Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator

Beerends, R. J.

doi:10.1090/s0002-9947-1991-1019520-3

Cited by 52 publications

(46 citation statements)

References 14 publications

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“…Hoffman and Withers (1988) presented a general folding construction. Characterization of such polynomials as eigenfunctions of differential operators is found in Beerends (1991) and Koornwinder (1974). Applications to the solution of differential equations are found in Munthe-Kaas (2006) and to triangle-based spectral element Clenshaw-Curtis-type quadratures in Ryland and Munthe-Kaas (2011).…”

Section: Multivariate Chebyshev Polynomialsmentioning

confidence: 99%

Topics in structure-preserving discretization

Christiansen¹,

Munthe-Kaas²,

Owren³

2011

Acta Numerica

123

115

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Appeared as section 5 in [17]. AbstractWe develop the theory of mixed finite elements in terms of special inverse systems of complexes of differential forms, defined over cellular complexes. Inclusion of cells corresponds to pullback of forms. The theory covers for instance composite piecewise polynomial finite elements of variable order over polyhedral grids. Under natural algebraic and metric conditions, interpolators and smoothers are constructed, which commute with the exterior derivative and whose product is uniformly stable in Lebesgue spaces. As a consequence we obtain not only eigenpair approximation for the Hodge-Laplacian in mixed form, but also variants of Sobolev injections and translation estimates adapted to variational discretizations.

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Section: Multivariate Chebyshev Polynomialsmentioning

confidence: 99%

Topics in structure-preserving discretization

Christiansen¹,

Munthe-Kaas²,

Owren³

2011

Acta Numerica

123

115

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“…Поскольку случай алгебры A 2 был детально изучен в работах [4], [5], мы сосре-доточимся на алгебрах A 1 ⊕ A 1 , B 2 и G 2 . В первом случае обобщение классических многочленов Чебышёва является тривиальным, поэтому здесь мы приведем только окончательный результат.…”

Section: явные построения для алгебр ранга R G =unclassified

Многочлены Чебышeва от многих переменных в терминах сингулярных элементов

Ляховский¹,

Lyakhovsky²

2013

ТМФ

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“…[5,7,2,13]. These polynomials also have a close connection to another well-known family of polynomials, namely the Schur polynomials.…”

Section: Introductionmentioning

confidence: 99%

Around a multivariate Schmidt–Spitzer theorem

Alexandersson

Shapiro

2014

Linear Algebra and its Applications

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Abstract. Given an arbitrary complex-valued infinite matrix A = (a ij ), i = 1, . . . , ∞; j = 1, . . . , ∞ and a positive integer n we introduce a naturally associated polynomial basis B A of C[x 0 , . . . , xn]. We discuss some properties of the locus of common zeros of all polynomials in B A having a given degree m; the latter locus can be interpreted as the spectrum of the m × (m + n)-submatrix of A formed by its m first rows and (m + n) first columns. We initiate the study of the asymptotics of these spectra when m → ∞ in the case when A is a banded Toeplitz matrix. In particular, we present and partially prove a conjectural multivariate analog of the well-known Schmidt-Spitzer theorem which describes the spectral asymptotics for the sequence of principal minors of an arbitrary banded Toeplitz matrix. Finally, we discuss relations between polynomial bases B A and multivariate orthogonal polynomials. IntroductionThe approach of this paper is motivated by the modern interpretation of the Heine-Stieltjes multiparameter spectral problem as presented in [9] and [10]. Let us recall some relevant results in the matrix set-up.Given integers m > 0 and n ≥ 0 consider the space M at(m, m + n) of complexvalued m × (m + n)-matrices. For s = 0, . . . , n define the s-th unit matrix(In what follows the sizes of matrices can be infinite.)Definition 1 (see [10]). Given a matrix A ∈ M at(m, m + n) define its eigenvalue locus E A asFor n = 0, E A coincides with the usual set of eigenvalues of a square matrix A. eigenvalue tuples (x 0 , x 1 , . . . , x n ) such that A − n s=0 x s I s has rank smaller than m. Remark 3. Notice that for n > 0, the locus E A is not a complete intersection since it is given by the vanishing of all maximal minors of A. (A similar phenomenon can be observed for common zeros of multivariate Schur polynomials, since Schur polynomials are given by determinant formulas.)2000 Mathematics Subject Classification. Primary 15B07; Secondary 34L20, 35P20.

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Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator

Cited by 52 publications

References 14 publications

Topics in structure-preserving discretization

Topics in structure-preserving discretization

Многочлены Чебышeва от многих переменных в терминах сингулярных элементов

Around a multivariate Schmidt–Spitzer theorem

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