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Results tagged with fourier-analysis
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user 59219
The representation of functions (or objects which are in some generalize the notion of function) as constant linear combinations of sines and cosines at integer multiples of a given frequency, as Fourier transforms or as Fourier integrals.
1
vote
Fourier transform of a function of bounded variation
If $f'$ is of bounded variation, then $\hat{f}$ will be integrable. To see this, note that by your assumptions, $f''$ (in the sense of tempered distributions) is a finite complex-valued measure, so th …
- 734
2
votes
0
answers
802
views
Existence of unbounded $M \subset \Bbb{R}$ of finite measure s.t. $1_M$ is $L^p$-Fourier mul...
I would like to know if there is a measurable set $M \subset \Bbb{R}$ such that
$M$ has finite Lebesgue measure $0 < \lambda(M) < \infty$,
$M$ is unbounded in the sense that $\lambda(M \setminus [-r …
- 734
5
votes
0
answers
209
views
Existence of $A\subset\Bbb{R}^n$ of finite measure and $\hat{1_A}\in\bigcap_{q>1}L^q$, but s...
I am interested in the following somewhat obscure question:
Is there some $n \in \Bbb{N}$, and a set $A \subset \Bbb{R}^n$ of finite measure such that the Fourier transform $\widehat{1_A}$ of its …
- 734
6
votes
2
answers
2k
views
Reverse Hausdorff Young for nonnegative functions
The classical Hausdorff-Young inequality states that
$$
\Vert \widehat{f} \Vert_{p'} \leq \Vert f \Vert_p \text{ for } 1 \leq p \leq 2.
$$
For $p=2$, we even have equality due to Plancherel.
If we …
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