# APPLICATION OF JACOBI POLYNOMIALS TO THE APPROXIMATE SOLUTION OF A SINGULAR INTEGRAL EQUATION WITH CAUCHY KERNEL ON THE REAL HALF-LINE

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In this paper, Jacobi and trigonometric polynomials are used to construct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

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Starting with a formula by Noble [Proc. Cambridge Philos. Soc. 59, 363 (1963), Eq. (16)] for a certain sum of products of Jacobi polynomials, another sum of this type is evaluated.

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We construct Jacobi-weighted orthogonal polynomials Pn,r(Î±,Î²Î³) (u ,v, w), Î±, Î², Î³ > -1, Î± + Î² + Î³ = 0, on the triangular domain T. We show that these polynomials Pn,r(Î±,Î²Î³) (u, v, w) over the triangular domain T satisfy the following properties:...

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Presents an algebraic interpretation of the continuous q-Jacobi polynomials. q deformation of the four-dimensional Euclidean algebra; q analog of the expansion of an exponential in Jacobi polynomials.

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Let {Pn (x)}[sup âˆž, sub n=0] be an orthogonal polynomial system relative to a compactly supported measure. We find characterizations for {P[sub n] (x)}[sup âˆž, sub n=0] to be a Bochner-Krall orthogonal polynomial system, that is, {P[sub n] (x)}[sup âˆž, sub n=0] are polynomial...

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In this paper we consider a large class of many-variable polynomials which contains generalizations of the classical Hermite, Laguerre, Jacobi and Bessel polynomials as special cases, and which occur as the polynomial part in the eigenfunctions of Calogeroâ€“Sutherland type operators and...

- On the construction of de la VallÃ©e Poussin means for orthogonal polynomials using convolution structures. Filbir, Frank; Themistoclakis, Woula // Journal of Computational Analysis & Applications;Oct2004, Vol. 6 Issue 4, p297
In this paper we construct a de la VallÃ©e Poussin approximation process for orthogonal polynomial expansions. Our construction is based on convolution structures which are established by the orthogonal polynomial system. We show that our approach leads to a natural generalization of the de la...

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We consider the classical extremal problem of estimating norms of higher order derivatives of algebraic polynomials when their norms are given. The corresponding extremal problem for general polynomials in uniform norm was solved by A. A. Markov, while Bernstein found the exact constant in the...

- CMV Matrices and Little and Big âˆ’1 Jacobi Polynomials. Derevyagin, Maxim; Vinet, Luc; Zhedanov, Alexei // Constructive Approximation;Dec2012, Vol. 36 Issue 3, p513
We introduce a new map from polynomials orthogonal on the unit circle to polynomials orthogonal on the real axis. This map is closely related to the theory of CMV matrices. It contains an arbitrary parameter Î» which leads to a linear operator pencil. We show that the little and big âˆ’1...