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Computing expansion coefficients in orthogonal bases of ultraspherical polynomials. (English) Zbl 0827.65021
The problem is the calculation of the coefficients in the approximation of a function by an expansion in Gegenbauer polynomials. The authors propose to replace the single integrals by double integrals. (This means that the integrand contains only elementary functions which are readily available). It is claimed that there are advantages in this. However the authors choose as basis for comparison two inappropriate methods which will clearly give poor results. The third method with which it is compared is the use of the appropriate Gaussian formulae. The results seem to show that the proposed method has little, if any improvement in accuracy over this.
It may be that their method would be useful in situations when the Gaussian formulae were not available. There is no discussion of the calculation of values of the gamma function which occur in the formulae.

65D20 Computation of special functions and constants, construction of tables
33C55 Spherical harmonics
65D32 Numerical quadrature and cubature formulas
Full Text: DOI
[1] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G., Higher transcendental functions, 2, (1953), McGraw-Hill New York, Bateman Manuscript Project · Zbl 0051.30303
[2] Laforgia, A., Sui nodi e le costanti di Christoffel di formule di quadratura gaussiana, Techn. report 1, (1977), 1st. Calcoli Numer., Torino
[3] Scanlon, P.J., An alternative form for the Legendre polynomial expansion coefficients, J. comput. phys., 69, 482-486, (1987) · Zbl 0609.33005
[4] Szego˝, G., Orthogonal polynomials, XXIII, (1939), Amer. Mathematical Soc., New York, Amer. Math. Soc. Colloq. Publ. · JFM 65.0278.03
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