Norm-preserving L−L integral transformations. (1985)

• Record Type: Journal Article
• Authors: Wei, Yu Chuen
• Is Part Of: International journal of mathematics and mathematical sciences. Volume 8:Number 3(1985)
• Page Start: 441
• Page End: 448
• Publication Date: 1985
• Subjects: ℓ−ℓ method. L−L integral transformation. absolutely continuity of integrals. Fubini-Tonelli theorem; Mathematics -- Periodicals; 510.5
• DOI: 10.1155/S0161171285000485
• Languages: English
• ISSNs: 0161-1712

Abstract

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.