Konvergenz mehrdimensionaler Interpolation

Summary This paper is concerned with convergence of interpolation polynomials in two variables. Some investigations have been made in the special cases of Lagrange and Fejér-Hermite interpolation. Here we give a general method to prove convergence of two-dimensional interpolation processes. The theory is based on the concept of tensor products of interpolation problems. Our main theorem shows that a two-dimensional interpolation process which is a tensor product of two one-dimensional processes converges for a functionh єC [a, b] C [, b] if both factors do for certain one-dimensional functions. In addition we prove how the order of convergence of the two-dimensional process depends on the one-dimensional processes.The converse theorem is not true. But we can give a sufficient condition which depends on the density of the interpolation nodes to assure the converse theorem in some cases.In an easy manner the convergence theorem may be extended to higher dimensions.

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Herrn Prof. Dr. Dr. h. c. L. Collatz zum 60. Geburtstag gewidmet