Approximation Problems in Analysis and Probability

By

  • M.P. Heble

This is an exposition of some special results on analytic or C-approximation of functions in the strong sense, in finite- and infinite-dimensional spaces. It starts with H. Whitney's theorem on strong approximation by analytic functions in finite-dimensional spaces and ends with some recent results by the author on strong C-approximation of functions defined in a separable Hilbert space. The volume also contains some special results on approximation of stochastic processes. The results explained in the book have been obtained over a span of nearly five decades.
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Book information

  • Published: January 1989
  • Imprint: NORTH-HOLLAND
  • ISBN: 978-0-444-88021-5

Reviews

...its treatment of approximation in infinite-dimensional spaces provides a welcome, different approach which supplements the existing body of work in this field.
R.M. Aron, Mathematical Reviews


The author has made important contributions...
Zentralblatt für Mathematik



Table of Contents

Weierstrass-Stone Theorem and Generalisations - A Brief Survey. Weierstrass-Stone Theorem. Closure of a Module - The Weighted Approximation Problem. Criteria of Localisability. A Differentiable Variant of the Stone-Weierstrass Theorem. Further Differentiable Variants of the Stone-Weierstrass Theorem. Strong Approximation in Finite-Dimensional Spaces. H. Whitney's Theorem on Analytic Approximation. C- Approximation in a Finite-Dimensional Space. Strong Approximation in Infinite-Dimensional Spaces. Kurzweil's Theorems on Analytic Approximation. Smoothness Properties of Norms in Lp-Spaces. C-Partitions of Unity in Hilbert Space. Theorem of Bonic and Frampton. Smale's Theorem. Theorem of Eells and McAlpin. Contributions of J. Wells and K. Sundaresan. Theorems of Desolneux-Moulis. Ck-Approximation of Ck by C - A Theorem of Heble. Connection Between Strong Approximation and Earlier Ideas of Bernstein-Nachbin. Strong Approximation - Other Directions. Approximation Problems in Probability. Bernstein's Proof of Weierstrass' Theorem. Some Recent Bernstein-Type Approximation Results. A Theorem of H. Steinhaus. The Wiener Process or Brownian Motion. Jump Processes - A Theorem of Skorokhod. Appendices: 1. Topological Vector Spaces. 2. Differential Calculus in Banach Spaces. 3. Differentiable Banach Manifolds. 4. Probability Theory. Bibliography. Index.