Weierstrass and Approximation Theory

@article{Pinkus2000WeierstrassAA,
  title={Weierstrass and Approximation Theory},
  author={Allan Pinkus},
  journal={Journal of Approximation Theory},
  year={2000},
  volume={107},
  pages={1-66}
}
  • A. Pinkus
  • Published 1 November 2000
  • Education
  • Journal of Approximation Theory
We discuss and examine Weierstrass’ main contributions to approximation theory. §1. Weierstrass This is a story about Karl Wilhelm Theodor Weierstrass (Weierstras), what he contributed to approximation theory (and why), and some of the consequences thereof. We start this story by relating a little about the man and his life. Karl Wilhelm Theodor Weierstrass was born on October 31, 1815 at Ostenfelde near Munster into a liberal (in the political sense) Catholic family. He was the eldest of four… 

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References

SHOWING 1-10 OF 126 REFERENCES

Men of Mathematics

TLDR
The amount of biographical details and of mathematics that the writer has compressed into a volume of 650 pages is extraordinary ; but he is never dull ; his style is lively, at times even 'snappy' ; he carries the reader along ; he whets the appetite.

Theory of Approximation

  • J. Cooper
  • Mathematics
    The Mathematical Gazette
  • 1960
Einfuhrung in Theorie und Anwendungen der Laplace-Transformation. By G. D oetsch . Pp. 301. Fr./D.M. 39.40. 1958. (Birkhauser, Basel und Stuttgart) This is an account of those properties of the

The origins of Cauchy's rigorous calculus

This book explores the background of a major intellectual revolution: the rigorous reinterpretation of the calculus undertaken by Augustin-Louis Cauchy and his contemporaries in the first part of the

Lecons sur l'approximation des Fonctions d'une Variable Reelle

are the best chapters in a book full of good things—a book which, while it is hardly suitable for a beginner working privately, is one that, if used under the guidance of an experienced teacher, can

The Hausdorff dimension of graphs of Weierstrass functions

The Weierstrass nowhere differentiable function, and functions constructed from similar infinite series, have been studied often as examples of functions whose graph is a fractal. Though there is a

Applications of the theory of Boolean rings to general topology

In an earlier paperf we have developed an abstract theory of Boolean algebras and their representations by algebras of classes. We now relate this theory to the study of general topology. The first

Lectures on complex approximation

I: Approximation by Series Expansions and by Interpolation.- I. Representation of complex functions by orthogonal series and Faber series.- 1. The Hilbert space L2(G).- A. Definition of L2(G).- B.

Introduction to approximation theory

Introduction: 1 Examples and prospectus 2 Metric spaces 3 Normed linear spaces 4 Inner-product spaces 5 Convexity 6 Existence and unicity of best approximations 7 Convex functions The Tchebycheff

An elementary proof of the Stone-Weierstrass theorem

In this note we give an elementary proof of the Stone-Weierstrass theorem. The proof depends only on the definitions of compactness ("each open cover has a finite subcover") and continuity ("the

Sur un point de la théorie des fonctions génératrices d’Abel

...