All Questions
9 questions
0
votes
1
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255
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Carleson's theorem: proof of a lemma
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. At the bottom of page 20 at the beginning of ...
- 75
3
votes
0
answers
204
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The inversion formula for the square root of a positive function
Let $f\in L^1(\mathbb{R})$. Suppose that $\hat{f}$, the Fourier transform of $f$, is a positive function in $C_0(\mathbb{R})$. Does there exists any function $g\in L^1(\mathbb{R})$ with $|\hat{g}|^2=\...
- 4,058
0
votes
1
answer
227
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Laplace transform injectivity for different values of $p$
Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty
).$ Assume that there exist $p_{0},p_{1}\in
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
,$ $p_{0}\neq ...
- 617
5
votes
3
answers
2k
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Fourier transform of periodic distributions
Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (...
- 595
3
votes
2
answers
287
views
An inequality for an integral transform of a function
Let
$$J_{f;y}(u):=3 u^3 \int_u^1\frac{dt}{t^4} \,e^{-i y t}f(t)- e^{-i u y}f(u),$$
where $y\in(0,\infty)$, $u\in(0,1)$, and
$$f(t):=t+\pi (1-t) t \cot (\pi t).$$
Here are the graphs of $f$ (black), ...
- 128k
1
vote
1
answer
390
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When are Fourier cosine coefficients convex?
In the question When are Fourier coefficients monotonic it was determined that, if a function $f$ is (the restriction to $[0,2\pi]$) of a completely monotone function, then its Fourier coefficients, ...
- 595
22
votes
2
answers
2k
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When are Fourier coefficients monotonic?
Given some sufficiently smooth function $f$ what conditions would be sufficient for its Fourier coefficients, as defined by
$$
\hat{f}(n) := \int_{0}^{2\pi}\cos(nx)f(x)\ dx, \quad \text{for } n = 1,2,\...
- 595
0
votes
0
answers
82
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A question about Fourier transform of a function defined by an integral
I have the function:
$$ G_k(x)_=\frac{1}{(4\pi)^{k/2}\Gamma(k/2)}\int_0^\infty e^{-\pi|x|^2/\delta}e^{-\delta/4\pi}\delta^{-(n-k)/2}\,\frac{d\delta}{\delta}, $$
for all $x\in\mathbb{R}^n$ and $k>0$....
- 339
1
vote
0
answers
151
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Fourier transforms exhibiting symmetries about their critical points
Upon looking at the graphs of various Fourier sine and cosine transforms (ones without Dirac deltas in their domain) I've noticed a pattern that is probably already known, but that I thought would be ...
- 113