Interpolation operators with applications. I. (English) Zbl 1171.41002

This is the first part of a useful survey on interpolation operators for the approximation of functions of one or more variables. After introducing the spaces of functions in which the interpolation problems are formulated, the paper describes the univariate interpolation operators with ranges in the class of polynomials or spline functions. The corresponding Lagrange, Hermite, and Birkhoff interpolation formulas are studied. Next, multivariate interpolation operators are considered in detail for the cases in which the domain of the interpolated functions is of rectangular type, simplex type, or arbitrary, with special emphasis on the bivariate case. Interpolation formulas based on product operators, Boolean sums, Steffensen and Stancu operators, and various types of Shepard operators are derived. Numerous examples of these interpolation procedures are provided.


41A05 Interpolation in approximation theory
41A10 Approximation by polynomials
41A15 Spline approximation
41A20 Approximation by rational functions