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Dimitrios Vavitsas

Publications and source records attributed to Dimitrios Vavitsas.

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

Boundary zeros of stable polynomials in the unit ball

Interpolation theory in the unit ball and semi-algebraic geometry yield explicit descriptions of the boundary zeros of stable polynomials. Given a polynomial $p\in \mathbb{C}[z_1, ...,z_n]$ that is zero-free in the unit ball and vanishes on the sphere along submanifolds of dimension at most one, we describe the boundary zeros $\mathcal{Z}(p)\cap\mathbb{S}_n$ in terms of peak sets for $A^\infty(\mathbb{B}_n)$. In particular, in the setting $n=2$, we achieve a characterization by proving that every accumulation point of $\mathcal{Z}(p)\cap\mathbb{S}_2$ lies in the relative interior of an one dimensional real analytic submanifold, and that these submanifolds form a foliation of the non-isolated part of $\mathcal{Z}(p)\cap\mathbb{S}_2$. As an application of the developed theory, we obtain a characterization of cyclic polynomials without weak essential singularities in the Dirichlet-type space $\mathcal{D}_{n-1/2}(\mathbb{B}_n)$. A theory for more general geometric settings of the boundary zeros is also developed, aiming to provide a starting point for further extensions.

math.CV↗
arXiv2026

Riesz $\alpha$-capacity of Cantor sets and cyclicity in Dirichlet-type spaces

We examine the threshold of the cyclicity for functions in Dirichlet-type spaces $\mathcal{D}_{\alpha}$, $\alpha\in(0,1]$. Given a fixed $\alpha^{*}\in(0,1]$, we construct a holomorphic function $f\in\mathcal{D}_{\alpha^{*}}$ which is cyclic in $\mathcal{D}_{\alpha}$ for all $\alpha<\alpha^{*}$, but fails to be cyclic in $\mathcal{D}_{\alpha^{*}}$. This function serves as a counterexample to the persistence of cyclicity at the critical index $\alpha^{*}$. Throughout the construction process, we work with generalized Cantor sets and study their Riesz $\alpha$-capacity.

math.CV↗
arXiv2024

Non-cyclicity and polynomials in Dirichlet-type spaces of the unit ball

We give a description of the intersection of the zero set with the unit sphere of a zero-free polynomial in the unit ball of $\mathbb{C}^n$. This description leads to the formulation of a conjecture regarding the characterization of polynomials that are cyclic in Dirichlet-type spaces in the unit ball of $\mathbb{C}^n$. Furthermore, we answer partially ascertaining whether an arbitrary polynomial is not cyclic.

math.CV↗