Intereting Posts

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When we have $1< p < 2$, for every function $f\in L^p(\mathbb{T})$ we can define the Fourier coeficients of $f$ as $$\hat{f}(k)= \frac{1}{2\pi}\int_{-\pi}^{\pi} f(t)e^{-int}dt,$$

which are well defined because $L^p(\mathbb{T})\subset L^1(\mathbb T)$.

Hausdorff-Young Theorem states that if $1< p \leq 2$ and $\frac{1}{p}+\frac{1}{p’}=1$, then

$$\big(\sum_{k\in\mathbb Z} |\hat{f}(k)|^{p’}\big)^{1/p’}\leq ||f||_p.$$

So if we define the Fourier operator for $1< p < 2$ as

\begin{align}\mathcal{F}:L^p(\mathbb{T})&\rightarrow \ell^{p’}(\mathbb Z)\\

f&\mapsto \{\hat{f}(k)\}_{k\in\mathbb Z}

\end{align}

this is a bounded operator, and it is 1-1, but it is not onto. I’m having problems to find an example of a sequence in $\ell^{p’}(\mathbb Z)$ ($p’>2)$ which is not the Fourier coeficients of a function in $L^p(\mathbb T)$ (or even in $L¹(\mathbb T)$).

The only examples I’ve found are of sequences in $C_0(\mathbb Z)$ which are not the Fourier coeficients of any $f\in L¹(\mathbb T),$ such as the sequence $$\bigg\{\frac{i\,sgn(k)}{\log|k|}\bigg\}_{|k|\geq 2}\in C_0(\mathbb Z).$$

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