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A MIXED-NORK BIVARIATE APPROXIMATION PROBLEM WITH APPLICATIONS TO LEWANOWICZ OPERATORS

IP.com Disclosure Number: IPCOM000128701D
Original Publication Date: 1977-Dec-31
Included in the Prior Art Database: 2005-Sep-16
Document File: 7 page(s) / 21K

Publishing Venue

Software Patent Institute

Related People

G. M. Phillips: AUTHOR [+5]

Abstract

This paper considers the approximation of f(x,y) by p(x,y) when the measure of accuracy is an X -norm in y and a supremum norm in x. Applications to integral transforms are given. This paper was presented at the Durham Working Symposium on Multivariate Approximation held at Durham University, England, 21-30 July 1977. Drs. Phillips and McCabe are at the University of St. Andrews, Scotland. CENTER FOR NUMERICAL ANALYSIS THE UNIVERSITY OF TEXAS AT AUSTIN

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THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

A MIXED-NORK BIVARIATE APPROXIMATION PROBLEM WITH APPLICATIONS TO LEWANOWICZ OPERATORS

by G. M. Phillips,, J. H. McCabe, E. W. Cheney

October 1977 ICNA 127

Abstract:

This paper considers the approximation of f(x,y) by p(x,y) when the measure of accuracy is an X -norm in y and a supremum norm in x. Applications to integral transforms are given.

This paper was presented at the Durham Working Symposium on Multivariate Approximation held at Durham University, England, 21-30 July 1977. Drs. Phillips and McCabe are at the University of St. Andrews, Scotland.

CENTER FOR NUMERICAL ANALYSIS THE UNIVERSITY OF TEXAS AT AUSTIN

1. INTRODUCTION.

We consider here a class of approximation problems having the following form. A convex set 9 of bivariate functions f(x,y) is pre-scribed, and it is desired to determine the quantity

(Equation Omitted)

Here 4 is a prescribed measure. It is desired also to obtain a "minimum solution", that is, a particular element f 0 in 9 for which

(Equation Omitted)

It will be observed that this problem involves both a supremum norm and an Z -norm.

One source of such approximation problems is in the study of integral transforms. It is sometimes necessary to approximate one integral transform by another one which is simpler. If the first has the form

(Equation Omitted)

and if the second has the form

(Equation Omitted)

This author was supported by the Science Research Council of the U.K. and the U.S. Army Research Office. then a measure of the distance between L' and L is given by

University of Texas Page 1 Dec 31, 1977

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A MIXED-NORK BIVARIATE APPROXIMATION PROBLEM WITH APPLICATIONS TO LEWANOWICZ OPERATORS

(Equation Omitted)

A specific application of this nature is considered in Section 3 below.

2. THE MAIN THEOREM.

In this section we formulate and prove a theorem which characterizes the solution of the problem outlined in the previous section.

Let X be a compact Hausdorff space, and let (Y,Z,~L) be a finite measure space. Let ~F be a prescribed convex set of functionsfrom Xx Y to R. About 97 we make the following assumptions.

(1) Equicontinuity. For each

(Equation Omitted)

an equicontinuous family in C(X). This means that

(Equation Omitted)

(2) Integrability. For each f e 9 and for each

(Equation Omitted)

belongs to the Lebesgue class

(Equation Omitted)

(3) Smoothness,. For each f e 9F and for each x e X$

(Equation Omitted)

Thus the zeros of f(x,-) form a set of measure 0. In particular, 0 91. The term "smoothness" is used here in accordance with established meaning in normed linear spaces. Thus, in a normed linear space, an element f is a smooth element if there is a unique hyperplane of support at f to the ball of radins 11f 11 - For each f e 9 there is a "set of critical points" defined by

(Equation Omitted)

MAIN THEOREM. For an element fo of 9, the following two conditions are equivalent:

(Equation Omitted)

(Equation Omitted)

PROOF. The idea...