arXiv subjects

David Walnut

Publications and source records attributed to David Walnut.

5 recordsLinked to original sources

arXiv2024

Cube tilings with linear constraints

We consider tilings $(\mathcal{Q},\Phi)$ of $\mathbb{R}^d$ where $\mathcal{Q}$ is the $d$-dimensional unit cube and the set of translations $\Phi$ is constrained to lie in a pre-determined lattice $A \mathbb{Z}^d$ in $\mathbb{R}^d$. We provide a full characterization of matrices $A$ for which such cube tilings exist when $\Phi$ is a sublattice of $A\mathbb{Z}^d$ with any $d \in \mathbb{N}$ or a generic subset of $A\mathbb{Z}^d$ with $d\leq 7$. As a direct consequence of our results, we obtain a criterion for the existence of linearly constrained frequency sets, that is, $\Phi \subseteq A\mathbb{Z}^d$, such that the respective set of complex exponential functions $\mathcal{E} (\Phi)$ is an orthogonal Fourier basis for the space of square integrable functions supported on a parallelepiped $B\mathcal{Q}$, where $A, B \in \mathbb{R}^{d \times d}$ are nonsingular matrices given a priori. Similarly constructed Riesz bases are considered in a companion paper.

math.CA↗

arXiv2024

Exponential bases for parallelepipeds with frequencies lying in a prescribed lattice

The existence of a Fourier basis with frequencies in $\mathbb{R}^d$ for the space of square integrable functions supported on a given parallelepiped in $\mathbb{R}^d$, has been well understood since the 1950s. In a companion paper, we derived necessary and sufficient conditions for a parallelepiped in $\mathbb{R}^d$ to permit an orthogonal basis of exponentials with frequencies constrained to be a subset of a prescribed lattice in $\mathbb{R}^d$, a restriction relevant in many applications. In this paper, we investigate analogous conditions for parallelepipeds that permit a Riesz basis of exponentials with the same constraints on the frequencies. We provide a sufficient condition on the parallelepiped for the Riesz basis case which directly extends one of the necessary and sufficient conditions obtained in the orthogonal basis case. We also provide a sufficient condition which constrains the spectral norm of the matrix generating the parallelepiped, instead of constraining the structure of the matrix.

math.CA↗

arXiv2022

Bases of complex exponentials with restricted supports

The complex exponentials with integer frequencies form a basis for the space of square integrable functions on the unit interval. We analyze whether the basis property is maintained if the support of the complex exponentials is restricted to possibly overlapping subsets of the unit interval. We show, for example, that if $S_1, \ldots, S_K \subset [0,1]$ are finite unions of intervals with rational endpoints that cover the unit interval, then there exists a partition of $\mathbb{Z}$ into sets $\Lambda_1, \ldots, \Lambda_K$ such that $\bigcup_{k=1}^K \{ e^{2\pi i \lambda (\cdot)} \chi_{S_k} : \lambda \in \Lambda_k \}$ is a Riesz basis for $L^2[0,1]$. Here, $\chi_S$ denotes the characteristic function of $S$.

math.CA↗

arXiv2021

Exponential bases for partitions of intervals

For a partition of $[0,1]$ into intervals $I_1,\ldots,I_n$ we prove the existence of a partition of $\mathbb{Z}$ into $\Lambda_1,\ldots, \Lambda_n$ such that the complex exponential functions with frequencies in $ \Lambda_k$ form a Riesz basis for $L^2(I_k)$, and furthermore, that for any $J\subseteq\{1,\,2,\,\dots,\,n\}$, the exponential functions with frequencies in $ \bigcup_{j\in J}\Lambda_j$ form a Riesz basis for $L^2(I)$ for any interval $I$ with length $|I|=\sum_{j\in J}|I_j|$. The construction extends to infinite partitions of $[0,1]$, but with size limitations on the subsets $J\subseteq \mathbb{Z}$; it combines the ergodic properties of subsequences of $\mathbb{Z}$ known as Beatty-Fraenkel sequences with a theorem of Avdonin on exponential Riesz bases.

math.FA↗

arXiv2015

Cornerstones of Sampling of Operator Theory

This paper reviews some results on the identifiability of classes of operators whose Kohn-Nirenberg symbols are band-limited (called band-limited operators), which we refer to as sampling of operators. We trace the motivation and history of the subject back to the original work of the third-named author in the late 1950s and early 1960s, and to the innovations in spread-spectrum communications that preceded that work. We give a brief overview of the NOMAC (Noise Modulation and Correlation) and Rake receivers, which were early implementations of spread-spectrum multi-path wireless communication systems. We examine in detail the original proof of the third-named author characterizing identifiability of channels in terms of the maximum time and Doppler spread of the channel, and do the same for the subsequent generalization of that work by Bello. The mathematical limitations inherent in the proofs of Bello and the third author are removed by using mathematical tools unavailable at the time. We survey more recent advances in sampling of operators and discuss the implications of the use of periodically-weighted delta-trains as identifiers for operator classes that satisfy Bello's criterion for identifiability, leading to new insights into the theory of finite-dimensional Gabor systems. We present novel results on operator sampling in higher dimensions, and review implications and generalizations of the results to stochastic operators, MIMO systems, and operators with unknown spreading domains.

cs.IT↗