Gaussian extended cubature formulae for polyharmonic functions
type: Item
instance of: scholarly article
title: Gaussian extended cubature formulae for polyharmonic functions
zbMATH Open document ID: 0965.31007
DOI: 10.1090/S0025-5718-00-01206-0
author: Borislav D. Bojanov
author: Dimitar K. Dimitrov
published in: Mathematics of Computation
publication date: 2001-02-19T00:00:00Z
Mathematics Subject Classification ID: 31B30
Mathematics Subject Classification ID: 65D32
zbMATH DE Number: 1566125
zbMATH Keywords: polyharmonic function
zbMATH Keywords: polyharmonic precision order
zbMATH Keywords: Gaussian extended cubature formula
MaRDI profile type: Publication
cites work: Q4897092
cites work: Q3660069
cites work: Integration of polyharmonic functions
cites work: Q4003612
cites work: Q4058005
cites work: Representation and uniqueness theorems for polyharmonic functions
cites work: Q4102261
cites work: Q3323132
cites work: Q4884008
review text: The purpose of this paper is to show certain links between univariate interpolation by algebraic polynomials and the representation of polyharmonic functions. For any positive \(r\), we denote by \(B(r)\), \(\overline{B}(r)\) and \(S(r)\) the open and the closed balls and the hypersphere with center 0 and radius \(r\) in \({\mathbb{R}}^n\). In case \(r=1\) the notation of the radius will be omitted. The iterates \(\Delta^m\) of the Laplace operator \(\Delta\) in \({\mathbb{R}}^n\) are defined recursively by \(\Delta^m=\Delta\Delta^{m-1}\). The function \(u\) is said to be polyharmonic of order \(m\), or \(m\)-harmonic, in \(B\) if \(u\) belongs to the space NEWLINE\[NEWLINE H^m(B):=\{u\in C^{2m-1}(\overline{B})\cap C^{2m}(B): \Delta^mu=0 \text{ on } B\} . NEWLINE\]NEWLINE The authors construct cubature formulae for multivariate functions having highest order of precision with respect to the class of polyharmonic functions. They obtain a Gauss type cubature formula that uses \(m\) values of linear functionals and is exact for all \(2m\)-harmonic functions, and consequently, for all algebraic polynomials of \(n\) variables of degree \(4m-1\). A univariate function \(\mu(t)\), defined on the interval \([a,b]\), is said to be a weight function on \([a,b]\) if \(\mu(t)\) is nonnegative there and all the moments \(\int_a^bt^s\mu(t) dt\), \(s=0,1,\dots\), exist. In this paper the following is proved. NEWLINENEWLINENEWLINETheorem 1. Let \(\mu(t)\) be any given weight function on \([a,b]\). There exist a unique sequence of distinct radii \(0<R_1<\cdots<R_m\leq _1\) and real weights \(A_k\), \(k=1,\cdots, m\), such that the extended cubature formula NEWLINE\[NEWLINE \int_B u(x) \mu(|x|) dx\approx\sum_{k=1}^mA_k\int_{S(R_k)}u(\xi) d\sigma(\xi)\tag{1}NEWLINE\]NEWLINE has polyharmonic order of precision \(2m\). Moreover, the radii \(R_k\) coincide with the positive zeros of the polynomial \(P_{2m}(t;\mu^*)\) of degree \(2m\), which is orthogonal on \([-1,1]\) with respect to the…
description: scientific article