Cyclotomy of symmetric polynomials
We prove some new identities for homogeneous symmetric polynomials when evaluated at roots of unity. Some of these formulas are applied to obtain new insights onto cyclotomic polynomials.
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arXiv subjects
Laura De Carli
Publications and source records attributed to Laura De Carli.
Block-equivalent finite Gabor frames
We study finite systems of vectors whose frame operator matrices are unitarily equivalent, via explicit and computationally efficient unitary transformations, to block-diagonal matrices. We call such systems block-equivalent. We show that a Gabor system $\mathcal{G}=\mathcal{G}(g,L\times K)\subset \mathbb C^N$ is block-equivalent when either the modulation set $L$ or the translation set $K$ is a subgroup of $\mathbb Z_N$. We also characterize situations in which the frame operator matrix becomes diagonal. Finally, we show that geometric conditions on subsets of $\mathbb Z_N$ force certain diagonals of the frame operator matrix of $\mathcal{G}$ to vanish, yielding additional sparsity and block structures.
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Concerning the stability of exponential systems and Fourier matrices
Fourier matrices naturally appear in many applications and their stability is closely tied to performance guarantees of algorithms. The starting point of this article is a result that characterizes properties of an exponential system on a union of cubes in $\mathbb{R}^d$ in terms of a general class of Fourier matrices and their extreme singular values. This relationship is flexible in the sense that it holds for any dimension $d$, for many types of exponential systems (Riesz bases, Riesz sequences, or frames) and for Fourier matrices with an arbitrary number of rows and columns. From there, we prove new stability results for Fourier matrices by exploiting this connection and using powerful stability theorems for exponential systems. This paper provides a systematic exploration of this connection and suggests some natural open questions.
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Divisibility criteria and coefficient formulas for cyclotomic polynomials
We establish necessary and sufficient conditions for a polynomial to be divisible by a cyclotomic polynomials and derive new formulas involving Ramanujan sums as an application of our results. Additionally, we provide new insights into the coefficients of cyclotomic polynomials and we propose a recursive relation between the coefficients of two cyclotomic polynomials whose indexes differ by a prime factor.
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Methods of construction of exponential bases on planar domains
In this paper, we survey and refine several results -- some previously established in the literature -- that facilitate the construction of exponential bases on planar domains with explicit control over the associated frame bounds. We apply our techniques to construct well-conditioned exponential bases on certain planar sets that multi-tile the plane.
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Egyptian fractions meet the Sierpinski triangle
We explore a novel link between two seemingly disparate mathematical concepts: Egyptian fractions and fractals. By examining the decomposition of rationals into sums of distinct unit fractions, a practice rooted in ancient Egyptian mathematics, and the arithmetic operations that can be performed using this decomposition, we uncover fractal structures that emerge from these representations.
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Frequency-dependent stability for Gabor frames
We prove new stability results for a class of regular Gabor frames $G(h; a,b )$ subject to frequency-dependent timing jitter under various conditions on the window function $h$.
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Approximating the divisor functions
We exploit the properties of a sequence of functions that approximate the divisor functions and combine them with an analytical formula of a delta-like sequence to give a new proof of a theorem of Gronwall on the asymptotic of the divisor functions.
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Lp simulation for measures
Being motivated by general interest as well as by certain concrete problems of Fourier Analysis, we construct analogs of the Lp spaces for measures. It turns out that most of standard properties of the usual Lp spaces for functions are extended to the measure setting. We illustrate the obtained results by examples and apply them to obtain a version of the uncertainty principle and an integrability result for the Fourier transform of a function of bounded variation.
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Applications of Lax-Milgram theorem to problems in frame theory
We apply Lax-Milgram theorem to characterize scalable and piecewise scalable frame in finite and infinite-dimensional Hilbert spaces. We also introduce a method for approximating the inverse frame operator using finite-dimensional linear algebra which, to the best of our knowledge, is new in the literature.
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Remarks on scalable frames
This paper investigates scalable frame in ${\mathbb R}^n$. We define the reduced diagram matrix of a frame and use it to classify scalability of the frame under some conditions. We give a new approach to the scaling problem by breaking the problem into two smaller ones, each of which is easily solved, giving a simple way to check scaling. Finally, we study the scalability of dual frames.
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Piecewise scalable frames
In this paper we define "piecewise scalable frames". This new scaling process allows us to alter many frames to Parseval frames which is impossible by the previous standard scaling. We give necessary and sufficient conditions for a frame to be piecewise scalable. We show that piecewise scalability is preserved under unitary transformations. Unlike standard scaling, we show that all frames in $\RR^2$ and $\RR^3$ are piecewise scalable. We also show that if the frame vectors are close to each other, then they might not be piecewise scalable. Several properties of scaling constants are also presented.
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Riesz bases from orthonormal bases by replacement
Given an orthonormal basis $ {\mathcal V}= \{v_j\} _{j\in N}$ in a separable Hilbert space $H$ and a set of unit vectors $ {\mathcal B}=\{w_j\}_{j\in N}$, we consider the sets $ {\mathcal B}_N$ obtained by replacing the vectors $v_1, ...,\, v_N$ with vectors $w_1,\, ...,\, w_N$. We show necessary and sufficient conditions that ensure that the sets $ {\mathcal B}_N$ are Riesz bases of $H$ and we estimate the frame constants of the $ {\mathcal B}_N$. Then, we prove conditions that ensure that $ {\mathcal B}$ is a Riesz basis. Applications to the construction of exponential bases on domains of $ R^d$ are also presented.
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Three problems on exponential bases
We consider three special and significant cases of the following problem. Let D be a (possibly unbounded) set of finite Lebesgue measure in R^d. Find conditions on D for which the standard exponential basis on the unit cube of R^d is a frame, a Riesz sequence, or a Riesz basis on L^2(D).
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p-Riesz bases in quasi shift invariant spaces
Let $ 1\leq p< \infty$ and let $ψ\in L^{p}(\R^d)$. We study $p-$Riesz bases of quasi shift invariant spaces $V^p(ψ;Y)$.
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Stability results for Gabor frames and the p-order hold models
We prove stability results for a class of Gabor frames in $ L^2(\R)$. We consider window functions in the Sobolev spaces $H^1_0(\R)$ and B-splines of order $p\ge 1$. Our results can be used to describe the effect of the timing jitters in the $p$-order hold models of signal reconstruction.
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Exponential bases on triangular domains
We construct explicit exponential bases on triangles in R^2 and on infinite unions of segments on the real line.
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Exponential bases on multi-rectangles in $\mathbb{R}^d$
We construct explicit exponential bases on finite unions of disjoint rectangles of $\mathbb{R}^d$ with rational vertices.
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