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We investigate affine Berkovich spaces over maximally complete fields and prove that they may be approximated by simpler spaces when the only functions we need to evaluate are polynomials of bounded degree. We derive applications to…
Functions on a bounded domain in scientific computing are often approximated using piecewise polynomial approximations on meshes that adapt to the shape of the geometry. We study the problem of function approximation using splines on a…
We study sampling and interpolation arrays with multiplicities for the spaces P_k of holomorphic polynomials of degree at most k. We find that the geometric conditions satisfied by these arrays are in accordance with the conditions…
In this paper, we study a specific system of Clifford-Appell polynomials and in particular their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions.…
In the present paper, we use a generalised shift operator in order to define a generalised modulus of smoothness. By its means, we define generalised Lipschitz classes of functions, and we give their constructive characteristics.…
This article offers a comprehensive treatment of polynomial functional regression, culminating in the establishment of a novel finite sample bound. This bound encompasses various aspects, including general smoothness conditions, capacity…
In this paper we provide a rigorous mathematical foundation for continuous approximations of a class of systems with piece-wise continuous functions. By using techniques from the theory of differential inclusions, the underlying piece-wise…
Using partial derivatives $\partial_zf$ and $\partial_{\ol z}f$, we introduce Besov spaces of polyanalytic functions on the unit disk and on the upper half-plane. We then prove that the dilatations of each function in polyanalytic Besov…
Polynomial reproduction plays a relevant role in deriving error estimates for various approximation schemes. Local reproduction in a quasi-uniform setting is a significant factor in the estimation of error and the assessment of stability…
The probability density function (PDF) associated with a given set of samples is approximated by a piecewise-linear polynomial constructed with respect to a binning of the sample space. The kernel functions are a compactly supported basis…
We prove that the Newton product of efficient polynomial projectors is still efficient. Various polynomial approximation theorems are established involving Newton product projectors on spaces of holomorphic functions on a neighborhood of a…
The approximation properties of the finite element method can often be substantially improved by choosing smooth high-order basis functions. It is extremely difficult to devise such basis functions for partitions consisting of arbitrarily…
In this article, I provide significant mathematical evidence in support of the existence of short-time approximations of any polynomial order for the computation of density matrices of physical systems described by arbitrarily smooth and…
The aim of this paper is to prove that a large class of quaternionic slice regular functions result to be (ramified) covering maps. By means of the topological implications of this fact and by providing further topological structures, we…
Generalizations of some known results on the best, best linear and best one-sided approxima- tions by trigonometric polynomials of the classes of 2\pi - periodic functions presented in the form of convolutions to the case of set-valued…
In this paper, a $k$-th generalized modulus of smoothness is defined based on an asymmetric operator of generalized translation and a theorem is proved about the coincidence of class of functions defined by this modulus and a class of…
In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to…
We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…
We give the theorem of coincidence of a class of functions defined by a generalised modulus of smoothness with a class of functions defined by the order of the best approximation by algebraic polynomials. We also prove the appropriate…
For a function $f$ that is piecewise analytic on a quasi-smooth arc $\mathcal{L}$ and any $0<\sigma<1$ we construct a sequence of "near-best" polynomials that converge at a rate $e^{-n^{\sigma}}$ at each point of analyticity of $f$ and are…