A PROPERTY OF L-L INTEGRAL TRANSFORMATIONS

The main result of this paper is the result that the collection of all integral transformations of the formF(x)=∫_0^∞ G(x,y)f(y)dy for all x ≥ 0, where f(y) is defined on [0,∞) and G(x,y) defined on D={(x,y): x ≥ 0, y ≥ 0} 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 f ε L, f(x)=∫_0^∞ G(x,y)f(y)dy for almost all x ≥ 0. 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).

並列關鍵字

L-L Integral Transformation ； Absolutely continuity of integrals ； Lebesgue measurable ； Lebesgue points

國際替代計量

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