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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

##### MSC:
 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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##### References:
 [1] Boehm, W.: Triangular spline algorithms. Comput. Aided Geometric Design.1, 148-162 (1984) · Zbl 0604.65006 [2] de Boor, C., DeVore, R.: Approximation by smooth multivariate splines. Trans. Amer. Math. Soc.276, 775-788 (1983) · Zbl 0529.41010 [3] de Boor, C., Höllig, K.: B-splines from parallelepipeds. J. Analyse Math.42, 99-115 (1982/83) · Zbl 0534.41007 [4] de Boor, C., Höllig, K.: Bivariate box splines and smooth pp functions on a three direction mesh. J. Comput. Appl. Math.9, 13-28 (1983) · Zbl 0521.41009 [5] de Boor, C., Höllig, K., Riemenschneider, S.: Bivariate cardinal interpolation by splines on a three-direction mesh. Illinois J. Math.29, 533-556 (1985) · Zbl 0586.41005 [6] Chui, C.K., Wang, R.-H.: Multivariate B-splines on triangulated rectangles. J. Math. Anal. Appl.92, 533-551 (1983) · Zbl 0523.41008 [7] Chui, C.K., Wang, R.-H.: Spaces of bivariate cubic and quartic splines on type-1 triangulations. J. Math. Anal. Appl.101, 540-554 (1984) · Zbl 0597.41013 [8] Dahmen, W., Micchelli, C.A.: Translates of multivariate splines. Linear Algebra Appl.52/53, 217-234 (1983) · Zbl 0522.41009 [9] Dahmen, W., Micchelli, C.A.: Recent progress in multivariate splines. Approximation Theory IV (C.K. Chui, L.L. Schumaker, J.D. Wards, eds.), 27-121. New York: Academic Press, 1983 · Zbl 0559.41011 [10] Dahmen, W., Micchelli, C.A.: Some results on box splines. Bull. Amer. Math. Soc.11, 147-150 (1984) · Zbl 0538.41017 [11] Gautschi, W.: Attenuation factors in practical Fourier analysis. Numer. Math.18, 373-400 (1972) · Zbl 0231.65101 [12] Gutknecht, M.H.: Two applications of periodic splines. Approximation Theory III (E.W. Cheney ed.), 467-472. New York: Academic Press, 1980 [13] Henrici, P.: Fast Fourier methods in computational complex analysis. SIAM Rev.21, 481-527 (1979) · Zbl 0416.65022 [14] Jia, R.-Q.: Linear independence of translates of a box spline. J. Approx. Theory40, 158-160 (1984) · Zbl 0534.41008 [15] Katznelson, Y.: An Introduction to Harmonic Analysis. New York: Wiley, 1968; Dover, 1976 · Zbl 0169.17902 [16] Locher, F.: Interpolation on uniform meshes by the translates of one function and related attenuation factors. Math. Comput.37, 403-416 (1981) · Zbl 0517.42004 [17] Prautzsch, H.: Unterteilungsalgorithmen für multivariate Splines ? Ein geometrischer Zugang. Diss., Technische Universität Braunschweig, 1983/84 · Zbl 0647.41015 [18] Sabin, M.: The use of piecewise forms for the numerical representation of shape. Diss., Hungarian Academy of Science, Budapest, 1977 [19] Schoenberg, I.J.: On spline interpolation at all integer points of the real axis. Mathematica (Cluj)10, 151-170 (1968) · Zbl 0183.33101 [20] Schoenberg, I.J.: Cardinal Spline Interpolation. CBMS Vol. 12, Philadelphia: Soc. Indust. Appl. Math., 1973
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