arXiv subjects

Sandra Saliani

Publications and source records attributed to Sandra Saliani.

7 recordsLinked to original sources

arXiv2026

Uniform Chebyshev approximations of functions satisfying a variation-type condition: reconstruction of time-varying signals on graphs

We prove uniform approximation theorems for Chebyshev expansions of continuous functions of two variables. Under a variation-type condition on the square $[-1,1]^2$, a continuous function admits a uniformly convergent Chebyshev expansion in the first variable whose coefficients are continuous functions of the second variable. These results are applied to the spectral graph wavelet transform of time-varying signals on finite weighted graphs: the scaling and wavelet kernels are approximated by expansions with time-varying coefficients whose degrees do not depend on time, the composition of the approximate transform with its adjoint admits the same explicit coefficient formulas as in the time-independent case, and reconstruction by the pseudoinverse is stable, with explicit bounds in terms of the uniform kernel errors. Numerical experiments on a sensor network confirm the convergence and stability estimates. In a denoising problem, soft thresholding of the graph wavelet coefficients with a time-varying transform parameter improves over its time-independent, fixed-parameter counterpart.

math.NA↗

arXiv2026

Beyond Single-Window Graph Fourier Analysis

We introduce a multi-windowed graph Fourier transform (MWGFT) for the joint vertex-frequency analysis of signals defined on graphs. Building on generalized translation and modulation induced by the graph Laplacian, the proposed framework extends the windowed graph Fourier transform by allowing multiple analysis and synthesis windows. Exact reconstruction formulas are derived for complex-valued graph signals, together with sufficient and computable conditions guaranteeing stable invertibility. The associated families of windowed graph Fourier atoms are shown to form frames for the space of graph signals. Numerical experiments on synthetic and real world graphs confirm exact reconstruction up to machine precision and demonstrate improved stability and vertex-frequency localization compared to single-window constructions, particularly on irregular graph topologies.

math.CA↗

arXiv2020

Free group representations from vector-valued multiplicative functions, III

Let $π$ be an irreducible unitary representation of a finitely generated nonabelian free group $Γ$; suppose $π$ is weakly contained in the regular representation. In 2001 the first and third authors conjectured that such a representation must be either odd or monotonous or duplicitous. In 2004 they introduced the class of multiplicative representations: this is a large class of representations obtained by looking at the action of $Γ$ on its Cayley graph. In the second paper of this series we showed that some of the multiplicative representations were monotonous. Here we show that all the other multiplicative representations are either odd or duplicitous. The conjecture is therefore established for multiplicative representations.

math.RT↗

arXiv2018

On Cannon cone types and vector-valued multiplicative functions for genus-two-surface-group

We consider Cannon cone types for a surface group of genus $g$, and we give algebraic criteria for establishing the cone type of a given cone and of all its sub-cones. We also re-prove that the number of cone types is exactly $8g(2g - 1)+1.$ In the genus $2$ case, we explicitly provide the $48\times 48$ matrix of cone types, $M,$ and we prove that $M$ is primitive, hence Perron-Frobenius. Finally we define vector-valued multiplicative functions and we show how to compute their values by means of $M$.

math.CO↗

arXiv2016

Linear independence of translates implies linear independence of affine Parseval frames on LCA groups

Motivated by Bownik and Speegle's result on linear independence of wavelet Parseval frames, we consider affine systems (analogous to wavelet systems) defined on a second countable, locally compact abelian group $G$, where the translations are replaced by the action of a countable, discrete subgroup $Γ$ of $ G$ acting as a group of unitary operators on $L^2(G)$. The dilation operation in the wavelet setting is replaced by integer powers of a unitary operator $δ$ onto $L^2(G)$. We show that, under some compatibility conditions between $δ$ and the action of the group $Γ$, the linear independence of the translates of any function in $L^2(G)$ by elements of $Γ$ implies the linear independence of affine Parseval frames in $L^2(G)$.

math.FA↗

arXiv2015

Free Group Representations from Vector-Valued Multiplicative Functions, II

Let $Γ$ be a non-commutative free group on finitely many generators. In a previous work two of the authors have constructed the class of multiplicative representations of $Γ$ and proved them irreducible as representation of $Γ\ltimes_λC(Ω)$. In this paper we analyze multiplicative representations as representations of $Γ$ and we prove a criterium for irreducibility based on the growth of their matrix coefficients.

math.RT↗

arXiv2014

$\ell^2$-Linear Independence for the System of Integer Translates of a Square Integrable Function

We prove that if the system of integer translates of a square integrable function is $\ell^2$-linear independent then its periodization function is strictly positive almost everywhere. Indeed we show that the above inference is true for any square integrable function since the following statement on Fourier analysis is true: For any (Lebesgue) measurable subset A of [0,1], with positive measure, there exists a non trivial square summable function, with support in A, whose partial sums of Fourier series are uniformly bounded.

math.CA↗