Polynomials orthogonal with sign-sensitive weight
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.).
nonsymmetrically orthonormalized polynomials
scientific article; zbMATH DE number 1037759
scientific article