Polynomials orthogonal with sign-sensitive weight
type: Item
instance of: scholarly article
title: Polynomials orthogonal with sign-sensitive weight
zbMATH Open document ID: 0879.42014
DOI: 10.1007/BF02308821
author: A.-R. K. Ramazanov
published in: Mathematical Notes
publication date: 1997-08-17T00:00:00Z
review text: Let \(\rho (x)\) be a sign-sensitive weight on \((a,b)\) [see \textit{E. P. Dolzhenko} and \textit{E. A. Sevast'yanov}, Dokl. Akad. Nauk, Ross. Akad. Nauk 332, No. 6, 686-689 (1994; Zbl 0818.41028)]. The author introduces the class \(L_{2,\rho }(a,b)\) of all measurable real functions \(f(x)\) for which \(|f|_{\rho }=(\int_a^b f(x)(f,\rho )(x)dx)^{1/2}\) is finite, where \((f,\rho )(x)=f^{+}(x)\rho_{+}(x) - f^{-}(x)\rho_{-}(x),\) \(f^{\pm }=(|f|\pm f)/2.\) The inner product for \(f, -f, g \in L_{2,\rho }(a,b)\) is defined by the relation \[ (f,g)_{\rho }=\int_a^b f(x)(g,\rho )(x)dx. \] A system of functions \(\phi_0(x),\phi_1(x),\ldots \) from \(L_{2,\rho }(a,b)\) is called (dual) nonsymmetrically orthonormalized with the sign-sensitive weight \(\rho (x)\) if \((\phi_n,\phi_m)_{\rho }=0\) for \(n>m \) (for \(n<m)\) and is equal to \(1\) for \(n=m.\) The existence of (dual) nonsymmetrically orthonormalized systems of algebraic polynomials is proved for a weight \(\rho (x)\) which satisfies the condition \[ \int_a^b \min \{ \rho_{-}(x),\rho_{+}(x)\}dx >0. \] Moreover, foundations of the classical theory of orthogonal polynomials are brought to the (dual) nonsymmetrically orthonormalized polynomials (for example, the properties of Fourier coefficients, quadrature formulas of Gauss type, etc.).
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1037759
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: sign-sensitive weight
zbMATH Keywords: nonsymmetrically orthonormalized polynomials
reviewed by: Alexei Lukashov
MaRDI profile type: Publication
cites work: Q4328118
cites work: Q4839164
cites work: Series in Faber polynomials and several generalizations
cites work: Q3264312
full work available at URL: https://doi.org/10.1007/bf02308821
OpenAlex ID: W1996301165
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description: scientific article; zbMATH DE number 1037759