CHEBYSHEV POLYNOMIALS OF THE 2ND, 3RD AND 4TH KINDS IN APPROXIMATION, INDEFINITE INTEGRATION, AND INTEGRAL-TRANSFORMS

[摘要] Chebyshev polynomials of the third and fourth kinds, orthogonal with respect to (1 + x)1/2(1 - x)-1/2 and (1 - x)1/2(1 + x)-1/2, respectively, on [-1, 1], are less well known than traditional first- and second-kind polynomials. We therefore summarise basic properties of all four polynomials, and then show how some well-known properties of first-kind polynomials extend to cover second-, third- and fourth-kind polynomials. Specifically, we summarise a recent set of first-, second-, third- and fourth-kind results for near-minimax constrained approximation by series and interpolation criteria, then we give new uniform convergence results for the indefinite integration of functions weighted by (1 + x)-1/2 or (1 - x)-1/2 using third- or fourth-kind polynomial expansions, and finally we establish a set of logarithmically singular integral transforms for which weighted first-, second-, third- and fourth-kind polynomials are eigenfunctions.

[发布日期] 1993-12-31

[效力级别]  Proceedings Paper

[关键词] CHEBYSHEV POLYNOMIALS;JACOBI POLYNOMIALS;ORTHOGONALITY;MINIMAX APPROXIMATION;NEAR-MINIMAX;CONSTRAINED;EXPANSION;INTERPOLATION;INDEFINITE INTEGRATION;INTEGRAL TRANSFORMS;SINGULAR;HYPERSINGULAR