2020

Semi-continuous g-frames in Hilbert spaces

In this paper, we introduce the concept of semi-continuous $g$-frames in Hilbert spaces. We first construct an example of semi-continuous $g$-frames using the Fourier transform of the Heisenberg group and study the structure of such frames. Then, as an application we provide some fundamental identities and inequalities for semi-continuous $g$-frames. Finally, we present a classical perturbation result and prove that semi-continuous $g$-frames are stable under small perturbations.

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arXiv2020

Uncertainty principles for the Opdam-Cherednik transform on modulation spaces

In this paper, we establish the Cowling--Price's, Hardy's and Morgan's uncertainty principles for the Opdam-Cherednik transform on modulation spaces associated with this transform. The proofs of the theorems are based on the properties of the heat kernel associated with the Jacobi-Cherednik operator and the versions of the Phragm{é}n-Lindl{ö}f type result for the modulation spaces.

math.FA↗
arXiv2020

Positive Weight Function and Classification of g-Frames

Given a positive weight function and an isometry map on a Hilbert spaces $\mathcal{H}$, we study a class of linear maps which is a $g$-frame, $g$-Riesz basis and a $g$-orthonormal basis for $\mathcal{H}$ with respect to $\mathbb{C}$ in terms of the weight function. We apply our results to study the frame for shift-invariant subspaces on the Heisenberg group.

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arXiv2020

Hilbert space valued Gabor frames in weighted amalgam spaces

Let $\mathbb{H}$ be a separable Hilbert space. In this paper we establish a generalization of Walnut's representation and Janssen's representation of the $\mathbb{H}-$valued Gabor frame operator on $\mathbb{H}-$valued weighted amalgam spaces $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$. Also we show that the frame operator is invertible on $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty$, if the window function is in the Wiener amalgam space $W_{\mathbb{H}}(L^{\infty},L^1_w)$. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on $W_{\mathbb{H}}(L^p,L^q_v)$, $1 \leq p, q \leq \infty,$ as a special case by choosing the appropriate Hilbert space $\mathbb{H}$.

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arXiv2018

On a problem by Hans Feichtinger

In this paper, we solve a spectral problem about positive semi-definite trace-class pseudodifferential operators on modulation spaces which was posed by H. Feichtinger. Later, C. Heil and D. Larson rephrased the problem in the broader setting of positive semi-definite trace-class operators on a separable Hilbert space. Our solution consists in constructing a counterexample that solves Hans Feichtinger's problem by first solving this second problem.

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arXiv2017

Approximation of the Inverse Frame Operator and Stability of Hilbert$-$Schmidt Frames

In this paper, we study the Hilbert$-$Schmidt frame (HS-frame) theory for separable Hilbert spaces. We first present some characterizations of HS-frames and prove that HS-frames share many important properties with frames. Then, we show how the inverse of the HS-frame operator can be approximated using finite-dimensional methods. Finally, we present a classical perturbation result and prove that HS-frames are stable under small perturbations.

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