Approximation Of Continuously Differentiable Functions

This self-contained book brings together the important results of a rapidly growing area. As a starting point it presents the classic results of the theory. The book covers such results as: the extension of Wells' theorem and Aron's theorem for the fine topology of order m; extension of Bernstein's and Weierstrass' theorems for infinite dimensional Banach spaces; extension of Nachbin's and Whitney's theorem for infinite dimensional Banach spaces; automatic continuity of homomorphisms in algebras of continuously differentiable functions, etc.

This book presents the evolution of uniform approximations of continuous functions. Starting from the simple case of a real continuous function defined on a closed real interval, i.e., the Weierstrass approximation theorems, it proceeds up to the abstract case of approximation theorems in a locally convex lattice of (M) type. The most important generalizations of Weierstrass’ theorems obtained by Korovkin, Bohman, Stone, Bishop, and Von Neumann are also included. In turn, the book presents the approximation of continuous functions defined on a locally compact space (the functions from a weighted space) and that of continuous differentiable functions defined on ¡n. In closing, it highlights selected approximation theorems in locally convex lattices of (M) type. The book is intended for advanced and graduate students of mathematics, and can also serve as a resource for researchers in the field of the theory of functions.

Papers and articles about periodic functions approximation.

​​​​​​​ This brief provides an elementary introduction to the theory of piecewise differentiable functions with an emphasis on differentiable equations. In the first chapter, two sample problems are used to motivate the study of this theory. The presentation is then developed using two basic tools for the analysis of piecewise differentiable functions: the Bouligand derivative as the nonsmooth analogue of the classical derivative concept and the theory of piecewise affine functions as the combinatorial tool for the study of this approximation function. In the end, the results are combined to develop inverse and implicit function theorems for piecewise differentiable equations. This Introduction to Piecewise Differentiable Equations will serve graduate students and researchers alike. The reader is assumed to be familiar with basic mathematical analysis and to have some familiarity with polyhedral theory.

This collection is the 14th in an ongoing series on differentiable functions of several variables, presenting recent contributions to a line of research begun by Sobolev in 1950. The papers study various spaces of differentiable functions of several real variables in Euclidean space, their imbeddings, equivalent normings, weighted estimates of derivatives, and traces on sets. Several questions of approximation in function spaces on the line, on a hyperboloid, and on Lobachevsky space are studied. Investigations of bilinear approximations are applied to estimates of the singular numbers of integral operators and widths. The authors also examine the asymptotics of the spectrum of elliptic systems, as well as the Dirichlet variational problem for a degenerate elliptic operator. Finally, a block method of solving Laplace's equation for nonanalytic boundary conditions is developed.

This book is intended as a self-contained introduction for non-specialists, or as a reference work for experts, to the particular area of approximation theory that is concerned with exact constants. The results apply mainly to extremal problems in approximation theory, which in turn are closely related to numerical analysis and optimization. The book encompasses a wide range of questions and problems: best approximation by polynomials and splines; linear approximation methods, such as spline-approximation; optimal reconstruction of functions and linear functionals. Many of the results are based on deep facts from analysis and function theory, such as duality theory and comparison theorems; these are presented in chapters 1 and 3. In keeping with the author's intention to make the book as self-contained as possible, chapter 2 contains an introduction to polynomial and spline approximation. Chapters 4 to 7 apply the theory to specific classes of functions. The last chapter deals with n-widths and generalises some of the ideas of the earlier chapters. Each chapter concludes with commentary, exercises and extensions of results. A substantial bibliography is included. Many of the results collected here have not been gathered together in book form before, so it will be essential reading for approximation theorists.