arXiv subjects

Kasso A. Okoudjou

Publications and source records attributed to Kasso A. Okoudjou.

At least 19 recordsLinked to original sources

arXiv2026

The HRT Conjecture for Symmetric Configurations and Real-Valued Functions

The Heil-Ramanathan-Topiwala conjecture asserted that every finite collection of distinct time-frequency shifts of a nonzero square-integrable function is linearly independent. Recent counterexamples with Schwartz windows leave open the classification of configurations and windows for which independence holds. We prove linear independence for every nonzero window in $L^2(\mathbb{R})$ and every configuration consisting of finitely many commensurably spaced points on one line together with two distinct points on a parallel line. As a consequence, we obtain independence for every nonzero real-valued window and every four point configuration $\{(0,0),(0,1),(s,0),(a,b)\}$ with $abs\ne0$. The configurations underlying the known four-point counterexamples can be put in this form by translation and dilation, so they cannot yield dependence with nonzero real-valued windows. The proof combines factorization of Laurent polynomials with product estimates along orbits of irrational rotations. Finally, for a fixed nonzero window and a fixed three-point system, we show that the set of time-frequency parameters producing a shift in its span is compact, has Lebesgue measure zero, has Hausdorff dimension at most one, and is totally disconnected.

math.FA↗

arXiv2026

On the Injectivity of STFT Phase Retrieval with super-exponential decaying window function

We investigate the uniqueness of short-time Fourier transform phase retrieval problems in $L^2(\mathbb{R})$. In particular, for underlying window functions whose Fourier transform decay faster than any exponential function, we derive sufficient conditions on discrete sampling sets for unique phase retrieval from the spectrogram. This result generalizes previous uniqueness guarantees on sampling sets for Gaussian windows.

math.FA↗

arXiv2025

Abelian and Dihedral equiangular tight frames of redundancy $2$

This paper studies group frames ($G$-frames) where the unitary group representation can be projective. When the group is abelian, for most combinations $N, n$, we show that $ETF(N,n)$ can only exist for genuinely projective group representations. In particular, cyclic-group frames for such parameters do not exist. We also give a characterization of all dihedral tight frames and dihedral $ETF(2n,n)$, using which, we conclude that regular dihedral $ETF(2n,n)$ must be genuinely projective. Following that, we give a characterization of regular dihedral $ETF(2n,n)$ in terms of certain structured skew Hadamard matrices. We then show that Paley $ETF(2n,n)$ and its doubling are both of this type. Finally, we classify all regular dihedral $ETF(2n,n)$ for $n\le 22$ up to switching equivalence.

math.CO↗

arXiv2025

Orthogonal Polynomials on Bubble-Diamond Fractals

We develop a theory of polynomials and, in particular, an analog of the theory of Legendre orthogonal polynomials on the bubble-diamond fractals, a class of fractal sets that can be viewed as the completion of a limit of a sequence of finite graph approximations. In this setting, a polynomial of degree $j$ can be viewed as a multiharmonic function, a solution of the equation $Δ^{j+1}u=0$. We prove that the sequence of orthogonal polynomials we construct obeys a three-term recursion formula.

math.FA↗

arXiv2024

The HRT conjecture for a class of meromorphic functions

The HRT conjecture states that any finite collection of time-frequency shifts of a non-zero square-integrable function on the real line is linearly independent. In this paper, we establish the linear independence of finite systems of time-frequency shifts of a non-zero meromorphic function. Consequently, we prove that the conjecture is true for any square-integrable function on $\mathbb{R}$ that can be extended to an analytic function on $\mathbb{C}$ except on finitely many points.

math.CV↗

arXiv2024

The HRT conjecture for two classes of special configurations

The HRT (Heil-Ramanathan-Topiwala) conjecture stipulates that the set of any finitely many time-frequency shifts of a non-zero square Lebesgue integrable function is linearly independent. The present work settles two special cases of this conjecture, namely, the cases where the set of time-frequency shifts has cardinality $N+1$ such that either $N$ of the points lie on some integer lattice and the last point is arbitrary, or $N$ of the points are on a line, while the last point does not belong this line. In both cases, we prove that the HRT conjecture holds appealing mainly to various forms of the ergodic theorem. We note that, in recent years, the latter case has been the subject of many investigations -- notably, the subcase where $N=3$ -- and our work completely resolves it.

math.FA↗

arXiv2022

Phase transitions for frame potentials]{Phase transitions for the minimizers of the $p^{th}$ frame potentials in $\mathbb{R}^2$

Given $N$ points $X=\{x_k\}_{k=1}^N$ on the unit circle in $\mathbb{R}^2$ and a number $0\leq p \leq \infty$ we investigate the minimizers of the functional $\sum_{k, \ell =1}^N |\langle x_k, x_\ell\rangle|^p$. While it is known that each of these minimizers is a spanning set for $\mathbb{R}^2$, less is known about their number as a function of $p$ and $N$ especially for relatively small $p$. In this paper we show that there is unique minimum for this functional for all $p\leq \log 3/\log 2$ and all odd $N\geq 3$. In addition, we present some numerical results suggesting the emergence of a phase transition phenomenon for these minimizers. More specifically, for $N\geq 3$ odd, there exists a sequence of number of points $\log 3/\log 2=p_1< p_2< \cdots < p_N\leq 2$ so that a unique (up to some isometries) minimizer exists on each sub-intervals $(p_k, p_{k+1})$. %In addition we conjecture that $\lim_{k\to \infty}p_{2k+1}=2$.

math.CO↗

arXiv2022

Quantitative approach to Grover's quantum walk on graphs

In this paper, we study Grover's search algorithm focusing on continuous-time quantum walk on graphs. We propose an alternative optimization approach to Grover's algorithm on graphs that can be summarized as follows: instead of finding specific graph topologies convenient for the related quantum walk, we fix the graph topology and vary the underlying graph Laplacians. As a result, we search for the most appropriate analytical structure on graphs endowed with fixed topologies yielding better search outcomes. We discuss strategies to investigate the optimality of Grover's algorithm and provide an example with an easy tunable graph Laplacian to investigate our ideas.

math-ph↗

arXiv2022

On the frame set of the second-order $B$-spline

The frame set of a function $g\in L^2(\mathbb{R})$ is the set of all parameters $(a, b)\in \mathbb{R}^2_+$ for which the collection of time-frequency shifts of $g$ along $a\mathbb{Z}\times b\mathbb{Z}$ form a Gabor frame for $L^2(\mathbb{R}).$ Finding the frame set of a given function remains a challenging open problem in time-frequency analysis. In this paper, we establish new regions of the frame set of the second order $B-$spline. Our method relies on the compact support of this function to partition a subset of the putative frame set and find an explicit dual window function in each of the partition regions. Numerical evidence indicates the existence of further points belonging to the frame set.

math.CA↗

arXiv2022

On root frames in $\mathbb{R}^d$

\emph{A root frame} for $\mathbb{R}^d$ is a finite frame whose vectors form a root system. In this note we establish some elementary properties of this class of frames and prove that root frames constitute a subclass of scalable frames. In addition, we show that root frames are examples of a larger class of frames called \emph{eigenframes}.

math.GM↗

arXiv2022

Spectral decimation of piecewise centrosymmetric Jacobi operators on graphs

We study the spectral theory of a class of piecewise centrosymmetric Jacobi operators defined on an associated family of substitution graphs. Given a finite centrosymmetric matrix viewed as a weight matrix on a finite directed path graph and a probabilistic Laplacian viewed as a weight matrix on a locally finite strongly connected graph, we construct a new graph and a new operator by edge substitution. Our main result proves that the spectral theory of the piecewise centrosymmetric Jacobi operator can be explicitly related to the spectral theory of the probabilistic Laplacian using certain orthogonal polynomials. Our main tools involve the so-called spectral decimation, known from the analysis on fractals, and the classical Schur complement. We include several examples of self-similar Jacobi matrices that fit into our framework.

math.SP↗

arXiv2021

Spectral decimation of a self-similar version of almost Mathieu-type operators

We introduce self-similar versions of the one-dimensional almost Mathieu operators. Our definition is based on a class of self-similar Laplacians instead of the standard discrete Laplacian, and includes the classical almost Mathieu operators as a particular case. Our main result establishes that the spectra of these self-similar almost Mathieu operators can be completely described by the spectra of the corresponding self-similar Laplacians through the spectral decimation framework used in the context of spectral analysis on fractals. In addition, the self-similar structure of our model provides a natural finite graph approximation model. This approximation is not only helpful in executing the numerical simulation, but is also useful in finding the spectral decimation function via Schur complement computations of given finite-dimensional matrices. The self-similar Laplacians used in our model were considered recently by Chen and Teplyaev who proved the emergence of singularly continuous spectra for specific parameters. We use this result to arrive at similar conclusions in the context of the self-similar almost Mathieu operators. Finally, we derive an explicit formula of the integrated density of states of the self-similar almost Mathieu operators as the weighted pre-images of the balanced invariant measure on a specific Julia set.

math.SP↗

arXiv2021

Towards a classification of incomplete Gabor POVMs in $\mathbb{C}^d$

Every (full) finite Gabor system generated by a unit-norm vector $g\in \mathbb{C}^d$ is a finite unit-norm tight frame (FUNTF), and can thus be associated with a (Gabor) positive operator valued measure (POVM). Such a POVM is informationally complete if the $d^2$ corresponding rank one matrices form a basis for the space of $d\times d$ matrices. A sufficient condition for this to happen is that the POVM is symmetric, which is equivalent to the fact that the associated Gabor frame is an equiangular tight frame (ETF). The existence of Gabor ETF is an important special case of the Zauner conjecture. It is known that generically all Gabor FUNTFs lead to informationally complete POVMs. In this paper, we initiate a classification of non-complete Gabor POVMs. In the process we establish some seemingly simple facts about the eigenvalues of the Gram matrix of the rank one matrices generated by a finite Gabor frame. We also use these results to construct some sets of $d^2$ unit vectors in $\mathbb{C}^d$ with a relatively smaller number of distinct inner products.

math.FA↗

arXiv2020

Optimal l-one Rank One Matrix Decompositions

In this paper we consider the decomposition of positive semidefinite matrices as a sum of rank one matrices. We introduce and investigate the properties of various measures of optimality of such decompositions. For some classes of positive semidefinite matrices we give explicitly these optimal decompositions. These classes include diagonally dominant matrices and certain of their generalizations, $2\times 2$, and a class of $3\times 3$ matrices.

math.FA↗

arXiv2019

Multivariate Tight Wavelet Frames with Few Generators and High Vanishing Moments

Tight wavelet frames are computationally and theoretically attractive, but most existing multivariate constructions have various drawbacks, including low vanishing moments for the wavelets, or a large number of wavelet masks. We further develop existing work combining sums of squares representations with tight wavelet frame construction, and present a new and general method for constructing such frames. Focusing on the case of box splines, we also demonstrate how the flexibility of our approach can lead to tight wavelet frames with high numbers of vanishing moments for all of the wavelet masks, while still having few highpass masks: in fact, we match the best known upper bound on the number of highpass masks for general box spline tight wavelet frame constructions, while typically achieving much better vanishing moments for all of the wavelet masks, proving a nontrivial lower bound on this quantity.

math.FA↗

arXiv2019

The Hartree-Fock equations in modulation spaces

We establish both a local and a global well-posedness theories for the nonlinear Hartree-Fock equations and its reduced analog in the setting of the modulation spaces on $\mathbb R^d$. In addition, we prove similar results when a harmonic potential is added to the equations. In the process, we prove the boundedeness of certain multilinear operators on products of the modulation spaces which may be of independent interest.

math.AP↗

arXiv2019

An invitation to Gabor analysis

Gabor analysis which can be traced back to Dennis Gabor's influential 1946 paper \emph{Theory of communication}, is concerned with both the theory and the applications of the approximation properties of sets of time and frequency shifts of a given window function. It re-emerged with the advent of wavelets at the end of the last century and is now at the intersection of many fields of mathematics, applied mathematics, engineering, and science. The goal of this paper is to give a brief introduction to Gabor analysis by elaborating on three open problems.

math.CA↗

arXiv2019

The Strong Maximum Principle for Schrödinger operators on fractals

We prove a strong maximum principle for Schrödinger operators defined on a class of fractal sets and their blowups without boundary. Our primary interest is in weaker regularity conditions than have previously appeared in the literature; in particular we permit both the fractal Laplacian and the potential to be Radon measures on the fractal. As a consequence of our results, we establish a Harnack inequality for solutions of these operators.

math.FA↗