# Fourier transform conjucture/theorem

1. Apr 20, 2008

### jostpuur

fourier transform conjecture/theorem

Is this a theorem?

Let $f\in L^2(\mathbb{R},\mathbb{R})$ be such function that f is continuous on some sets $]x_0-\delta, x_0[$ and $]x_0, x_0+\delta[$ with $\delta >0$, and

$$\lim_{x\to x_0^-} f(x) \neq \lim_{x\to x_0^+}f(x).$$

(So we cannot choose a continuous g such that [g]=[f] in $L^2$ sense)

There does not exist such $M,C,\epsilon>0$, that

$$|(\mathcal{F} f)(k)| < C \frac{1}{|k|^{1+\epsilon}},\quad \forall |k|> M.$$

Last edited: Apr 20, 2008