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International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 3, Pages 441-448
http://dx.doi.org/10.1155/S0161171285000485

Norm-preserving LL integral transformations

Department of Mathematics, University of Wisconsin-Oshkosh, Oshkosh 54901, Wisconsin, USA

Received 22 March 1984

Copyright © 1985 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we consider an LL integral transformation G of the form F(x)=0G(x,y)f(y)dy, where G(x,y) is defined on D={(x,y):x0,y0} and f(y) is defined on [0,). The following results are proved: For an LL integral transformation G to be norm-preserving, 0|G*(x,t)|dx=1 for almost all t0 is only a necessary condition, where G*(x,t)=limh0inf1htt+hG(x,y)dy for each x0. For certain G's. 0|G*(x,t)|dx=1 for almost all t0 is a necessary and sufficient condition for preserving the norm of certain f ϵ L. In this paper the analogous result for sum-preserving LL integral transformation G is proved.