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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.2012-11-15

2 Answers 2

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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} $


Complete Answer for a: The $L^2$ norm for a Fourier Multiplier Operator is the $L^\infty$ norm of the multiplier. This follows easily by Plancherel's Theorem. Since $\left\|1-|\zeta|\right\|_{L^\infty[-1,1]}=1$, the $L^2$ norm of $T$ is $1$.


Complete Answer for b: By the Convolution Theorem, $\widehat{m\hat{f}}=\widehat{m}\ast\tilde{f}$, where $\tilde{f}(x)=f(-x)$. Since the Fourier Transform of $(1-|\zeta|)1_{[-1,1]}(\zeta)$ is $\frac{1-\cos(2\pi\xi)}{2\pi^2\xi^2}$ and $ \left\|\frac{1-\cos(2\pi\xi)}{2\pi^2\xi^2}\right\|_{L^1}=1 $ Young's Inequality for Convolutions shows that the $L^p$ norm for this multiplier operator is $1$ for all $p$.

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    @robjohn Could you provide a thorough a solution, I'm still confused....2017-02-26
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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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    For (a), how do you find the $f$ such that the estimate is sharp. So far I've calculate that the constant times $\| f \|^2$ is 2.2017-02-24