Inversion and Representation Theorems for the Laplace Transformation

[摘要] A study is made of the Laplace transformation on Banach-valuedfunctions of a real variable, with particular reference to inversionand representation theories. First a new type of integral for Banach-valuedfunctions of a real variable, the "Improper Bochner" integral is defined.The relations between the Bochner, Improper Bochner,Riemann-Graves, and Riemann-Stieltjes integrals are studied. Next,inversion theorems are proved for a new "real" inversion operatorwhen the integral in the Laplace transformation is each of the abovementioned types. Lastly, representation of Banach-valued functions byLaplace integrals of functions in B_p([0,∞);¥), 1 ≤ p < ∞, is studied, and theorems are very like those proved, for numerically-valuedfunctions, by D. V. Widder in his book "The Laplace Transform"(Princeton, 1941) page 312. The classes H_p(α ; ¥), 1 ≤ p < ∞, arealso studied in this section as is the representation of numerically-valuedfunctions by Laplace-Stieltjes integrals.

[发布机构] University:California Institute of Technology;Department:Physics, Mathematics and Astronomy

[关键词] Mathematics