International Journal of Mathematics and Mathematical Sciences
Volume 8 (1985), Issue 3, Pages 441-448
doi:10.1155/S0161171285000485
Norm-preserving L−L integral transformations
Department of Mathematics, University of Wisconsin-Oshkosh, Oshkosh 54901, Wisconsin, USA
Received 22 March 1984
Copyright © 1985 Yu Chuen Wei. 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 L−L integral transformation G of the form F(x)=∫0∞G(x,y)f(y)dy, where G(x,y) is defined on D={(x,y):x≥0,y≥0} and f(y) is defined on [0,∞). The following results are proved: For an L−L integral transformation G to be norm-preserving, ∫0∞|G*(x,t)|dx=1 for almost all t≥0 is only a necessary condition, where G*(x,t)=limh→0inf1h∫tt+hG(x,y)dy for each x≥0. For certain G's. ∫0∞|G*(x,t)|dx=1 for almost all t≥0 is a necessary and sufficient condition for preserving the norm of certain f ϵ L. In this paper the analogous result for sum-preserving L−L integral transformation G is proved.