- #1
mathmari
Gold Member
MHB
- 5,049
- 7
Hey!
In $\mathbb{R}$ with Lebesgue measure, we take $f\in L^1$ and we set $\hat{f}(t)=\int f(x) e^{ixt} dx$, for each $x$ $\ \ \ (i^2=-1)$
Show that:
1. $\hat{f}$ is continuous
2. $\lim_{t\rightarrow \pm \infty}\hat{f}(t)=0$
3. $||\hat{f}||_{\infty}\leq ||\hat{f}||_{1}$
Could you give me some hints how I could do that?? (Wondering)
In $\mathbb{R}$ with Lebesgue measure, we take $f\in L^1$ and we set $\hat{f}(t)=\int f(x) e^{ixt} dx$, for each $x$ $\ \ \ (i^2=-1)$
Show that:
1. $\hat{f}$ is continuous
2. $\lim_{t\rightarrow \pm \infty}\hat{f}(t)=0$
3. $||\hat{f}||_{\infty}\leq ||\hat{f}||_{1}$
Could you give me some hints how I could do that?? (Wondering)