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This work is a continuation of the recent study by the authors on approximation theory over the sphere and the ball. The main results define new Sobolev spaces on these domains and study polynomial approximations for functions in these…
In many applications it is important to understand the sensitivity of eigenvalues of a matrix polynomial to perturbations of the polynomial. The sensitivity commonly is described by condition numbers or pseudospectra. However, the…
Efficient and stable algorithms for the calculation of spectral quantities and correlation functions are some of the key tools in computational condensed matter physics. In this article we review basic properties and recent developments of…
This note presents an online pseudospectral method for system identification using Chebyshev polynomial basis under aperiodic sampling. The system dynamics are approximated piecewise by introducing a sliding time window. The number of…
We develop a spectral method for solving univariate singular integral equations over unions of intervals by utilizing Chebyshev and ultraspherical polynomials to reformulate the equations as almost-banded infinite-dimensional systems. This…
The focus of this article is the approximation of functions which are analytic on a compact interval except at the endpoints. Typical numerical methods for approximating such functions depend upon the use of particular conformal maps from…
In recent years, the use of sparse recovery techniques in the approximation of high-dimensional functions has garnered increasing interest. In this work we present a survey of recent progress in this emerging topic. Our main focus is on the…
A piecewise Chebyshevian spline space is good for design when it possesses a B-spline basis and this property is preserved under knot insertion. The interest in such kind of spaces is justified by the fact that, similarly as for polynomial…
Many quantum algorithms for ground-state preparation and energy estimation require the implementation of high-degree polynomials of a Hamiltonian to achieve better convergence rates. Their circuit implementation typically relies on quantum…
Unions of graph multiplier operators are an important class of linear operators for processing signals defined on graphs. We present a novel method to efficiently distribute the application of these operators. The proposed method features…
In this paper, we derive optimality conditions (Chebyshev approximation) for multivariate functions. The theory of Chebyshev (uniform) approximation for univariate functions is very elegant. The optimality conditions are based on the notion…
A collection of algorithms in object-oriented MATLAB is described for numerically computing with smooth functions defined on the unit ball in the Chebfun software. Functions are numerically and adaptively resolved to essentially machine…
Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order piecewise polynomials, multilevel schemes, or other types…
We present an amelioration of current known algorithms for optimal spectral partitioning problems. The idea is to use the advantage of a representation using density functions while decreasing the computational time. This is done by…
This article is concerned with an extension of univariate Chebyshev polynomials of the first kind to the multivariate setting, where one chases best approximants to specific monomials by polynomials of lower degree relative to the uniform…
Computation of the trace of a matrix function plays an important role in many scientific computing applications, including applications in machine learning, computational physics (e.g., lattice quantum chromodynamics), network analysis and…
In this manuscript, we analyze the expansions of functions in orthogonal polynomials associated with a general weight function in a multidimensional setting. Such orthogonal polynomials can be obtained by Gram-Schmidt orthogonalization.…
This paper presents an efficient spectral method for solving the fractional Fredholm integro-differential equations. The non-smoothness of the solutions to such problems leads to the performance of spectral methods based on the classical…
In this paper, we consider a class of nonconvex and nonsmooth fractional programming problems, that involve the sum of a convex, possibly nonsmooth function composed with a linear operator and a differentiable, possibly nonconvex function…
The aim of this article is to study the role of piecewise implementation of Pad\'e-Chebyshev type approximation in minimising Gibbs phenomena in approximating piecewise smooth functions. A piecewise Pad\'e-Chebyshev type (PiPCT) algorithm…