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zu Castell, W.; Filbir, F.

Radial basis functions and corresponding zonal series expansions on the sphere. (English) Zbl 1063.42017

J. Approximation Theory 134, No. 1, 65-79 (2005).
Summary: Since radial positive definite functions on \(\mathbb R^d\) remain positive definite when restricted to the sphere, it is natural to ask for properties of the zonal series expansion of such functions which relate to properties of the Fourier-Bessel transform of the radial function. We show that the decay of the Gegenbauer coefficients is determined by the behavior of the Fourier-Bessel transform at the origin.

Cited in 14 Documents

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42A82 Positive definite functions in one variable harmonic analysis
33C10 Bessel and Airy functions, cylinder functions, \({}_0F_1\)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)

Keywords:

radial basis functions; zonal basis functions; positive definite functions; conditionally positive definite functions; Bessel functions; Gegenbauer polynomials; kriging; Fourier-Bessel transform
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References:

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