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I am having some trouble with some basic properties of a given operator.

Firstly, the operator T is defined as taking the fourier inverse transform of the function $(1-|\zeta|)1_{[-1,1]}(\zeta)\hat{f}(\zeta)$.

(a) Show T is bounded on $L^2(R)$, and compute the operator norm.

(b) Further, show that $T$ is a bounded operator on $L^p(R)$. The hint of b is that T is in fact convolution with a function g s.t $|g(x)|< C/1+x^2$. Lp convolution inequality is needed.

Just guess a) requires plancherel theorem to show $||T||_2 = ||u||_∞$ ,where u is $(1-|\zeta|)1_{[-1,1]}(\zeta)$. But cant figure out how to do it . Also, I dont know how to do with (b).

Could someone help with it? Thanks.

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I rolled back the last edit since it removed the statement of the problem. –  robjohn Nov 15 '12 at 8:12
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2 Answers

Hint for a:

${\large|}1-|\zeta|{\large|}\le1$ on $[-1,1]$

Hint for b: $$ \begin{align} \int_{-1}^1(1-|x|)e^{-2\pi ix\xi}\,\mathrm{d}x &=2\int_0^1(1-x)\cos(2\pi x\xi)\,\mathrm{d}x\\ &=\frac{\sin(2\pi\xi)}{\pi\xi}-2\int_0^1x\cos(2\pi x\xi)\,\mathrm{d}x\\ &=\frac{\sin(2\pi\xi)}{\pi\xi}-\frac1{\pi\xi}\int_0^1x\,\mathrm{d}\sin(2\pi x\xi)\\ &=\frac{\sin(2\pi\xi)}{\pi\xi}-\frac{\sin(2\pi\xi)}{\pi\xi}+\frac1{\pi\xi}\int_0^1\sin(2\pi x\xi)\,\mathrm{d}x\\ &=\frac{1-\cos(2\pi\xi)}{2\pi^2\xi^2} \end{align} $$

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thanks , I can show T is bounded in L2 ? but I cant calulate the norm. Any help? –  user 763 Nov 15 '12 at 5:27
    
Use the hint for a and Plancherel. You seem to say as much in your guess. –  robjohn Nov 15 '12 at 8:16
    
but I am stuck on ||T||>||u||∞ –  user 763 Nov 15 '12 at 10:26
    
I wanna have ||Tf||=||uf|| –  user 763 Nov 15 '12 at 10:44
    
Plancherel says that $\|f\|_2=\|\hat{f}\|_2$. The definition of $T$ says that $\widehat{Tf}=u\hat{f}$. Cogitate. –  robjohn Nov 15 '12 at 12:03
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Since this is homework, only hints:

(a) Start with $\|Tf\|_2^2$, use Plancherel and estimate by "a constant times $\|f\|^2$ (in fact, use Plancherel twice). This constant is an upper bound for $\|T\|$. Then, find some $f$ such that the estimate is sharp.

(b) I presume that $M$ is $T$? However, this hint you have is already saying quite a lot. You also need to use the convolution theorem (twice), i.e. that the Fourier transform takes convolutions to pointwise products.

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