Computing expansion coefficients in orthogonal bases of ultraspherical polynomials.

*(English)*Zbl 0827.65021The 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.

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.

Reviewer: D.Kershaw (Lancaster)

##### MSC:

65D20 | Computation of special functions and constants, construction of tables |

33C55 | Spherical harmonics |

65D32 | Numerical quadrature and cubature formulas |

##### Keywords:

expansion coefficients; orthogonal bases; ultraspherical polynomials; Gegenbauer polynomials
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\textit{F. Sartoretto} and \textit{R. Spigler}, J. Comput. Appl. Math. 56, No. 3, 295--303 (1994; Zbl 0827.65021)

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##### References:

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