A Differential Analogue of Favard's Theorem

12/14/2020
by   Arieh Iserles, et al.
0

Favard's theorem characterizes bases of functions {p_n}_n∈ℤ_+ for which x p_n(x) is a linear combination of p_n-1(x), p_n(x), and p_n+1(x) for all n ≥ 0 with p_0≡1 (and p_-1≡ 0 by convention). In this paper we explore the differential analogue of this theorem, that is, bases of functions {φ_n}_n∈ℤ_+ for which φ_n'(x) is a linear combination of φ_n-1(x), φ_n(x), and φ_n+1(x) for all n ≥ 0 with φ_0(x) given (and φ_-1≡ 0 by convention). We answer questions about orthogonality and completeness of such functions, provide characterisation results, and also, of course, give plenty of examples and list challenges for further research. Motivation for this work originated in the numerical solution of differential equations, in particular spectral methods which give rise to highly structured matrices and stable-by-design methods for partial differential equations of evolution. However, we believe this theory to be of interest in its own right, due to the interesting links between orthogonal polynomials, Fourier analysis and Paley–Wiener spaces, and the resulting identities between different families of special functions.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
09/27/2017

A Comparison Between Laguerre, Hermite, and Sinc Orthogonal Functions

A series of problems in different fields such as physics and chemistry a...
research
03/22/2023

Gyroscopic polynomials

Gyroscopic alignment of a fluid occurs when flow structures align with t...
research
12/21/2020

Sparse spectral methods for partial differential equations on spherical caps

In recent years, sparse spectral methods for solving partial differentia...
research
06/19/2019

Sparse spectral and p-finite element methods for partial differential equations on disk slices and trapeziums

Sparse spectral methods for solving partial differential equations have ...
research
12/28/2020

Exploring tropical differential equations

The purpose of this paper is fourfold. The first is to develop the theor...
research
07/31/2019

Müntz Sturm-Liouville Problems: Theory and Numerical Experiments

This paper presents two new classes of Müntz functions which are called ...
research
12/14/2017

Statistical Inference for SPDEs: an overview

The aim of this work is to give an overview of the recent developments i...

Please sign up or login with your details

Forgot password? Click here to reset