منابع مشابه
Old and New Geronimus Type Identities for Real Orthogonal Polynomials
Let be a positive measure on the real line, with orthogonal polynomials fpng and leading coe¢ cients f ng. The Geronimus type identity 1 jIm zj Z 1 1 P (t) jzpn (t) pn 1 (t)j dt = n 1 n Z P (t) d (t) ; valid for all polynomials P of degree 2n 2 has known analogues within the theory of orthogonal rational functions, though apparently unknown in the theory of orthogonal polynomials. We present ne...
متن کاملStieltjes-type Polynomials on the Unit Circle
Stieltjes-type polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of co...
متن کاملStieltjes-type polynomials on the unit circle
Stieltjes-type polynomials corresponding to measures supported on the unit circle T are introduced and their asymptotic properties away from T are studied for general classes of measures. As an application, we prove the convergence of an associated sequence of interpolating rational functions to the corresponding Carathéodory function. In turn, this is used to give an estimate of the rate of co...
متن کاملApplications of New Geronimus Type Identities for Real Orthogonal Polynomials
Let be a positive measure on the real line, with associated orthogonal polynomials fpng. Let Im a 6= 0. Then there is an explicit constant cn such that for all polynomials P of degree at most 2n 2, cn Z 1 1 P (t) jpn (a) pn 1 (t) pn 1 (a) pn (t)j dt = Z P d : In this paper, we provide a self-contained proof of the identity. Moreover, we apply the formula to deduce a weak convergence result, a d...
متن کاملStieltjes polynomials and Lagrange interpolation
Bounds are proved for the Stieltjes polynomial En+1, and lower bounds are proved for the distances of consecutive zeros of the Stieltjes polynomials and the Legendre polynomials Pn. This sharpens a known interlacing result of Szegö. As a byproduct, bounds are obtained for the Geronimus polynomials Gn. Applying these results, convergence theorems are proved for the Lagrange interpolation process...
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ژورنال
عنوان ژورنال: Journal of Computational and Applied Mathematics
سال: 1988
ISSN: 0377-0427
DOI: 10.1016/0377-0427(88)90263-4