Multivariate Bernoulli splines and the periodic interpolation problem

Periodic spline interpolation in Euclidian spaceRd is studied using translates of multivariate Bernoulli splines introduced in [25]. The interpolating polynomial spline functions are characterized by a minimal norm property among all interpolants in a Hilbert space of Sobolev type. The results follow from a relation between multivariate Bernoulli splines and the reproducing kernel of this Hilbert space. They apply to scattered data interpolation as well as to interpolation on a uniform grid. For bivariate three-directional Bernoulli splines the approximation order of the interpolants on a refined uniform mesh is computed.

[1]  Martin H. Gutknecht,et al.  Attenuation factors in multivariate fourier analysis , 1987 .

[2]  P. Müller.  H. Meschkowski, Hilbertsche Räume mit Kernfunktion. VIII + 256 S. m. 11 Abb. Berlin/Göttingen/Heidelberg 1962. Springer‐Verlag. Preis geb. DM 58,– , 1963 .

[3]  C. D. Boor,et al.  On splines and their minimum properties , 1966 .

[4]  K. Ritter.  Two Dimensional Splines and their Extremal Properties , 1969 .

[5]  Gregory M. Nielson,et al.  Bivariate spline functions and the approximation of linear functionals , 1973 .

[6]  H. G. ter Morsche,et al.  Attenuation Factors and multivariate periodic spline interpolation , 1987, Topics in Multivariate Approximation.

[7]  The Bernoulli Spline and Approximation by Trigonometric Blending Polynomials , 1989 .

[8]  H. Ehlich,et al.  Untersuchungen zur numerischen Fourieranalyse , 1966 .

[9]  Herbert Meschkowski,et al.  Hilbertsche Räume mit Kernfunktion , 1962 .

[10]  J. L. Walsh,et al.  The theory of splines and their applications , 1969 .

[11]  Lois Mansfield.  Optimal approximation and error bounds in spaces of bivariate functions , 1972 .

[12]  I. J. Schoenberg.  On Interpolation by Spline Functions and its Minimal Properties , 1988 .

[13]  W. Schempp,et al.  Einführung in die harmonische Analyse , 1980 .

[14]  Lois Mansfield,et al.  On the variational characterization and convergence of bivariate splines , 1972 .

[15]  Werner Haussmann,et al.  On multivariate spline systems , 1974 .

[16]  Franz-Jürgen Delvos,et al.  On the variational characterization of bivariate interpolation methods , 1975 .

[17]  Arthur Sard.  Optimal approximation: an addendum , 1968 .

[18]  Michael Golomb,et al.  APPROXIMATION BY PERIODIC SPLINE INTERPOLANTS ON UNIFORM MESHES. , 1968 .

[19]  Franz-Jürgen Delvos.  Periodic interpolation on uniform meshes , 1987 .

[20]  On optimal periodic spline interpolation , 1975 .