It is known that smooth functions with exponential decay at $\pm\infty$ are functions whose Fourier transform have analytic continuation in some suited complex strip. I was wondering what happens if we ask for exponential decay from only one side. More precisely :
The context : In Fourier Analysis, Self-Adjointness by Reed and Simon (Methods of Modern Mathematical Physics, Vol. 2), Theorem IX.14 tells us that (I'll take dimension 1 for simplicity) :
if $T$ is a tempered distribution on
$\mathbb R$ such that :
- $\hat T$ has an analytic continuation to $|\Im z| < a$ for
some $a > 0$- on each slice $\mathbb R + i\eta$ with $|\eta| < a$, $\hat T$ is
integrable- for each $0 < b < a$, the supremum of the integrals on the slices
$\mathbb R + i\eta$ with $|\eta| < b$
is finitethen $T$ is a bounded continuous
function and for any $0 < b < a$, there
exists $C_b \geq 0$ s.t. $$|T(x)| \leq C_b e^{-b|x|}$$
This result can be "desymmetrised" easily : by assuming analytic continuation in $-a_- < |\Im z| < a_+$ with $a_-,a_+ > 0$ we get decay faster than $e^{-a_+x}$ as $x\to -\infty$ and than $e^{a_-x}$ as $x\to+\infty$.
In http://arxiv.org/abs/math/0007097, authors mention a similar result which is an equivalence in $\mathcal S(\mathbb R)$ morally saying
Functions with such kind of decay are functions whose Fourier transform are of rapid decrease and admit a holomorphic continuation in the associated strip such that the continuation on each slice of the strip is of rapid decrease.
The question :
I was wondering if anyone would knew what happens if we only ask for exp. decay at $-\infty$ : for instance for a $L^\infty$ function, is there some reasonable equivalent condition on the Fourier transform (in the sense of tempered distributions) for it to have $e^{ax}$ decay as $x\to-\infty$ ?
Not assuming exp. decay at $+\infty$ makes that such functions can have "ugly" Fourier transforms (since they're not in $L^1$ or $L^2$ in general), for instance $e^x H(-x) + H(x)$ where $H$ denotes the Heaviside step function yields $\delta_0 + i \text{pv}\frac{1}{\xi} – \frac{i}{\xi-i}$. Nonetheless, the "functional part" (in some kind of bad and undefined sense…) of this transform indeed has holomorphic continuation in the half-plane $\Im z < 1$ (but is not $L^1$ on any slice). This is the kind of observations that decided me to ask this question here 🙂
Best Answer
Let me summarize what has been already said. If $f$ is a locally integrable function, you can break it into two parts: $f=f_1+f_2$, where $f_1$ is supported on the positive ray, and $f_2$ on the negative ray. Then you consider two "halves" of the Fourier transform: $$F^-(z)=\int_0^\infty e^{-izt}f_1(t)dt$$ and $$F^+(z)=\int_{-\infty}^0 e^{-itz}f_2(t)dt.$$ The first function is analytic in the lower half-plane, and the second is analytic in the upper half-plane. This pair of analytic functions is called "generalized Fourier transform in the sense of Carleman", or "Fourier transform in the sense of hyperfunctions". If "usual" Fourier transform exists in some sense, like when $f\in L^2$ or $f\in L^1$, or $f$ is a temperate distribution, then it is recovered as the sum of boundary values of these two analytic functions. Boundary values must be understood in the appropriate sense, dependng on your conditions on $f$. Now if $f$ decreases exponentially, say on the negative ray, this means that $F^+$ has an analytic continuation from the upper half-plane to a larger half-plane $\Im z>-a$. That is Fourier transform on the real line is a sum of of a function analytic in the lower half-plane and another one analytic in the larger upper half-plane $\Im z>-a$. This is the characterization you asked for.
In general, a temperate distribution on the real line has a canonical representation as sum of the boundary values of two analytic functions, one in the upper half-plane, another in the lower half-plane. So this property (analytic continuation of one of these functions to a larger half-plane) is an intrinsic property which exactly characterizes the exponential decrease of the original on one of the semiaxes.
References:
Carleman, L'Intégrale de Fourier et Questions que s'y Rattachent, Publications Scientifiques de l'Institut Mittag-Leffler, 1. Uppsala, 1944.
A. Kaneko, Introduction to hyperfunctions, Kluwer, Dordrecht, Tokyo, 1988.