All Questions
Tagged with fourier-transform ca.classical-analysis-and-odes
43 questions
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votes
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If $u \in H^2(\mathbb{R}^3)$, does $r^{-1} u \in H^{\alpha}(\mathbb{R}^3)$ for some $\alpha > 0$?
Let $u$ belong to the Sobolev space $H^1(\mathbb{R}^3)$. We have the classical Hardy inequality
\begin{equation*}
\int_{\mathbb{R}^3} \frac{|u|^2}{|x|^2} dx \le 4\int_{\mathbb{R}^3} |\nabla u(x)|^2 dx,...
- 39k
0
votes
1
answer
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 ...
0
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0
answers
88
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Convolution of $\mathscr{F}\{ \log \}(x) * \mu$ with compactly supported measure $\mu$
As I read in this post the Fourier transform of $\psi(\lambda) = \log{|\lambda|}$ must be interpreted in distributional sense and it is given by:
$$\mathscr{F}\{\psi\}(x)=-2\pi \gamma \delta(x)-\pi \...
- 141
2
votes
1
answer
495
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Fourier transform of a function of bounded variation
I know if $f\in L^2(\mathbb R)$ is two times continuously differentiable, then we must have that the Fourier transform is integrable. Is there any more relaxed condition than this? For example if $f$ ...
20
votes
1
answer
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Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$
Can one show that Fourier transform of
$$ f_a(x) = a^{-2} \exp(-|x|^a), \qquad a \in (0,2)$$
is decreasing in $a$?
I have a solution for $a \in (0,1]$ which cannot be used for $a\in (1,2)$.
CommunityBot
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1
vote
0
answers
73
views
$L^p$ norm of Fourier transform of function composed with a diffeomorphism
Suppose $f$ is a compactly supported smooth function from $\mathbb{R}^n$ to $\mathbb{C}$ and $A$ is a diffeomorphism on $\mathbb{R}^n$, do we have any theorems relating the $L^p$ norm of $\hat{f}$ and ...
- 265
1
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3
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307
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Fourier transform of a generalized function on the plane
Is there an explicit formula for the Fourier transform of the generalized function of 2 variables
$$\frac{1}{x+y^2+i0}?$$
Remark. Equivalent question: consider the Schroedinger equation one the ...
- 188k
3
votes
0
answers
106
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A new arranging of discrete sine transform
Let $n$ be even and consider the discrete sine transform of type 5 which is the matrix
$$S=\left(\sin(k+1)(l+1)\frac{\pi}{n+\frac12}\right)_{k,l=0}^{n-1}$$
Let us denote by $s_{-,l}$ the $l^{\text{...
- 12.9k
4
votes
1
answer
326
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When does the sine transform result in a positive function?
For each $x>0$ is, $$\int_0^\infty \tanh(t)e^{-\cosh(t)} \sin(x t) dt > 0\ \ \ ?$$
In general, it is known that if the kernel is decreasing, then the sine transform is positive. Note that this ...
- 402
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votes
0
answers
79
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For $\Phi$ a majorant of $1_{[-1/2,1/2]}$, how small can the total variation of $\widehat\Phi$ be?
Let $\Phi:\mathbb{R}\to \mathbb{R}$ be a real-valued, symmetric, non-negative function such that $\Phi(t)\geq 1$ for $|t|\leq 1/2$. Assume furthermore that $\Phi$ and $\widehat\Phi$ are both in $L^1\...
- 6,367
4
votes
2
answers
549
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A proof of Bernstein's inequality
I'm studying the Meyer's book, "Wavelets and operators", and I'm confused about a proof of Bernstein's inequality at page 47, which is stated below:
"The function $\frac{\xi^\beta}{|\xi|...
- 6,035
1
vote
1
answer
203
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Explanation of a step in a work by C. E. Kenig and A.D. Ionescu
I am studying the work
Ionescu, A. D.; Kenig, C. E., Local and global wellposedness of periodic KP-I equations, Bourgain, Jean (ed.) et al., Mathematical aspects of nonlinear dispersive equations. ...
- 4,729
1
vote
1
answer
507
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Fourier transform of the fractional Poisson kernel
Recall that the extension of function from $u:\mathbb{R}^n\to \mathbb{R}$ can be defined using the Poisson Kernel as follows:
$$u^{\mathrm{e}}(\mathbf{x}):=\gamma_{n} \int_{\mathbb{R}^{n}} \frac{x_{n+...
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1
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1
answer
364
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What is the 3-dimensional Fourier transform of $1/k^4$?
In electrostatics, we often encounter the following 3-dimensional integral:
\begin{equation}
V=\int d^{3}\vec{k}\,\dfrac{e^{i\vec{k}.\vec{r}}}{|\vec{k}|^{2}}
\end{equation}
which yields the Coulomb ...
- 63.9k
1
vote
0
answers
74
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Calculation of a multi-dimensional Fourier transform
I am interested in the following multi-dimensional Fourier transform:
$$
\int_{\mathbb{R}^{p}} \mathrm{d} \vec{r}_{\parallel}\int_{\mathbb{R}^{q}} \mathrm{d} \vec{r}_\perp \, e^{-\mathrm{i}\, \vec{p}...
- 373
3
votes
0
answers
204
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The inversion formula for the square root of a positive function
Let $f\in L^1(\mathbb{R})$. Suppose that $\hat{f}$, the Fourier transform of $f$, is a positive function in $C_0(\mathbb{R})$. Does there exists any function $g\in L^1(\mathbb{R})$ with $|\hat{g}|^2=\...
- 63.9k
1
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278
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Vector convolution?
I am working on a research problem which leads to the following optimization problem:
\begin{equation}
\hat{M} = \operatorname*{arg\,max}_M \Bigl\lVert\sum_{k=0}^{M-1} {\mathbf y}_k \exp\left(-j 2\pi ...
- 1,848
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0
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102
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How to construct non-abelian functions?
I have found some functions $t_g, g \in G$ for cyclic groups $G=C_n$ which seem to satisfy the following convolution identity:
$$t_g(x+y) = \sum_{h \in G} t_{gh^{-1}}(x) t_h(y)$$
Example of such ...
- 3,064
1
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1
answer
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2-dimensional Fourier transform
Let $k=(k_1,k_2) \in \mathbb{Z}^2$. Let $\lambda=(\lambda_1,\lambda_2)\in [0,2\pi]^2$ and $F(\lambda)$ be a bounded real function of $\lambda\in [0,2\pi]^2$.
I am interested in the following equation:...
- 128k
2
votes
2
answers
432
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What is multidimensional Fourier transform of $e^{-\mathrm{abs}(x)}$?
What is multidimensional Fourier transform of $e^{-\mathrm{abs}(x)}$?
I know the answer for 1-dimension case. However, I cannot do integration for higher dimensional case.
I used spherical coordinate ...
- 91.8k
0
votes
1
answer
227
views
Laplace transform injectivity for different values of $p$
Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty
).$ Assume that there exist $p_{0},p_{1}\in
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
,$ $p_{0}\neq ...
- 617
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votes
3
answers
723
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Fourier cosine transform from Erdélyi's Tables of Integral Transforms
I’d like confirmation that
$$ \frac{\cos(b \sqrt{a^2+x^2})}{(a^2+x^2)^{3/2}} $$
has the Fourier cosine transform
$$ \frac{\pi}{2 a} \, \exp(-ay) \qquad \text{if $y>a$,}
$$
as found in Tables of ...
- 99
3
votes
2
answers
287
views
An inequality for an integral transform of a function
Let
$$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).$$
Here are the graphs of $f$ (black), ...
- 6,367
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votes
3
answers
2k
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Fourier transform of periodic distributions
Following M. Ruzhansky and V. Turunen's book Pseudo-Differential Operators and Symmetries, in Chapter 3, Definition 3.1.25 (page 304), the space of periodic distributions is defined as follows (...
- 23k
1
vote
1
answer
389
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When are Fourier cosine coefficients convex?
In the question When are Fourier coefficients monotonic it was determined that, if a function $f$ is (the restriction to $[0,2\pi]$) of a completely monotone function, then its Fourier coefficients, ...
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votes
2
answers
2k
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When are Fourier coefficients monotonic?
Given some sufficiently smooth function $f$ what conditions would be sufficient for its Fourier coefficients, as defined by
$$
\hat{f}(n) := \int_{0}^{2\pi}\cos(nx)f(x)\ dx, \quad \text{for } n = 1,2,\...
- 6,035
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
0
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1
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88
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Integration against a certain Fourier transform
I asked the following question on mathstack but didn't receive any answers. I suspect that this question has a simple answer but I haven't thought about Fourier transforms in a while so am being ...
- 17.2k
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0
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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 $k>0$....
- 105k
1
vote
0
answers
103
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Integrability of Fourier transform of truncated fractional power
Is the Fourier transform of the function $f$ which agrees with $1_{[-1.1]}|x|^\alpha$ on $[-1,1]$ and then decays very fast to zero to become a compactly supported continuous function, is in $L^1(\...
- 1,879
3
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0
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140
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Decay of Laplace (or Mellin) transform beyond region of convergence?
Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can ...
- 91.8k
1
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0
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146
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Functional equation with Fourier transform
What are the continuous functions $f$ such that on $\mathbb{R}^{+*}$:
$$f(x) - \frac{C}{x} \hat{f}(\frac{1}{x}) =x^{\alpha}$$
Where $\hat{f}$ is the Fourier transform of $f(|x|)$ and $C$ a constant....
- 1,199
8
votes
1
answer
667
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Fourier transform that is almost a brick wall - but why?
Let $$g(x) := \sqrt{1+x^2},$$ and $$h(x) := g^{-3/2}(x) \exp(-i2\pi g(x)).$$
I can observe that the Fourier transform $|H(f)|$ is almost flat if $|f|<1$, and $H(f)\approx 0, \; |f|>1$.
This ...
- 188k
41
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6
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87k
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Fourier vs Laplace transforms
In solving a linear system, when would I use a Fourier transform versus a Laplace transform? I am not a mathematician, so the little intuition I have tells me that it could be related to the boundary ...
- 103
3
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0
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94
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Asymptotics of fast decaying oscillating integrals
Let $g(t):\mathbb{R}\rightarrow \mathbb{C}$ be a smooth even function such that for each positive integer $n$ the function $g^{(n)}(t)$ tends to 0 when $t$ goes to infinity and $\int_0^\infty |g^{(n)}(...
- 109k
3
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1
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About Fourier transforms of piecewise linear functions. [closed]
Consider a function $f$ which is $0$ for $x< 1$ and is say $x-1$ for $x >1$.
Consider a function $g$ which is $0$ for $x <2$ and is say $x -2$ for $x>2$.
Now using some kind of ...
- 188k
3
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0
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651
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Decay of the Fourier transform of a surface area measure
Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\...
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0
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443
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What is the Fourier transform of this function?
Consider the function
$$
f(x_1,x_2)=|x_1x_2|^{-\alpha/2}\int_{\mathbb{R}} \frac{e^{it(x_1+u)}-1}{i(x_1+u)} \frac{e^{it(x_2-u)}-1}{i(x_2-u)} |u|^{-\beta}du.
$$
It is known that $f(x_1,x_2)\in L^2(\...
- 3,574
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1
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Estimate a Fourier Transform [closed]
I'm reading an article which claims the following result (p.9): if $f : \mathbb{R}^{2} \to \mathbb{R}$ is of the form $f(x_1,x_2) = \sin (N x_{1}) h (g^{-1}(x))$, where $g$ is a diffeomorphism and $h$ ...
- 5,169
2
votes
1
answer
112
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How to relate this summation to standard discrete cosine transformation?
The standard type III discrete cosine transformation (DCT) is defined as follows:
$${X_k} = \frac{1}{2}{x_0} + \sum\limits_{n = 1}^{N - 1} {{x_n}} \cos \left[ {\frac{\pi }{N}n\left( {k + \frac{1}{2}} ...
- 123
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0
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122
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Resolvent of the operator
Consider the Laplace operator defined on the biggest possible subset of$L^{2}(R^{2})$:
$T= - \partial^{2}_{x} -\partial^{2}_{y}+x^{2}+y^{2}+ 2.i(x \frac{\partial}{\partial y}-y\frac{\partial}{\...
- 135
3
votes
0
answers
151
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Meaning of fractional Fourier transform with imaginary iteration count
(I'm reposting this from math.stackexchange as I didn't get an answer there and thought it might be "advanced" enough for this site.)
As one may know, the Fourier Transform
$$F[f](\nu) = \int_{-\...
- 1,687
0
votes
1
answer
489
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Littlewood-Paley theory and norm estimation
In the paper "A Convolution Inequality Concerning Cantor-Lebesgue Measures", the Littlewood-Paley theory is used to estimate the norm of multiplier operator in Lemma 1.
It is claimed that Lemma 2 is ...
- 153