arXiv subjects

A. Antony Selvan

Publications and source records attributed to A. Antony Selvan.

8 recordsLinked to original sources

arXiv2026

Bernoulli--Strang--Fix Conditions: Approximation and Prediction by Sampling Kantorovich Operators

In this paper, we introduce the Bernoulli--Strang--Fix conditions and their generalized versions for a vector-valued generator $φ=(φ_0,\dots,φ_{ρ-1})$ and a periodic nonuniform sampling set $X$. We use these conditions to establish exact and asymptotic polynomial reproduction properties of sampling Kantorovich operators associated with $(φ,X)$ up to a prescribed degree. We analyze the approximation properties and convergence behavior of these operators in detail. Furthermore, we demonstrate their application to signal prediction from a finite number of past local average samples, showing that sampling Kantorovich operators can also serve as effective prediction operators. Finally, we present numerical examples based on Gaussian functions and B-splines to illustrate and validate the theoretical approximation and prediction results.

cs.IT↗

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↗

arXiv2024

Construction of irregular complete interpolation sets for shift-invariant spaces

For several shift-invariant spaces, there exists a real number $a\in\mathbb{R}$ such that the set $a+\mathbb{Z}$ is a complete interpolation set. In this paper, we characterize the complete interpolation property of the set $(a+\mathbb{N}_0)\cup(α+a+\mathbb{N}^{-})$ for shift-invariant spaces using Toeplitz operators. Using this characterization, we determine all $α$ for which the sample set $\mathbb{N}_0\cupα+\mathbb{N}^{-}$ forms a complete interpolation set for transversal-invariant spaces. We introduce a new recurrence relation for exponential splines, examines the zeros of these splines, and explores the zero-free region of the doubly infinite Lerch zeta function. Consequently, we demonstrate that $\left\langle\frac{m}{2}\right\rangle+\mathbb{N}_0\cupα+\left\langle\frac{m}{2}\right\rangle+\mathbb{N}^{-}$ is a complete interpolation set for a shift-invariant spline space of order $m\geq 2$ if and only if $|α|<1/2$.

math.FA↗

arXiv2024

Derivative sampling expansions in shift-invariant spaces with error estimates covering discontinuous signals

This paper is concerned with the problem of sampling and interpolation involving derivatives in shift-invariant spaces and the error analysis of the derivative sampling expansions for fundamentally large classes of functions. A new type of polynomials based on derivative samples is introduced, which is different from the Euler-Frobenius polynomials for the multiplicity $r>1$. A complete characterization of uniform sampling with derivatives is given using Laurent operators. The rate of approximation of a signal (not necessarily continuous) by the derivative sampling expansions in shift-invariant spaces generated by compactly supported functions is established in terms of $L^p$- average modulus of smoothness. Finally, several typical examples illustrating the various problems are discussed in detail.

math.FA↗

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↗

arXiv2017

A new sampling density condition for shift-invariant spaces

Let $X=\{x_i:i\in\mathbb{Z}\}$, $\dots 0$. Under certain conditions on $ϕ$, it is proved that if there exists a positive integer $ν$ such that $$δ_ν:=\sup\limits_{i\in\mathbb{Z}}(x_{i+ν}-x_i)<\dfracν{2π}\left(\dfrac{c_{k}^2}{M_{2k}}\right)^{\frac{1}{4k}},$$ then every function belonging to a shift-invariant space $V(ϕ)$ can be reconstructed stably from its nonuniform sample values $\{f^{(j)}(x_i):j=0,1,\dots, k-1, i\in\mathbb{Z}\}$, where $c_k$ is a Wirtinger-Sobolev constant and $M_{2k}$ is a constant in Bernstein-type inequality of $V(ϕ)$. Further, when $k=1$, the maximum gap $δ_ν<ν$ is sharp for certain shift-invariant spaces.

math.CA↗

arXiv2015

Separation of zeros and a Hermite interpolation based frame algorithm for band limited functions

It is shown that if a non-zero function $f\in B_σ$ has infinitely many double zeros on the real axis, then there exists at least one pair of consecutive zeros whose distance apart is greater than $\dfracπστ^{1/4}$, $τ\approx5.0625$. A frame algorithm is provided for reconstructing a function $f\in B_σ$ from its nonuniform samples $\{f^{(j)}(x_i):j=0,1,\dots, k-1, i\in\mathbb{Z}\}$ with maximum gap condition, $\sup\limits_i(x_{i+1}-x_i)=δ<\dfrac{1}σc_k^{1/2k}$, where $c_k$ is a Wirtinger-Sobolev constant, using Hermite interpolation.

math.CA↗