Fourier transform of Analytic Functions

Question

Forgive me if this question does not meet the bar for this forum. But i would really appreciated some help.

I'm trying to construct a function according to some conditions in the frequency domain of the Fourier transformation. I want the function to be analytic and real when I transform it back to the time domain. The Fourier transformation of $f$ has of course some symmetry criteria to make $f$ real. But what about the Analytic property. As an analytic function imply some convergent power series expansion, and the Fourier transform of a polynomial is a sum of derivatives of Delta functions, I assume that there is a corresponding criteria of the Fourier transformation.

So the question is: If a function $f:\mathbb{R}\rightarrow \mathbb{R}$ is assumed to be analytic, what is the corresponding criteria for the Fourier transform of the function $\mathcal{F}[f] (k)$?

Edit: what I am trying to construct is probability distribution with the following condition

$f(x/\mu)/\mu=\frac{2}{3} f(x) + \frac{1}{3} (f\ast f)(x)\quad$ where $\ast$ mark the convolution, and $\mu=\frac{4}{3}$. $f$ is positive and real for $x\in [0,\infty)$

Taking the fourier transformation make the condition simpler:

$\tilde f(\mu k) = \frac{2}{3}\tilde f(k) + \frac{1}{3}\tilde f^2(k)$

So my problem is to construct $f$ (I am in particular interested in the tail behavior) and I try to use the properties of $\tilde f$. I posted a similar problem a while ago (see here). Julián Aguirre answered how to construct $\tilde f$ if it is analytic. But the inverse transformation of the power expansion is an infinite sum of derivatives of Delta functions, and is of little help.

Answer 1

What is sufficient (though not necessary) is that the Fourier transform decays exponentially at $\infty$ (if you want just analyticity on the line) or faster than any exponent (if you want your original function to be entire). In particular, anything with compact support will do. If this is too restrictive for your construction, you'd better just tell what exactly you are trying to construct.

Answer 2

Heh. I grabbed my copy of Reed and Simon, volume 2, and by chance opened it to exactly the right page. The short answer is "exponential decay". Section IX.3 has several relevant theorems. For instance:

Theorem IX.13. Let $f$ be in $L^2(\mathbb{R}^n)$. Then $e^{b|x|} f \in L^2(\mathbb{R}^n)$ for all $b < a$ if and only if $\hat{f}$ has an analytic continuation to the set $\{\zeta : |\mathrm{Im} \zeta| < a\}$ with the property that for each $\eta \in \mathbb{R}^n$ with $|\eta| < a$, $\hat{f}(\cdot + i\eta) \in L^2(\mathbb{R}^n)$ and for any $b < a$, $\sup_{|\eta|\le b} \lVert \hat{f}(\cdot + i\eta)\rVert_2 < \infty$.

MR0493420 (58 #12429b) Reed, Michael; Simon, Barry Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. xv+361 pp. (Reviewer: P. R. Chernoff) 47-02 (81.47)

Answer 3

If you need an "if and only if" result, you should use a generalization of the Fourier transform called the FBI transform. There's a nice theorem linking the real analyticity of a function to the decay of its FBI transform. On page 137 of A Primer of Real Analytic Functions, Krantz and Parks state it this way:

Fix $x_0 \in \mathbb{R}$. An integral function $f$ is real analytic at $x_0$ if and only if $f$ satisfies condition $RA(x_0)$.

A function satisfies "condition $RA(x_0)$" if its FBI transform decays exponentially as you vary the parameters in a certain way.

I suspect the FBI transform might be overkill for what you're trying to do, but if you're interested, check out Section 5.3 of A Primer of Real Analytic Functions.

Answer 4

The following basic result needs to be quoted on these matters of analyticity: the Paley-Wiener-Schwartz theorem gives a characterization of distributions with compact support. Let $u$ be a tempered distribution ; $u$ is compactly supported in a ball of center $0$ and radius $R$ if an only if $\hat u$ is an entire function such that $$\exists C_0, \exists N_0,\forall \zeta\in \mathbb C^d,\quad \vert\hat u(\zeta)\vert\le C_0(1+\vert \zeta\vert)^{N_0}e^{R\vert\Im\zeta\vert}. $$ Something analogous allows a characterization of $C^\infty$ functions with compact support.

This leads to the following characterization of the analytic wave front set, due to Bros and Iagolnitzer. Let $v\in \mathcal E'(\mathbb R^d)$. We define the Fourier-Bros-Iagolnitzer transform $Tv$ of $v$ by the following formula, where the integral is in fact a bracket of duality, $$ (Tv)(z,\lambda)=\int_{\mathbb R^d} e^{-\pi\lambda(z-x)^2} v(x) dx,\qquad z\in \mathbb C^d, \lambda >0. $$ Let $\Omega$ be an open subset of $\mathbb R^d$; let us note $\Omega\times(\mathbb R^d\backslash \{0\})$ by $\dot T^*(\Omega)$ and by $dL(z)$ the Lebesgue measure on $\mathbb C^d$. Let $u\in \mathcal D'(\Omega)$. The analytic wave-front-set of $u$, denoted by $WF_{A}(u)$, is the complement in $\dot T^*(\Omega)$ of the set of points $(x_{0},\xi_{0})$ such that $$ \exists W_{0}\in \mathscr V_{x_{0}-i\xi_{0}}, \exists \chi_{0}\in C^\infty_c(\Omega), \chi_{0}(x)=1\ \text{near $x_{0}$}, \exists \epsilon_{0}>0\quad \text{with} $$ $$ \sup_{\lambda\ge 1, z\in W_{0}}e^{\epsilon_{0}\lambda} \vert{(T\chi_{0} u)(z,\lambda)}\vert e^{-\pi\lambda(\Im z)^2} <+\infty. $$ The first projection of $WF_A(u)$ is the analytic singular support.