Smoothing and Approximation of Functions[MatSciRep:55]

Abstract

These notes are based on lectures given at the Institute during January and February of 1967. In essence they constitute an introductory course in the theory of approximation(degree of approximation). The discussion is limited to functions on the real line, and (Except for few introductory theorems) to uniform, or sup norm approximation. Much of the theory presented here can quite easily be extended both to L^p norms and to several variables; To prepare the way for such extensions, it is sought to present the results in the most unified possible way. The treatment given here is not altogether Orthodox, and has some theoretical pretensions. Trigonometric approximation is treated on the Line group, rather than the circle group. This is not altogether trivial change, since the unrolling of the circle onto the line gives rise to different kernels and integral formulas often simpler ones(this device was noted by la vallee Poussin, as well as later authors, especially Bochner, Akhieser, and Butzer, yet not, so far as it is known, fully exploited). Most of the material in chapter 5 is new and may be thought of as carrying out a program begun by P.L. Butzer in a series of papers as listed in the reference. The theorems given here are the first known Inverse theorems for general kernels. The Tauberian formulation of inverse problems given here is so general that it reveals both direct and inverse problems, as part of a single general problem - solved in large measure. Thematic unity in presentation is given, and the generation of approximations by convolution integrals is considered as the main theme. The only specialized knowledge required for reading these notes is the elements of Fourier Transforms. The author wishes to thank Dr. K.R. Unni, Mr.G.N. KeshavaMurthy and Mr.M.R. Subrahmanya for their kind assistance in the preparation of these notes.