Related papers: Fourier Frames for the Cantor-4 Set
We prove qualitative and quantitative results concerning the asymptotic density in dilates of centered convex bodies of the frequency vectors of orthogonal exponential bases and frames associated to bounded domains in Euclidean space.
The measure on generalized solenoids constructed using filters by Dutkay and Jorgensen is analyzed further by writing the solenoid as the product of a torus and a Cantor set. Using this decomposition, key differences are revealed between…
In a previous paper [1] it was discussed the viability of functional analysis using as a basis a couple of generic functions, and hence vectorial decomposition. Here we complete the paradigm exploiting one of the analysis methodologies…
We examine Fourier frames and, more generally, frame measures for different probability measures. We prove that if a measure has an associated frame measure, then it must have a certain uniformity in the sense that the weight is distributed…
This survey synthesizes the principal descriptive set-theoretic perspectives on deterministic Cantor sets on the real line and charts directions for future study. After recounting their historical genesis and compiling an up-to-date…
For some fractal measures it is a very difficult problem in general to prove the existence of spectrum (respectively, frame, Riesz and Bessel spectrum). In fact there are examples of extremely sparse sets that are not even Bessel spectra.…
A distinctive problem of harmonic analysis on $\R$ with respect to a Borel probability measure $\mu$ is identifying all $t\in\R$ such that both \[\left\{e^{-2\pi i\lambda x}: \lambda\in\Lambda\right\}\quad\text{and}\quad \left\{e^{-2\pi…
Depending on a natural parameter $l$, we study the topological, metric, and fractal properties of the homogeneous self-similar set $$K_{l}=\left\{\sum_{i=1}^{\infty} \frac{\varepsilon_i}{(2l+2)^i} : (\varepsilon_i) \in \{0, 2, 4, \dots, 2l,…
In this paper, we prove that the partial sum process of general orthogonal series is a geometric 2-rough process under the same condition as in Menshov-Rademacher Theorem. For Fourier series, the condition can be improved, and an equivalent…
It is a consensus in signal processing that the Gaussian kernel and its partial derivatives enable the development of robust algorithms for feature detection. Fourier analysis and convolution theory have central role in such development. In…
We establish a version of the fractal uncertainty principle, obtained by Bourgain and Dyatlov in 2016, in higher dimensions. The Fourier support is limited to sets $Y\subset \mathbb{R}^d$ which can be covered by finitely many products of…
We study matrix forms of quaternionic versions of the Fourier Transform and Convolution operations. Quaternions offer a powerful representation unit, however they are related to difficulties in their use that stem foremost from…
We present a complex frame of eleven vectors in 4-space and prove that it defines injective measurements. That is, any rank-one $4\times 4$ Hermitian matrix is uniquely determined by its values as a Hermitian form on this collection of…
We study the Falconer distance set problem in Euclidean space and obtain improved dimensional estimates under natural Fourier analytic assumptions cast in terms of the Fourier dimension and spectrum. Interestingly, under reasonably mild…
We give a construction of radial Fourier interpolation formulas in dimensions 3 and 4 using Maass--Poincar\'e type series. As a corollary we obtain explicit formulas for the basis functions of these interpolation formulas in terms of what…
We show that every frame for a Hilbert space H can be written as a (multiple of a) sum of three orthonormal bases for H. A result of N.J. Kalton is included which shows that this is best possible in that: A frame can be represented as a…
We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of $\mathscr{L}_2(\mathbb{R})$. This allows us to derive explicit expansions on the real…
The construction of a multiresolution analysis starts with specification of a scale function. The Fourier transform of this function is defined by an infinite product. The convergence of this product is usually discussed in the context of…
We use a restriction theorem for Fourier transforms of fractal measures to study projections onto families of planes in R^3 whose normal directions form nondegenerate curves.
In this work, the concept of mutually unbiased frames is introduced as the most general notion of unbiasedness for sets composed by linearly independent and normalized vectors. It encompasses the already existing notions of unbiasedness for…