Complex orthogonality on the semicircle with respect to Gegenbauer weight: Theory and applications. (English) Zbl 0732.65014
Topics in mathematical analysis, Vol. Dedicated Mem. of A. L. Cauchy, Ser. Pure Math. 11, 695-722 (1989).
Summary: [For the entire collection see Zbl 0721.00014.]
Complex polynomials \(\{\pi_ k\}\), \(\pi_ k(z)=z^ k+...,\) orthogonal with respect to a complex-valued inner product \((f,g)=\int^{\pi}_{0}f(e^{i\theta})g(e^{i\theta})w(e^{i\theta})d \theta,\) \(w(z)=(1-z^ 2)^{\lambda -{1\over 2} }, \lambda >-{1\over 2},\) are considered and used in construction of Gauss-Gegenbauer quadrature formulas over the semi-circle. Some numerical results regarding error bounds for these formulas, applied to analytic functions, are given. Applications are discussed involving numerical differentiation and the computation of Cauchy principal value integrals with Gegenbauer weights.
Complex polynomials \(\{\pi_ k\}\), \(\pi_ k(z)=z^ k+...,\) orthogonal with respect to a complex-valued inner product \((f,g)=\int^{\pi}_{0}f(e^{i\theta})g(e^{i\theta})w(e^{i\theta})d \theta,\) \(w(z)=(1-z^ 2)^{\lambda -{1\over 2} }, \lambda >-{1\over 2},\) are considered and used in construction of Gauss-Gegenbauer quadrature formulas over the semi-circle. Some numerical results regarding error bounds for these formulas, applied to analytic functions, are given. Applications are discussed involving numerical differentiation and the computation of Cauchy principal value integrals with Gegenbauer weights.
MSC:
| 65D32 | Numerical quadrature and cubature formulas |
| 65D25 | Numerical differentiation |
| 41A55 | Approximate quadratures |
| 33C45 | Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) |
