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PROJECTION OPERATORS IN APPROXIMATION THEORY

IP.com Disclosure Number: IPCOM000128699D
Original Publication Date: 1976-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 36 page(s) / 76K

Publishing Venue

Software Patent Institute

Related People

E. W. Cheney: AUTHOR [+3]

Abstract

This is an expository article addressed to nonspecialists, explaining how linear operators occur in approximation theory and how certain natural extremum. problems involving them arise.

This text was extracted from a PDF file.
This is the abbreviated version, containing approximately 8% of the total text.

Page 1 of 36

THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

PROJECTION OPERATORS IN APPROXIMATION THEORY

E. W. Cheney

October 1976 CNA-115

Abstract.

This is an expository article addressed to nonspecialists, explaining how linear operators occur in approximation theory and how certain natural extremum. problems involving them arise.

CENTER FOR NUMERICAL ANALYSIS

THE UNIVERSITY OF TEXAS AT AUSTIN

CONTENTS

1. A Reconnaissance of Approximation Theory 2. Linear Approximation Operators . . . .

3. Projection Constants . . . . . . . . . . .. . . . . . . . . . . . . . . .

4. Minimal Projections . . . . . . . . . . . .. . . . . . . . . . . . . . . . 12 5. Projections Having Finite Carrier
. . . . .. . . . . . . . . . . . . . . . 20 6. Orthogonal Projections in C(T) . . . . . . . . . . . . . . . . . . . . . 26 7.
Projections in e 1- spaces . . . . . . . . . . . . . . . . . . . . . . . . 28 8. Open Problems . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 32 9. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . o . 33

1. A Reconnaisance of Approximation Theory

Approximation Theory traditionally concerns itself with the approximation of functions, primarily functions of a real or complex variable. As we shall see, some newer aspects of the subject concern the approximation of other objects, particularly operators.

A typical situation within the purview of approximation theory is the follow-ing. One is presented with a metric space (X,d) whose elements are the objects to be approximated. Within X a subset Y is prescribed. The elements of Y are the "approximators." For any x e X and any y e Y. d(x.,y) is interpreted as the error or discrepancy when y is used as an approximation to x. The quantity

(Equation Omitted)

measures how well x is capable of being approxi- yey mated by elements of Y. For a fixed pair Y c-- X, a theory of approximation would attempt to answer such questions as the following: (1) Does there exist., for an arbitrary x in X, a "best" approximation y in Y, i.e., an element for which

University of Texas Page 1 Dec 31, 1976

Page 2 of 36

PROJECTION OPERATORS IN APPROXIMATION THEORY

(Equation Omitted)

Is it possible to estimate

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from a knowledge of special properties of x (as, for example,, its differentiability if it is a function)? (3) Does there exist a continuous "proximity map"

(Equation Omitted)

i.e., a map such that

(Equation Omitted)

for all x e X? (4) Is it true that each x has a unique best approximation in Y? (5) For a fixed x in X, how are its best approximations in Y characterized? (6) What algorithms (numerical procedures) can be devised for producing best approximations? (7) Do there exist simple and convenient substitutes for the proximity map? In particular,, do such maps

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exist satisfying

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with e small?

The above questions concern a fixed pair, Y~-- X. If there is a nested se-quence of sets,

(Equation Omitted)

then some ques...