Fourier coefficients for Laguerre–Sobolev type orthogonal polynomials

Alejandro Molano

doi:10.1108/ajms-07-2021-0164

Abstract

Purpose In this paper, the authors take the first step in the study of constructive methods by using Sobolev polynomials.Design/methodology/approach To do that, the authors use the connection formulas between Sobolev polynomials and classical Laguerre polynomials, as well as the well-known Fourier coefficients for these latter.Findings Then, the authors compute explicit formulas for the Fourier coefficients of some families of Laguerre–Sobolev type orthogonal polynomials over a finite interval. The authors also describe an oscillatory region in each case as a reasonable choice for approximation purposes.Originality/value In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product. As far as the authors know, this particular problem has not been addressed in the existing literature.

Highlights

Within the framework of spectral approximation, and to recover values of smooth functions with exponential accurate, it is customary to use Fourier series for periodic problems and series of classical orthogonal polynomials for nonperiodic problems
In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product
If it deals with piecewise smooth function, estimates by means of partial sums are unhealthy; oscillations do not decrease near discontinuities with partial sums of higher order; and far of them, convergence order is low

Summary

Introduction

Within the framework of spectral approximation, and to recover values of smooth functions with exponential accurate, it is customary to use Fourier series for periodic problems and series of classical orthogonal polynomials for nonperiodic problems. [1, 2], the problem to construct piecewise smooth function values with exponential accuracy at all points is solved by means of approximations with Fourier–Gegenbauer coefficients expansions These are the so-called Gegenbauer reconstruction methods where the expansion of Gegenbauer polynomials in its Fourier series. In order to take the first step in the study of constructive methods by using Sobolev polynomials, this paper deals with Fourier coefficients for certain families of polynomials orthogonal with respect to the Sobolev type inner product (1.1) when μ0 is the classical and absolutely continuous Laguerre measure on [0, ∞). Since the orthogonality interval for Laguerre polynomials is unbounded, we will turn special attention to oscillation regions for the Sobolev polynomials

Preliminaries Let P be the space of polynomials with real coefficients

Quasi-orthogonality and zeros