Gaussian extended cubature formulae for polyharmonic functions
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 weight function \(\mu^*(t)=|t|^{n-1}\mu(|t|)\) to any polynomial of degree \(2m-1\). There is no extended cubature formula of the form \((1)\) with polyharmonic order of precision \(>2m\). NEWLINENEWLINENEWLINEThe coefficients \(\{ A_k\}\) are explicitly determined as integrals of univariate polynomials and they are positive. Formula (1) can be considered as a polyharmonic extension of the first Green formula.
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