2
votes
1answer
80 views

Smooth function becomes analytic

Let $f$ be a smooth function ,defined on unit interval $[0,1]$.Moreover $\Vert f^{(k)}\Vert_2\leq \alpha,\:\forall k\in\mathbb{N}_o$. Can we conclude that $f$ is analytic. More generally when ...
2
votes
0answers
84 views

Explicit example of a function with Fourier transform in $C_0^\infty(\mathbb{R})$

Is anyone aware of an explicit example of a (Schwartz, real-analytic, extending to an entire function with suitable decay properties along imaginary directions as for the Paley–Wiener-Schwartz ...
1
vote
0answers
23 views

Is it possible to extend a given function to be real analytic if its analytic wave front set consists of finitely many covectors at each point?

Specifically, suppose you have a function $f:\mathbb{R}^{2} \to \mathbb{R}$ and you assume that its analytic wave front set $\mathrm{WF}_{A}(f)$ contains at most finitely many covectors $\{(x, ...
2
votes
1answer
143 views

A Paley-Wiener like theorem in real-analysis

I try to identify conditions for the Fourier-transformation $\mathcal{F}(f)$ of some function $f \in L^1(\mathbb{R}^n)$ to be real-analytic. Namely I want to show that one of the following two ...
1
vote
1answer
227 views

analytic continuation of Fourier transform

let be the Fourier transform $$h(u)= \int_{-\infty}^{\infty} g(x) \exp(iux) \, \text{d}x$$ then let us suppose that for $ |x| \to \infty $ the function $g$ goes as $$g(x) = \exp(-ax) \text{ for ...