Questions tagged [fourier-analysis]
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.
1,389
questions
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How to choose some $h$ so its Fourier transform supported in some set?
Suppose that $K=[-N, N]$ is some compact subset of $\mathbb R$, for some $N>2.$
Can we expect to choose $h$ such that $h=1$ on $K$ and the support of the Fourier transform of $\widehat{h}$ ...
- 195
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vote
0
answers
46
views
A parametrix construction for heat boundary value problem using Fourier transformation
Let $\Omega$ be a smooth bounded open subset in $\mathcal{R}^{d}$, with $d \geqslant 3
$ and $T>0$. Consider the linear parabolic initial Dirichlet boundary value problem with $f\in H^{-1}(\Omega)$...
- 11
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votes
0
answers
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views
Sobolev estimates on domain with boundary
Could someone point me to a reference for the proof of the following Sobolev estimate
$$
\|u\|_{L^{2 d /(d-2)}(\Omega)} \leqslant C(\|f\|_{L^{2 d /(d+2)}(\Omega)} + \|g\|_{(\partial\Omega)})
$$
for ...
- 11
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votes
0
answers
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Can the best constants in harmonic analysis be approximated in principle?
Consider the trivial example of Holder's inequality $\|f\|_p\,\|g\|_q\geq |fg|_1$ if $\frac{1}{p}+\frac{1}{q}=1, p,q\geq 1$ and $f,g$ are functions on $\mathbb{R}^n$. Let's suppose we don't know how ...
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$\|\hat{f}\|_{L^q}< \infty \implies \left\| \|\chi_{n+(-1/2, 1/2]} \widehat{f}\|_{L^p_{\xi}} \right\|_{\ell^q_n}<\infty $
Suppose that support of $f:\mathbb R \to \mathbb R$ is compact set $K\subset \mathbb R.$ Assume that $ \int_{\mathbb R} |\widehat{f}|^q d\xi <\infty.$ ($\widehat{\cdot}$ denote the Fourier ...
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0
answers
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Positive definitness of $f(|x|^\gamma)$, $0<\gamma<1$
Let $f(x)$ be a positive definite function on $x \in R^d$. Assume $f(x)$ is radial ,
so $f(x)$ is a function of $|x|$, let's say $g(|x|):=f(x)$. How can I show that
$g(|x|^\gamma)$ is positive ...
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0
answers
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Question about stationary phase with Hessian close to $0$
Let $\phi$ be a smooth real function in one variable and say $w$ is a smooth function with compact support say $[- 1, 1]$. Let me define
$$
I_{\lambda} = \int_{\mathbb{R}} w(t) e^{i \lambda \phi(t)} ...
- 3,373
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votes
0
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Expand White Noise and Brownian Motion in Haar basis: which version of Haar basis?
Start with the Haar basis of $L^2(\mathbb{R})$, namely, the functions
$$
\chi(t-k) \text { and } 2^{j / 2} h\left(2^j t-k\right), j \geq 0, k \in \mathbb{Z}, \quad \quad \quad (1)
$$
where $\chi(t)$ ...
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When does the Fourier transform of a measure decay?
Let $\mu$ be a Borel measure on $\Bbb R^d$.
It is well known that $\mu= |f|dx$ with $f\in L^1(\Bbb R^d)$ then its Fourier transform satisfies
$$\widehat{\mu}(\xi)\to0,\qquad \xi\to\infty.$$
However if ...
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votes
1
answer
180
views
Riemann-Liouville integral of $f$ is zero implies $f =0$ a.e
The Riemann-Liouville integral is defined by
$$
I^\alpha f(x)=\frac{1}{\Gamma(\alpha)} \int_a^x f(t)(x-t)^{\alpha-1} d t
$$
where $\Gamma$ is the gamma function and $a$ is an arbitrary but fixed base ...
- 103
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votes
0
answers
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What is the "best" good kernel?
A family of functions $k_n(x):[-\pi,\pi]\to \mathbb R$ for $n\in \mathbb N$ is said to be a good kernel if all the following are satisfied:
$\frac{1}{2\pi }\int_{-\pi}^\pi k_n(x) \, \mathrm d x=1$,
$...
- 2,862
1
vote
1
answer
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views
Why complex conjugate in definition of the Fourier transform?
Let $G$ be a locally compact abelian group and $f:G \to \mathbb{C}$ a function. Its Fourier transform (when it exists) is defined to be
$$\widehat{f}(\chi) = \int_G f(g) \bar{\chi}(g) \mathrm{d} g,$$
...
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4
votes
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answer
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Inequality for Fourier transform of a power exponential function
Let
$$
f_{\alpha}(x)=\phi_1(\alpha) \mathrm{e}^{-\frac{|x|^\alpha}{\phi_2(\alpha) }},
x \in \mathbb{R}, 0<\alpha<2,
$$
where
$\phi_1(\alpha)=\frac{\alpha}{2}
\left\{{\{\Gamma(3/\alpha)\}^{1/...
- 148
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votes
0
answers
67
views
Positive definiteness with nonnegative weights
Is there a simple criterion to certify if some function $f: \mathbb{R} \to \mathbb{R}$ satisfies that $\sum_{i,j=1}^n c_ic_jf(x_i-x_j) \ge 0$ for all $x_i \in \mathbb{R}$ and $c_i \ge 0$?
Note that if ...
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votes
1
answer
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views
$\Lambda f\ge 0\iff f\ge 0$ if $\Lambda$ is a Gaussian convolution kernel?
Consider $\Lambda f(x)=\int_{\mathbb R} f(x-y) e^{-y^2} dy$. Suppose that for a bounded function $f$, $\Lambda f(x)\ge 0$ for all $x\in\mathbb R$. Does it imply that $f\ge 0$ almost surely?
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Inequality for sums of sines with similar frequency
Let $c>0$ be a very small constant and $N \in \mathbb N$ very large.
Assume we have a function $f(x)$ for $x \in S^1$ defined as
$$
f(x) = \sum_{k=\lfloor N/(1+c) \rfloor}^{N} c_k \sin(kx+b_k)
$$
...
- 353
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votes
0
answers
128
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Direct analytic proof of positive definiteness of stable characteristic functions
Is there a direct analytic proof that the function
$$
f ( t ) =
\exp\left(-|t|^\alpha \big[ \lambda + i \theta \operatorname{sign} ( t ) \big]\right),
\qquad
\lambda > 0, \quad
|\theta| < \...
- 109
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3
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If the Fourier coefficient $\hat{f}(k)$ of $f\in C^1(\mathbb T)$ is zero for all $|k|<N$, then $\|f\|_{L^\infty}\leq \frac CN \|f'\|_{L^1}$?
Let $f\in C^1(\mathbb T)=C^1(\mathbb R/\mathbb Z)$ be a function such that
$$\hat f(k):=\int_{\mathbb T}f(x)e^{-2\pi ikx}\,dx=0,\qquad \forall k\in\{-N+1,\cdots,-1,0,1,\cdots, N-1\}.$$
Do we have $\|f\...
- 337
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votes
0
answers
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views
Systems of parabolic equations -- Petrovskii's condition
Consider the flat torus $\mathbf{T}^d:=\mathbf{R}^d/\mathbf{Z}^d$ and define the corresponding periodic-parabolic cylinder $Q_T:=(0,T)\times\mathbf{T}^d$.
Given a matrix field $A:Q_T\rightarrow\text{M}...
- 2,410
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votes
1
answer
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Littlewood-Paley characterisation of Hölder regularity
I am going through Terence Tao's "Nonlinear Dispersive Equations (Local & Global Analysis)" and trying to work through some of his exercises. However, I find myself being stumped by ...
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votes
1
answer
125
views
The asymptotic behaviour of a singular integral
Given $0<\alpha, \beta<1$, $a,b>0$, $a^2+b^2<1$.
I am trying to determine the asymptotic behaviour of
$$F(a,b):=\int_{\substack{a/2<x<2a\\\\b/\sqrt{2}<\sqrt{1-x^2}<\sqrt{2}b}}\...
- 391
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0
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Bounding $\widehat{G_{m+1}}\ast\widehat{H_m}(0)$ when $\frac{1}{2}\leq\widehat{H_m}(0)\leq\frac{3}{2}$ and $H_m,G_{m+1}$ are smooth over $\mathbb{T}$
To put this question into proper context, what I am asking is related to the construction of smooth function $H_m$ over the torus $\mathbb{T}$ such that
$$\left|\widehat{H_m}(k)\right|\leq C\log(\left|...
- 121
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votes
0
answers
92
views
Is this formula for 2D Fourier integral of diffraction kernel correct?
Well I have a function parametrized by $z$
$$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$
where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
- 141
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votes
1
answer
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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 \...
- 571
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votes
0
answers
47
views
Eigenfunction expansion theorem for general manifold for smooth functions
The question starts with the well known facts that: if $f$ is a smooth function on $S^1$, then its Fourier series converges to it in smooth topology.
This must be true in more general setting. I have ...
- 31
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votes
2
answers
257
views
Calculating the Fourier dimension of a real interval $\left[a, b\right]$
(Preliminaries:) 1.) Let $S\subset\mathbb{R}^n$ and define $\mathcal{M}(S) = \{\text{$\mu$ a Borel measure}: \text{$0 < \mu(S) < \infty$ and $\mathrm{support}(\mu)\subset S$}\}$.
2.) Define the ...
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Function on $\mathbb{Z}/p^k \mathbb{Z}$ with small Fourier transform?
For $f:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$, define the Fourier transform $\widehat{f}:\mathbb{Z}/p^k \mathbb{Z}\to \mathbb{C}$ in the usual way, viz., $\widehat{f}(\xi) = \sum_x f(x) e(-\xi x/p^k)...
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Existence of a specific family of functions on an abelian group with vanishing properties on rank 2 subgroups
Fix a prime $p$, and let $W_0\subset W$ be an inclusion of a codimension one $\mathbb{F}_p$ vector spaces. Let $W_e$ denote a fixed nontrivial coset of $W_0$ in $W$.
The question is whether there ...
- 1,607
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votes
1
answer
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views
Approximation of Hölder functions by Fourier series
Let $Q$ be a bounded domain in $\mathbf R^N$ with smooth boundary. Let $f\in C^a(\overline{Q})$, $0<a<1$.
Denote $\psi_k(x)$ normalized eigenfunctions and $\lambda_k$ eigenvalues ($k=0,1,2\...
- 21
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votes
2
answers
170
views
Well-defined distribution and its singular support
Let $f$ be a smooth function on $X$, an open subset of $\mathbb{R}^n$, with $Im(f) \geq 0$. Let us fix an $\epsilon > 0$.
Let $T_{\epsilon} := \frac{1}{f(x)+i\epsilon} $ in $D’(X)$.
Now if we ...
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vote
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answers
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Localize functions in the Hardy space $\mathcal H^1(\mathbb R^n)$
Let $f$ belong to the Hardy space $\mathcal H^1(\mathbb R^n)$, $B\subset \mathbb R^n$ be the unit ball. Does there exist a $\bar f\in \mathcal H^1(\mathbb R^n)$ with compact support such that $\bar f=...
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votes
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Vanishing periodizations $\sum_{k \in \mathbb Z} f(t+ak)$ of a function $f$ for different values of $a$ implies $f=0$?
Consider a continuous function $f : \mathbb R \to \mathbb C$ with rapid decay (e.g. $|f(t)| < e^{-t^2}$). For a constant $a>0$ let
$$
F_a(t) = \sum_{k \in \mathbb Z} f(t+ak)
$$
be the ...
- 105
5
votes
1
answer
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views
Real-analytic analogue of Schwartz functions
Consider the space $\mathcal{S}'$ of functions $\mathbb{R}^n\to\mathbb C$ that are (real-)analytic and with exponential decay at infinity. This is an analogue of Schwartz space, but real-analytic ...
- 53
1
vote
0
answers
98
views
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 $...
- 141
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vote
0
answers
100
views
Question on the existence of a certain decomposition method for real square matrices
I was working around with the decomposition of the multidimensional linear canonical transform (which is not even continuous w.r.t. the parameters) into a few fractional Fourier transforms (and other ...
- 151
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votes
0
answers
181
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Main ideas behind the proof of the Carleson theorem
I tried to read a few years ago the book "Pointwise Convergence of Fourier Series" (Springer, Juan Arias De Reyna) which is a detailed proof of the Carleson theorem, but I was lost after a ...
- 577
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votes
2
answers
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When I know self convolution of the complex function can I recover function itself or its modulus?
I have a function $A : \mathbb{R} \to \mathbb{C}$.
I know there exists unknown function $u: \mathbb{R} \to \mathbb{C}$, such that $A$ is convolution of $u$ and its complex conjugate $A = u * u^*$.
I ...
- 141
6
votes
0
answers
153
views
Detailed examples of induction on scale
I'm trying to understand the induction on scale argument in harmonic analysis. On this abstract it's mentioned that induction on scale can be used to prove Cauchy Schwartz inequality, Beckner's tight ...
- 203
1
vote
0
answers
47
views
A convergence problem in the space of tempered distributions
Let $K(x):=|x|^{-\alpha}$ be a function on $\mathbb{R}^{n}\setminus\{0\}$ with $0<\alpha<n$. Suppose $\phi$ is a $C^{\infty}_{c}(\mathbb{R}^n)$ function such that
$$\text{(i)}\quad \text{supp}\...
- 391
1
vote
1
answer
108
views
Why does failure of boundedness of this operator for $p<q$ implies its failure for $p>q^{\prime}$?
I am reading the paper "P.Sjolin, Convolution with Oscillating Kernels, Indiana University Mathematics Journal Vol. 30, No. 1 (1981), pp. 47-55" where $L^p-L^p$ boundedness of the operator
$...
- 391
2
votes
1
answer
256
views
Fourier series of Eisenstein series — elegant and very good approximation
I played around with the Fourier series of the Eisenstein series resp. divisor sums and did some calculations, see below. Although the deduction is not rigorous / wrong (as the power series for the ...
- 386
0
votes
1
answer
123
views
Fourier series of an arbitrary function of a cosine function
Is there a general expression for the Fourier series of the function $f(a\cos(\omega t))$ in terms of the derivatives of $f$? Obviously, the function can be expressed as a Maclaurin series $f(0)+af'(0)...
0
votes
0
answers
80
views
Closed formula for iterated Fourier series
I'm trying to obtain a closed formula for the following integral.
\begin{align}
I_n = {} & \int_0^h \Bigr[\sum_{r_1=1}^\infty a_{1,r} \cos\left(\frac{2\pi}{h} r_1t_1\right) \\[6pt]
& {}+ b_{1,...
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votes
1
answer
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views
A geometric interpretation of the fractional Fourier transform
I was reading Joe Polchinsky’s autobiography which contains the following anecdote from his time at Caltech (page 18):
Once a week, Feynman led Physics X, where freshman and sophomores could ask ...
- 51
4
votes
1
answer
159
views
How to compute the asymptotics of this oscillatory integral?
I posted this on Stackexchange but got no responses or comments.
Consider the following integral, for $\epsilon\ne 0:$
$$\displaystyle\frac{1}{(2\pi)^2\epsilon^4}\int_{\Omega}yb\,e^{\frac{i}{\epsilon}[...
- 885
2
votes
1
answer
106
views
Vertical Fourier decomposition for skew-Hermitian 1-forms
In an arXiv preprint [2108.05125v1], the authors use the following vertical Fourier decomposition (page 7 therein).
Let $(M,g)$ be a Riemannian surface and $SM$ be its unit tangent bundle. Denote by $...
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votes
2
answers
336
views
$L^p-L^q$ boundedness of this simple singular oscillatory integral operator
Let $0<\alpha<1$ and define
$$Tf(x):=\int e^{\dot{\imath} x y} \frac{f(y)}{|x-y|^{\alpha}}dy.$$
The Hardy-Littlewood-Sobolev inequality characterizes $L^p-L^q$ boundedness of
$Hf(x):=\int \frac{...
- 391
1
vote
1
answer
106
views
Uniqueness of Fourier–Stieltjes transform for finite complex valued measures
Let $\mu$ be a finite complex valued measure on $\mathbb{R}$ and let $\hat{\mu}$ be it's Fourier–Stieltjes transform
$$
\hat{\mu}(\omega)= \int_{\mathbb{R}} e^{it\omega} d \mu(t)
$$
Question: Does $\...
- 611
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votes
2
answers
666
views
Is it possible to obtain the inequality $\|\nabla f\|_{L^{2p}} \leq C (\|f\|_{L^\infty} \|f\|_{W^{2, p}})^{1/2}$ from interpolation/harmonic analysis?
Nirenberg's paper On elliptic PDEs claims that if a function $f$ on $\mathbb{R}^n$ tends to zero at infinity or is in $L^q$ for any $q < \infty$ then the "interpolation" inequality
$$
\...
- 1,081
0
votes
1
answer
148
views
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_{\...
- 3,764