Inversion and Representation Theorems for the Laplace Transformation

• Main Author: Rooney, Paul George
• Format: Others
• Published: 1952
• Online Access: https://thesis.library.caltech.edu/10602/2/Rooney_PG_1952.pdf; Rooney, Paul George (1952) Inversion and Representation Theorems for the Laplace Transformation. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/1WJR-ZQ41. https://resolver.caltech.edu/CaltechTHESIS:12122017-142045730 <https://resolver.caltech.edu/CaltechTHESIS:12122017-142045730>

A study is made of the Laplace transformation on Banach-valued functions of a real variable, with particular reference to inversion and representation theories. First a new type of integral for Banach-valued functions 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 operator when the integral in the Laplace transformation is each of the above mentioned types. Lastly, representation of Banach-valued functions by Laplace integrals of functions in B_p([0,∞);¥), 1 &#8804; p &#60; ∞, is studied, and theorems are very like those proved, for numerically-valued functions, by D. V. Widder in his book "The Laplace Transform" (Princeton, 1941) page 312. The classes H_p(α ; ¥), 1 &#8804; p &#60; ∞, are also studied in this section as is the representation of numerically-valued functions by Laplace-Stieltjes integrals.