# Approximation Problems in Analysis and Probability

**By**

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 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.