Alternation points and bivariate Lagrange interpolation

Lawrence A. Harris

doi:10.1016/j.cam.2018.02.014

Alternation points and bivariate Lagrange interpolation

Lawrence A. Harris

Mathematics

Research output: Contribution to journal › Article › peer-review

4 Scopus citations

Abstract

Given m+1 strictly decreasing numbers h₀,h₁,…,h_m, we give an algorithm to construct a corresponding finite sequence of orthogonal polynomials p₀,p₁,…,p_m such that p₀=1, p_j has degree j and p_m−j(h_n)=(−1)ⁿp_j(h_n) for all j,n=0,1,…,m. Using these polynomials, we construct bivariate Lagrange polynomials and cubature formulas for nodes that are points in R² where the coordinates are taken from given finite decreasing sequences of the same length and where the indices have the same (or opposite) parity.

Original language	English
Pages (from-to)	43-52
Number of pages	10
Journal	Journal of Computational and Applied Mathematics
Volume	340
DOIs	https://doi.org/10.1016/j.cam.2018.02.014
State	Published - Oct 1 2018

Bibliographical note

Publisher Copyright:
© 2018 Elsevier B.V.

Keywords

Bézout identity
Christoffel–Darboux formula
Cubature
Orthogonal polynomials

ASJC Scopus subject areas

Computational Mathematics
Applied Mathematics

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10.1016/j.cam.2018.02.014

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title = "Alternation points and bivariate Lagrange interpolation",

abstract = "Given m+1 strictly decreasing numbers h0,h1,…,hm, we give an algorithm to construct a corresponding finite sequence of orthogonal polynomials p0,p1,…,pm such that p0=1, pj has degree j and pm−j(hn)=(−1)npj(hn) for all j,n=0,1,…,m. Using these polynomials, we construct bivariate Lagrange polynomials and cubature formulas for nodes that are points in R2 where the coordinates are taken from given finite decreasing sequences of the same length and where the indices have the same (or opposite) parity.",

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author = "Harris, {Lawrence A.}",

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year = "2018",

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doi = "10.1016/j.cam.2018.02.014",

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N2 - Given m+1 strictly decreasing numbers h0,h1,…,hm, we give an algorithm to construct a corresponding finite sequence of orthogonal polynomials p0,p1,…,pm such that p0=1, pj has degree j and pm−j(hn)=(−1)npj(hn) for all j,n=0,1,…,m. Using these polynomials, we construct bivariate Lagrange polynomials and cubature formulas for nodes that are points in R2 where the coordinates are taken from given finite decreasing sequences of the same length and where the indices have the same (or opposite) parity.

AB - Given m+1 strictly decreasing numbers h0,h1,…,hm, we give an algorithm to construct a corresponding finite sequence of orthogonal polynomials p0,p1,…,pm such that p0=1, pj has degree j and pm−j(hn)=(−1)npj(hn) for all j,n=0,1,…,m. Using these polynomials, we construct bivariate Lagrange polynomials and cubature formulas for nodes that are points in R2 where the coordinates are taken from given finite decreasing sequences of the same length and where the indices have the same (or opposite) parity.

KW - Bézout identity

KW - Christoffel–Darboux formula

KW - Cubature

KW - Orthogonal polynomials

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DO - 10.1016/j.cam.2018.02.014

M3 - Article

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SN - 0377-0427

VL - 340

SP - 43

EP - 52

JO - Journal of Computational and Applied Mathematics

JF - Journal of Computational and Applied Mathematics

ER -

Alternation points and bivariate Lagrange interpolation

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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