Bivariate cardinal interpolation by splines on a three-direction mesh

Abstract

Abstract : In a series of monographs Schogenberg developed a comprehensive theory of univariate cardinal splines. His results strongly influenced the analysis of totally positive matrices. In this report the author extend two of his basic results on cardinal interpolation to bivariate box-splines. They show that, for functions of exponential type, cardinal interpolation is a rapidly convergent approximation process as the degree tends to infinity. Being not restricted to a tensor product mesh gives a greater flexibility, and because of the exponential decay of the Lagrange functions, spline interpolation is suitable, e.g., for data smoothing. They also expect that bivariate cardinal splines have a similar significance for theoretical questions as in the univariate case.