Norm-preserving L−L integral transformations. (1985) - Wei, Yu Chuen
Abstract : In this paper we consider anL − L integral transformationG of the formF ( x ) = ∫ 0 ∞ G ( x, y ) f ( y ) d y, whereG ( x, y ) is defined onD = { ( x, y ) : x ≥ 0, y ≥ 0 } andf ( y ) is defined on[ 0, ∞ ) . The following results are proved: For anL − L integral transformationG to be norm-preserving, ∫ 0 ∞ | G * ( x, t ) | d x = 1 for almost allt ≥ 0 is only a necessary condition, whereG * ( x, t ) = lim h → 0 inf 1 h ∫ t t + h G ( x, y ) d y for eachx ≥ 0 . For certainG 's.∫ 0 ∞ | G * ( x, t ) | d x = 1 for almost allt ≥ 0 is a necessary and sufficient condition for preserving the norm of certainf  ϵ  L . In this paper the analogous result for sum-preservingL − L integral transformationG is proved.