Norm-preserving L − L integral transformations

Yu Chuen Wei

International Journal of Mathematics and Mathematical Sciences, 1985, vol. 8, 1-8

Abstract:

In this paper we consider an L − L integral transformation G of the form F ( x ) = ∫ 0 ∞ G ( x , y ) f ( y ) d y , 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 ) | d x = 1 for almost all t ≥ 0 is only a necessary condition, where G * ( x , t ) = lim h → 0 inf 1 h ∫ t t + h G ( x , y ) d y for each x ≥ 0 . For certain G 's. ∫ 0 ∞ | G * ( x , t ) | d x = 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.

Date: 1985
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jijmms:146587

DOI: 10.1155/S0161171285000485

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