All Questions
Tagged with fourier-transform harmonic-analysis
84 questions
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Show that the kernel $|x -y|^{-1}$ on $\mathbb{R}^3 \times \mathbb{R}^3$ is Hilbert Schmidt with respect to a weighted $L^2$ space
Let $\langle x \rangle := (1 + |x|^2)^{1/2}$, $x \in \mathbb{R}^3$. For $s > 1$, consider the weighted convolution operator
\begin{equation*}
T_s \varphi = \langle x \rangle^{-s} \int_{\mathbb{R}^3}...
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289
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Closed sets in the space of Fourier transforms $\mathcal{F}L^{1}$
Consider the space of all Fourier transforms of $L^{1}(\mathbb R),$ that is,
$$\mathcal{F}L^{1}=\mathcal{F}L^{1}(\mathbb R):= \{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}(\mathbb R)\},$$
with the ...
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1
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359
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Relating the R-transform in free probability to noncommutative group representations
In traditional (commutative) probability theory, sums of random variables correspond to convolutions of distribution functions, which plays well with the Fourier Transform.
In free (noncommutative) ...
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484
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When one can expect $\widehat{(fg)} = \hat{f} \ast \hat{g}$; $f, g\in L^{1} (G)$?
Let $f, g \in L^{1}(\mathbb T)= L^{1} ([-\pi, \pi))$. We define, the Fourier transform of $f$ as follows:
$$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt, \ (n\in \mathbb Z).$$
It is ...
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152
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Is $\int_{\mathbb{R}} \int_{\mathbb{R}^n} \alpha w(t) e(\alpha (a_1t_1 + \dotsb + a_n t_n)) dt\,d \alpha = 0$?
Let $a_i$ be a nonzero real number for each $1 \leq i \leq n$. $w$ a smooth nonnegative with compact support. I would like to understand the following integral.
$$
I = \int_{\mathbb{R}} \int_{\mathbb{...
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385
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Interchanging Integration Order involving Fourier Transform
$$f(\omega,u):=\frac1{\omega+iu}$$
where $i$ is the imaginary unit number. We see that the integral of a Fourier transform
$$\int_1^\infty du\int_{-\infty}^\infty d\omega\,f(\omega,u)\,e^{-i\omega x}=...
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1
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229
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Result of Beurling concerning absolute convergence of Fourier series of |f|
Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$ and we put,
$$A(\mathbb T):= \{f\in L^{1}(\...
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367
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How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ (periodic on $\mathbb R$)? [closed]
We put, $M(\mathbb R)= $The set of bounded complex Borel measure $\mu$ on $\mathbb R$ and for $\mu \in M(\mathbb R)$, we define $||\mu||:= |\mu| (\mathbb R) = \text {total variation of } \ \mu $; and
...
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180
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Annihilator property dual
Let $G$ be a locally compact group and $\phi$ be in $ L^{\infty}(G)$ that annihilates $I$, where $I$ is a closed ideal of $ L^1(G)$, so by duality we have:
$$\int_G f(y)\phi(y)dy=0$$
for all $f\in I$....
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0
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52
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Sufficient conditions for boundedness of Fourier transform
This should be a well studied topic: I am looking for sufficient conditions on a function $u(x)$ on $\mathbb{R}$ ensuring that its Fourier transform is bounded. Of course one such condition is $u\in L^...
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72
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Positive eigenfunctions of the discrete Fourier transform
Let $G$ be a finite cyclic group of order $n$ ($n$ need not be prime) and $\mathcal{F}$ the normalized discrete Fourier transform defined on $G$.
Is there a canonical way to construct an eigenfunction ...
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73
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$L^p$ norm of Fourier transform of function composed with a diffeomorphism
Suppose $f$ is a compactly supported smooth function from $\mathbb{R}^n$ to $\mathbb{C}$ and $A$ is a diffeomorphism on $\mathbb{R}^n$, do we have any theorems relating the $L^p$ norm of $\hat{f}$ and ...
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108
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Recovering phase function using Fourier decomposition
I have a function $\phi(x): \mathbb{R} \to [0, 2 \pi)$, which describes phase of another function
$$f = e^{i \phi(x)}. $$
I am interested in the following problem. If I know the function/distribution $...
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79
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A problem arising from Wiener-Levy theorem on the real line
Theorem (Wiener-Levy). Let $A(\mathbb{T})$ be the Fourier-algebra on the unit circle $\mathbb{T}$. Let $f$ be in $A(\mathbb{T})$ and suppose that $F$ is an analytic function on the range of $f$. Then $...
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244
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On $L^2$ spaces which have an orthogonal basis of characters (complex exponentials)
Suppose $\Omega \subset \mathbb{R}^n$. What conditions on $\Omega$ make it so there exists a countable set $\Lambda$ such that $\{e^{2\pi i\lambda t} \}_{\lambda \in \Lambda}$ form an orthogonal basis ...
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668
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Asymptotics of a function from its Fourier transform
My question is: given a Fourier transform $\hat f$ of a function $f$, is it possible to estimate its asymptotic behaviour without performing the inverse transform?
Let me give a concrete example.
...
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124
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Inequality about the Fourier transform: $\Vert u \Vert_{L^k} \le \Vert \mathcal{F}(u) \Vert_{L^m}$ (where $1 \le m \le 2$ and $m,k$ Holder conjugates)
How can I prove the following inequality about the Fourier transform?
$$\Vert u \Vert_{L^k(\mathbb{R}^N)} \le \Vert \mathcal{F}(u) \Vert_{L^m(\mathbb{R}^N)}$$ for $1 \le m \le 2$ and $m,k$ Holder ...
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327
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If $\mathcal{F}$ is the Fourier transform, what can be said about $\mathcal{F}(L^1(\mathbb{R})) \cap L^1(\mathbb{R})$?
The Fourier transform gives a map of the Schwartz space to itself which turns out to be a linear homeomorphism of period 4.
However, when the domain is extended to $L^1(\mathbb{R})$, the situation is ...
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157
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Technical question about a Fourier transform
I would like to know if there is an explicit expression for the Fourier transform of the following function:
$$f(x)=\mathbb{1}_{(0,\infty)}e^{-x-ix^2},$$
or to know where I can find some techniques to ...
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181
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How Fourier transform behaves if we kills the oscillation?
Let $a, b \in \mathbb R$ such that $ab> 1$ ; put
$$L^{1}_{a}(\mathbb R)= \{ f:\mathbb R\to \mathbb C \ \text {measurable} : ||(1+|x|)^{a}f||_{L^{1}(\mathbb R)}< \infty \},$$
and
$$FL^{1}_{b}(...
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2
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227
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Does this distribution exist?
Assume there is a distribution in two variables $\mathcal{W}\in\mathcal{S}'(\mathbb{R}^2)$ with Fourier transform $\hat{\mathcal{W}}(\alpha,\beta)\equiv \int_{-\infty}^\infty e^{i(\alpha x+\beta y)} \...
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1
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105
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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,...
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1
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126
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Clarification on the Interpretation of Fourier Coefficients in the Context of Fourier Projections
I am currently studying a paper (Section 3.4.3 of Lanthaler, Mishra, and Karniadakis - Error estimates for DeepONets: a deep learning framework in infinite dimensions) where the authors define an ...
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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 ...
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170
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When some Fourier coefficients are fixed, can we control the extremals of the function?
Let $n$ be a odd number. Does there exist any $2\pi$-periodic continuous function $f
:\mathbb{R}\to \mathbb{R}$ such that the following points simultaneously hold?
1- $-n\lneqq f_{\min}$ (where $f_{\...
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1
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226
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Transformation of Fourier Transform
Suppose that $f$ is a function with a Fourier transform, and that $g:\mathbb{R}\rightarrow \mathbb{R}$ is a smooth function such that $g\circ f$ has a Fourier transform also.
Is there an expression ...
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1
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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 ...
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0
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23
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Is there a classification of 2D projective convolution kernels?
Is there any classification of all distributions on $\mathbb{R}^2$ such that they are equal to the convolution with themselves? i.e. given a distribution $\gamma$ under which conditions
$$ \gamma\star\...
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0
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75
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$|\partial $ as Fourier multiplier
I have the following nonlinear dispersive PDEs
$$i \partial_t u- \partial_x^2 u =|\partial_x| |u|^2$$
where $f$ is some nice complex-valued function.
I am trying to use the ansatz $u(t,x) = e^{i \...
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0
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166
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Parseval-Plancherel identity involving absolute value
Let $\hat{f}$ be the fourier transform of $f$.
By Parseval-Plancherel identity, for suitable $f,g$, we have
$$\left\|\hat{f}*\hat{h}\right\|_{L^2_{\xi}}^2=\left\|f\cdot h\right\|_{L^2_{x}}^2.$$
Let ...
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0
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79
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Is Wiener amalgam spaces $W^{2,1}(\mathbb R)\subset C_0(\mathbb R)$?
I have been learning Wiener amalgam spaces.
In Wiener amalgam spaces $W(X, L^2)$, I am taking $X=\mathcal{F}L^{1}=\{f\in L^{\infty}(\mathbb R):\hat{f}\in L^{1}\},$ and $m(x)=1.$
Take $f(x)= \chi_{\...
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1
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1k
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A question about pointwise convergence of Fourier transform in $N$-dimensions
I am retreating back on this statement, after some explorations and calculation
Bow to Willie and others who were skeptical on this. Main difficulty can be seen in this reference. But I must mention ...
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1
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213
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Building a smooth function from a rapidly decreasing sequence
Is it possible to build a 1-priodic smooth function from a rapidly decreasing sequence such that the sequence be the Fourier coefficients of the function?
More precisely:
Let $\lbrace c_k\rbrace _{k \...
-3
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1
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230
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$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?
Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at $\...