Refrence Request: Laplace Transform (and Inverse Transform) for Mathematicians

Question

I have the standard pure math background in ODEs, analysis (up to Papa Rudin), and linear algebra, and am familiar with the "mechanics" of Laplace transforms as a control systems engineer. I'd like to learn about the formal theory of Laplace transforms such as books on modern Fourier theory cover the Fourier transforms, but can only find mechanical expositions for engineers on the internet. Does anyone have a good reference for this?

Answer 1

If you are working with square integrable time functions $f: [0,\infty)\rightarrow\mathbb{C}$, then the Laplace transform is fully characterized in terms of the Fourier transform. The Laplace transform $$ \mathscr{L}\{f\}(s)=\int_{0}^{\infty}e^{-st}f(t)dt $$ is a function in $H^2(\Pi^+)$; $H^2$ is the Hardy space of holomorphic functions $g(z)$ on the right half-plane for which $$ \|g\|^2_{H^2}= \sup_{y > 0}\int_{-\infty}^{\infty}|g(x+iy)|^2dy < \infty. $$ The holomorphic function $g$ has an $L^2$ boundary function on the imaginary axis, and has the Cauchy integral representation $$ g(z) = \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{g(iu)}{iu-z}du, \\ \|g\|_{H^2(\Pi^+)} = \|g(it)\|_{L^2(\mathbb{R})}. $$ This corresponds with the Bromwich inversion integral for the Laplace transform: $$ f(z) = \frac{1}{2\pi i}\int_{\gamma-i\infty}^{\gamma+i\infty}e^{sz}\mathscr{L}\{f\}(s)ds. $$ To get from one to the other is accomplished using the identity $$ \int_{0}^{\infty}e^{iut}e^{-tz}dt=-\frac{1}{iu-z} $$ So, for at least this class of functions, the Laplace transform theory combines $L^2$ Fourier transforms and holomorphic functions on the right half plane.

The Functional Analysis context is operator theory on $H^2$. Wiener-Hopf integral equations were the precursor of this; many of the early results were due to Norbert Wiener. Wiener used his technique of splitting forward and backward equations to solve convolution equations that others could not. Weiner was working on filtering and prediction for the US Government when he came up with his results. The $L^2$-$H^2$ results started with Paley and Wiener. Very often the results are cast in terms of Hardy spaces on the unit disk. In any case, Holomorphic function spaces $H^2(\mathbb{\Pi}^+)$ or $H^2(\mathbb{D})$ ($D$=unit disk) and operators on these spaces become a way to study the Laplace transform for $L^2$ time functions on $[0,\infty)$. The techniques of discrete and continuous tranforms come together in this context.

Unfortunately, I don't know of a good place to tell you to begin, except to take a look at the book by Wiener and Paley, which is rather old.

https://www.amazon.com/Fourier-Transforms-Complex-Colloquium-Publications/dp/0821810197/ref=sr_1_1?ie=UTF8&qid=1481837633&sr=8-1&keywords=paley+wiener