arXiv subjects

Riya Ghosh

Publications and source records attributed to Riya Ghosh.

5 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 $\theta=\pi/2$.

math.FA↗

arXiv2026

Gabor Frames for a Rational Window with Symmetric Poles

We investigate the Gabor frame problem for a rational window $$g(x)= \frac{x}{(x^2+1)(x^2+4)}.$$ This window falls outside the classes covered by existing frame set characterizations for rational Herglotz functions and ratios of exponential polynomials. For rational densities ($\alpha\beta = p/q$), we leverage the Zak transform to map the Gabor frame condition directly to a finite-dimensional equivalence criterion, which is completely determined by the rank of an associated polynomial matrix. Applying this framework, we establish new frame results for every rational density $$\alpha\beta=\frac{p}{q}\in \left(0,\frac{1}{2}\right)\cup\left(\frac{1}{2},\frac{2}{3}\right)\cup \left(\frac{2}{3},\frac{5}{7}\right].$$

math.FA↗

arXiv2025

Signal Prediction by Derivative Samples from the Past via Perfect Reconstruction

This paper investigates signal prediction through the perfect reconstruction of signals from shift-invariant spaces using nonuniform samples of both the signal and its derivatives. The key advantage of derivative sampling is its ability to reduce the sampling rate. We derive a sampling formula based on periodic nonuniform sampling (PNS) sets with derivatives in a shift-invariant space. We establish the necessary and sufficient conditions for such a set to form a complete interpolating sequence (CIS) of order $r-1$. This framework is then used to develop an efficient approximation scheme in a shift-invariant space generated by a compactly supported function. Building on this, we propose a prediction algorithm that reconstructs a signal from a finite number of past derivative samples using the derived perfect reconstruction formula. Finally, we validate our theoretical results through practical examples involving cubic splines and the Daubechies scaling function of order 3.

cs.IT↗

arXiv2023

Obstructions for Gabor frames of the second order B-spline

For a window $g\in L^2(\mathbb{R})$, the subset of all lattice parameters $(a, b)\in \mathbb{R}^2_+$ such that $\mathcal{G}(g,a,b)=\{e^{2πib m\cdot}g(\cdot-a k) : k, m\in\mathbb{Z}\}$ forms a frame for $L^2(\mathbb{R})$ is known as the frame set of $g$. In time-frequency analysis, determining the Gabor frame set for a given window is a challenging open problem. In particular, the frame set for B-splines has many obstructions. Lemvig and Nielsen in \cite{counter} conjectured that if \begin{align} a_0=\dfrac{1}{2m+1},~ b_0=\dfrac{2k+1}{2},~k,m\in \mathbb{N},~k>m,~a_0b_0<1,\nonumber \end{align} then the Gabor system $\mathcal{G}(Q_2, a, b)$ of the second order B-spline $Q_2$ is not a frame along the hyperbolas \begin{align} ab=\dfrac{2k+1}{2(2m+1)},\text{ for }b\in \left[b_0-a_0\dfrac{k-m}{2}, b_0+a_0\dfrac{k-m}{2}\right],\nonumber \end{align} for every $a_0$, $b_0$. Nielsen in \cite {Nielsenthesis} also conjectured that $\mathcal{G}(Q_2, a,b)$ is not a frame for $$a=\dfrac{1}{2m},~b=\dfrac{2k+1}{2},~k,m\in \mathbb{N},~k>m,~ab<1\text{ with }\gcd(4m,2k+1)=1.$$ In this paper, we prove that both conjectures are true.

math.FA↗

arXiv2023

On Gabor frames generated by B-splines, totally positive functions, and Hermite functions

The frame set of a window $ϕ\in L^2(\mathbb{R})$ is the subset of all lattice parameters $(α, β)\in \mathbb{R}^2_+$ such that $\mathcal{G}(ϕ,α,β)=\{e^{2πiβm\cdot}ϕ(\cdot-αk) : k, m\in\mathbb{Z}\}$ forms a frame for $L^2(\mathbb{R})$. In this paper, we investigate the frame set of B-splines, totally positive functions, and Hermite functions. We derive a sufficient condition for Gabor frames using the connection between sampling theory in shift-invariant spaces and Gabor analysis. As a consequence, we obtain a new frame region belonging to the frame set of B-splines and Hermite functions. For a class of functions that includes certain totally positive functions, we prove that for any choice of lattice parameters $α, β>0$ with $αβ<1,$ there exists a $γ>0$ depending on $αβ$ such that $\mathcal{G}(ϕ(γ\cdot),α,β)$ forms a frame for $L^2(\mathbb{R})$.

math.FA↗