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## Description

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.## 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 L

^{p}-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. C

^{k}-Approximation of C

^{k}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.

## Details

- No. of pages:
- 244

- Language:
- English

- Copyright:
- © 1989

- Published:
- 1st January 1989

- Imprint:
- North Holland

- Print ISBN:
- 9780444880215

- Electronic ISBN:
- 9780080872704

## Reviews

@from:R.M. Aron
@qu:...its treatment of approximation in infinite-dimensional spaces provides a welcome, different approach which supplements the existing body of work in this field.
@source:Mathematical Reviews
@qu:The author has made important contributions...
@source:Zentralblatt für Mathematik