arXiv subjects

Kasso Okoudjou

Publications and source records attributed to Kasso Okoudjou.

11 recordsLinked to original sources

arXiv2026

On the structure of the Gram matrix for Gabor systems generated by B-splines

We consider the Gabor system $\mathcal{G}(g,a\mathbb{Z}\times b\mathbb{Z})$ generated by a continuous, compactly supported function $g$ over the time-frequency lattice generated by the parameters $a$ and $b$. We show that, under an appropriate ordering of the Gabor elements, certain submatrices of the Gram matrix of $\mathcal{G}(g,a\mathbb{Z}\times b\mathbb{Z})$ exhibit a block-Toeplitz structure. This structural property enables us to derive spectral results for finite sub-blocks of the Gram matrix by appealing to the spectral theory of Toeplitz matrices. In particular, we apply our results to the Gram matrix of Gabor systems generated by the $N$th-order B-spline.

math.GM↗

arXiv2024

Tight Frames Generated By A Graph Short-Time Fourier Transform

A graph short-time Fourier transform is defined using the eigenvectors of the graph Laplacian and a graph heat kernel as a window parametrized by a non-negative time parameter $t$. We show that the corresponding Gabor-like system forms a frame for $\mathbb{C}^d$ and give a description of the spectrum of the corresponding frame operator in terms of the graph heat kernel and the spectrum of the underlying graph Laplacian. For two classes of algebraic graphs, we prove the frame is tight and independent of the window parameter $t$.

math.SP↗

arXiv2022

Gradient Flows for Frame Potentials on the Wasserstein Space

In this paper we bring together some of the key ideas and methods of two disparate fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular, we construct gradient flows in the Wasserstein space $P_2(\mathbb{R}^d)$ for a new potential, the tightness potential, which is a modification of the probabilistic frame potential. It is shown that the potential is suited for the application of a gradient descent scheme from optimal transport that can be used as the basis of an algorithm to evolve an existing frame toward a tight probabilistic frame.

math.FA↗

arXiv2021

Sobolev Orthogonal Polynomials on the Sierpinski Gasket

We develop a theory of Sobolev orthogonal polynomials on the Sierpiński gasket ($SG$). These orthogonal polynomials arise through the Gram-Schmidt orthogonalisation process applied on the set of monomials on $SG$ using several notions of a Sobolev inner products. After establishing some recurrence relations for these orthogonal polynomials, we give estimates for their $L^2$, $L^\infty$ and Sobolev norms, and study their asymptotic behaviour. Finally, we study the properties of zero sets of polynomials and develop fast computational tools to explore applications to quadrature and interpolation.

math.CA↗

arXiv2020

Finding duality for Riesz bases of exponentials on multi-tiles

It is known that if $Ω\subset \mathbb{R}^{d}$ belongs to a class of multi-tiling domains when translated by a lattice $Λ$, there exists a Riesz basis of exponentials for $L^{2}(Ω)$ constructed using $k$ translates of the dual lattice $Λ^*$. In this paper, we give an explicit construction of the corresponding biorthogonal dual Riesz basis. We also extend the iterative reconstruction algorithm introduced in prior work to this setting.

math.CA↗

arXiv2019

Universal optimal configurations for the $p$-frame potentials

Given $d, N\geq 2$ and $p\in (0, \infty]$ we consider a family of functionals, the $p$-frame potentials FP$_{p, N, d}$, defined on the set of all collections of $N$ unit-norm vectors in $\mathbb R^d$. For the special case $p=2$ and $p=\infty$, both the minima and the minimizers of these potentials have been thoroughly investigated. In this paper, we investigate the minimizers of the functionals FP$_{p, N, d}$, by first establishing some general properties of their minima. Thereafter, we focus on the special case $d=2$, for which, surprisingly, not much is known. One of our main results establishes the unique minimizer for big enough $p$. Moreover, this minimizer is universal in the sense that it minimizes a large range of energy functions that includes the $p$-frame potential. We conclude the paper by reporting some numerical experiments for the case $d\geq 3$, $N=d+1$, $p\in (0, 2)$. These experiments lead to some conjectures that we pose.

cs.IT↗

arXiv2017

Duality and Geodesics for Probabilistic Frames

Probabilistic frames are a generalization of finite frames into the Wasserstein space of probability measures with finite second moment. We introduce new probabilistic definitions of duality, analysis, and synthesis and investigate their properties. In particular, we formulate a theory of transport duals for probabilistic frames and prove certain properties of this class. We also investigate paths of probabilistic frames, identifying conditions under which geodesic paths between two such measures are themselves probabilistic frames. In the discrete case, this is related to ranks of convex combinations of matrices, while, in the continuous case, this is related to the continuity of the optimal transport plan.

math.FA↗

arXiv2015

Finite two-distance tight frames

A finite collection of unit vectors $S \subset \mathbb{R}^n$ is called a spherical two-distance set if there are two numbers $a$ and $b$ such that the inner products of distinct vectors from $S$ are either $a$ or $b$. We prove that if $a\ne -b,$ then a two-distance set that forms a tight frame for $\mathbb{R}^n$ is a spherical embedding of a strongly regular graph, and every strongly regular graph gives rise to two-distance tight frames through standard spherical embeddings. Together with an earlier work by S. Waldron on the equiangular case ({\em Linear Alg. Appl.}, vol. 41, pp. 2228-2242, 2009) this completely characterizes two-distance tight frames. As an intermediate result, we obtain a classification of all two-distance 2-designs.\

math.FA↗

arXiv2010

Dilation properties for weighted modulation spaces

In this paper we give a sharp estimate on the norm of the scaling operator $U_λf(x)=f(λx)$ acting on the weighted modulation spaces $\M{p,q}{s,t}(\R^{d})$. In particular, we recover and extend recent results by Sugimoto and Tomita in the unweighted case. As an application of our results, we estimate the growth in time of solutions of the wave and vibrating plate equations, which is of interest when considering the well posedeness of the Cauchy problem for these equations. Finally, we provide new embedding results between modulation and Besov spaces.

math.FA↗

arXiv2007

Weak uncertainty principle for fractals, graphs and metric measure spaces

We develop a new approach to formulate and prove the weak uncertainty inequality which was recently introduced by Okoudjou and Strichartz. We assume either an appropriate measure growth condition with respect to the effective resistance metric, or, in the absence of such a metric, we assume the Poincare inequality and reverse volume doubling property. We also consider the weak uncertainty inequality in the context of Nash-type inequalities. Our results can be applied to a wide variety of metric measure spaces, including graphs, fractals and manifolds.

math.FA↗

arXiv2006

Unimodular Fourier multipliers for modulation spaces

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol $e^{i|ξ|^α}$, where $α\in[0, 2]$, are bounded on all modulation spaces, but, in general, fail to be bounded on the usual $L^p$-spaces. As a consequence, the phase-space concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers $|ξ|^{-δ} \sin (|ξ|^α)$ for $ 0 \leq δ\leq α$.

math.FA↗