International Journal of Mathematics and Mathematical Sciences
Volume 7 (1984), Issue 3, Pages 497-501

A property of LL integral transformations

Yu Chuen Wei

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

Received 11 April 1984

Copyright © 1984 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.


The main result of this paper is the result that the collection of all integral transformations of the form F(x)=0G(x,y)f(y)dy for all x0, where f(y) is defined on [0,) and G(x,y) defined on D={(x,y):x0,y0} has no identity transformation on L, where L is the space of functions that are Lebesgue integrable on [0,) with norm f=0|f(x)|dx. That is to say, there is no G(x,y) defined on D such that for every fL, f(x)=0G(x,y)f(y)dy for almost all x0. In addition, this paper gives a theorem that is an improvement of a theorem that is proved by J. B. Tatchell (1953) and Sunonchi and Tsuchikura (1952).