Intereting Posts

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in a topological space only finite subsets are compact sets
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Showing that $l^p(\mathbb{N})^* \cong l^q(\mathbb{N})$
bézier to f(x) polynomial function
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Finding the error in a proof
Maximal ideals in polynomial rings
Errors in math research papers
Lebesgue non-measurable function
Find the sum of this series
What is the equation describing a three dimensional, 14 point Star?
How many subsets of $\mathbb{N}$ have the same cardinality as $\mathbb{N}$?
For any smooth manifold, is it true that for any two points on the manifold, there exists a chart that covers the two points?
Proof of the formula $1+x+x^2+x^3+ \cdots +x^n =\frac{x^{n+1}-1}{x-1}$

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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- Fundamental solution to the Poisson equation by Fourier transform
- Problem on Big Rudin about Fourier Transform
- Conceptual/Graphical understanding of the Fourier Series.

- Proving that every non-negative integer has an unique binary expansion with generating functions
- Integration of $\int \frac{x^2+20}{(x \sin x+5 \cos x)^2}dx$
- Calculating $\iint (x+y) \, dx \, dy$
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- Given any nine integers show that it is possible to choose, from among them, four integers a, b, c, d such that a + b − c − d is divisible by 20.
- Examples of problems that are easier in the infinite case than in the finite case.
- Loop space suspension/adjunction