Attenuation factors in multivariate Fourier analysis. (English) Zbl 0639.65079

W. Gautschi [ibid. 18, 373-400 (1972; Zbl 0231.65101)] presented a general theory of attenuation factors for families of periodic functions in one variable and F. Locher [Math. Comput. 37, 403-416 (1981; Zbl 0517.42004)] studied the attenuation factors of interpolants from the linear space spanned by the translates h(x-j/N), \(j\in Z\) of a periodic generating function h(x) on a uniform mesh of size N. In the present paper the above results are extended to the multivariate case where, for simplicity, 1-periodicity but different meshsize is assumed in each coordinate direction. The multivariate version of the basic characterization theorem is proved under somewhat weaker assumptions than in the Gautschi’s paper and special results are stated for tensor product families. In the case of multivariate interpolation by translates of a periodic generating function h(x) some interesting relations are derived under the assumption that h itself is the sum of translates of a nonperiodic germ function H, i.e. \(h(x)=\sum_{j\in Z^ D}H(x-j).\) The latter is applied to box splines where explicit formulas for the computation of the attenuation factors could be obtained.
Reviewer: V.Veselý


65T40 Numerical methods for trigonometric approximation and interpolation
65D05 Numerical interpolation
65D07 Numerical computation using splines
42B05 Fourier series and coefficients in several variables
41A15 Spline approximation
41A63 Multidimensional problems
42A10 Trigonometric approximation
42A15 Trigonometric interpolation
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