Monomial and Rodrigues orthogonal polynomials on the cone
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Publication:6408985
DOI10.1016/J.JMAA.2022.126977zbMath1533.33011arXiv2208.12954MaRDI QIDQ6408985
Ana Foulquié Moreno, Yuan Xu, Amílcar Branquinho, Rabia Aktaş
Publication date: 27 August 2022
Abstract: We study two families of orthogonal polynomials with respect to the weight function $w(t)(t^2-|x|^2)^{mu-frac12}$, $mu > -frac 12$, on the cone ${(x,t): |x| le t, , x in mathbb{R}^d, t >0}$ in $mathbb{R}^{d+1}$. The first family consists of monomial polynomials $mathsf{V}_{mathbf{k},n}(x,t) = t^{n-|mathbf{k}|} x^mathbf{k} + cdots$ for $mathbf{k} in mathbb{N}_0^d$ with $|mathbf{k}| le n$, which has the least $L^2$ norm among all polynomials of the form $t^{n-|mathbf{k}|} x^mathbf{k} + mathsf{P}$ with $deg mathsf{P} le n-1$, and we will provide an explicit construction for $mathsf{V}_{mathbf{k},n}$. The second family consists of orthogonal polynomials defined by the Rodrigues type formulas when $w$ is either the Laguerre weight or the Jacobi weight, which satisfies a generating function in both cases. The two families of polynomials are partially biorthogonal.
Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) (33C45) Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis (42C05) General harmonic expansions, frames (42C15) Approximation by polynomials (41A10)
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