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.

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69 views

Zygmund class, Schwartz class and Littlewood-Paley projection operators

I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions: Consider the Zygmund class of functions defined as ...
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1answer
75 views

Estimate of Hölder Norms (Littlewood–Paley theory)

I'm currently studying Littlewood–Paley theory and its application to norm estimate/PDEs by reading Muscalu and Schlag's textbook, where I encountered the following norm estimate problem: Recall that ...
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2answers
154 views

Vanishing convolution between density and compactly supported function

Find a pair of functions $f,g:\mathbb{R}\to\mathbb{R}$ such that: $f$ is smooth and compactly supported (say, on $[0,1]$ but this isn't crucial), $g(x)>0$ for all $x\in\mathbb{R}$, $\int g(x)\,dx=...
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1answer
285 views

Does there exist an upper bound on the Fourier coefficients of the reciprocal theta function $\frac {1}{\theta}$?

Define the theta function as $$ \theta(x) = \sum_{n=-\infty}^\infty e^{-\gamma(x+n)^2} $$ where $\gamma>0$. Clearly, $\theta$ is 1-periodic, non-zero and smooth. Therefore, the reciprocal map $x \...
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48 views

Inverse of a Function Using its Fourier Series

Let ${f}:{\mathbb{{{R}}}}\to{\mathbb{{{R}}}}$ be a real, isomorphic function defined on the interval ${\left[{0},{l}\right]}$ yielding a meromorphic analytic continuation to the complex plane ${\...
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1answer
123 views

Scaling of double convolution

I am interested in the scaling of $$F(x_1,x_4)=\int_{\mathbb R^2} e^{-\vert x_1 -x_2 \vert -\varepsilon \vert x_2 -x_3 \vert- \vert x_3 -x_4 \vert } \ dx_2 dx_3 $$ In particular, I suspect that $$F(...
3
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1answer
117 views

A convolution type singular integral operator with log

Define a convolution type operator $T_m$ by $$T_m(f) = p.v.\int_\mathbb{R}f(x-y)\frac{\log^m|y|}{y}dy.$$ Here $m\ge0$ is an integer. Consider $f \in H^s (s > 0)$ which is the usual Sobolev space. ...
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2answers
175 views

What is the distribution of the following limit?

Assume $x \in \mathbb{R}$. We already know that $$\lim_{\epsilon \to 0+} \frac{1}{x-i\epsilon} - \frac{1}{x+i\epsilon} = 2\pi i \delta_x.$$ Here $\delta_x$ denotes the Dirac distribution. If we ...
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2answers
175 views

Decay estimate of Fourier transform of a compactly supported function

Assume $f(x), x \in \mathbb{R}$ is a function with a compact support such that its Fourier transform $\hat{f}(\xi)$ has a decay rate $$\hat{f}(\xi) \lesssim \frac{1}{|\xi|^\gamma + 1}$$ for some $\...
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59 views

Half-integer Fourier transform

Suppose we want to carry out the (generalized) Fourier transform of a function defined in the domain $\mathbb{T}\times\tfrac12\mathbb{Z}$, i.e. dependent on the arguments $\phi\in\left(-\pi,+\pi\right]...
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33 views

Hamiltonian of Benjamin-Ono equation

Could someone please tell me why the Hamiltonian of the Benjamin-Ono equation $$\partial_t u=H \partial_x ^2 u -\partial_x(u^2),$$ is given, on the Torus $\mathbb R / 2\pi \mathbb Z$ , by $$\mathscr{H}...
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1answer
312 views

Schwartz regularity for the density of a stochastic process

Let $B$ be a standard Brownian motion in $\mathbb R$. Define the variables $$\begin{align*} X &= B_1, & Y &= \int_0^1B_s\mathrm ds, & Z&= \int_0^1B_s^2\mathrm ds. \end{align*}$$ It ...
5
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1answer
213 views

Generalization of Bernstein’s inequality

I'm using Muscalu and Schlag's textbook to study harmonic analysis and I encountered the following claim: Given some function $f \in \mathcal{S}(\mathbb{R}^{d})$, where $\mathcal{S}(\mathbb{R}^{d})$ ...
2
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1answer
68 views

Uniform convergence of Eigenfunction decomposition on Riemannian sphere?

Let $\{u_k\}_{k=1}^\infty$ be a sequence of ($L^2$ normalized) mutually orthogonal eigenfunctions of the operator $-\Delta$ on the sphere $\mathbb{S}^n$ (here $\Delta$ is the Laplace Beltrami operator)...
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94 views

$L^p$ estimate of a multiplier operator

I'm studying harmonic analysis by myself and I encountered the following claim about multipliers: consider a sequence of complex numbers $\{m_{n}\}_{n \in \mathbb{Z}}$ that satisfies: $$\sum_{n \in \...
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1answer
84 views

Is the Besov space $B_{\infty,1}^0(\mathbb{R}^d)$ a multiplication algebra?

Let $s\in\mathbb{R}$ and $1\leq p,q\leq\infty$. Consider the Besov scale of spaces $B_{p,q}^s(\mathbb{R}^d)$ defined by the norm $$\|f\|_{B_{p,q}^s} := (\sum_{j=0}^\infty \|P_{j} f\|_{L^p}^q)^{1/q},$$ ...
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38 views

Can I write this series in a recursive way?

I would like to know given the following definition of the function X(n) if it is possible to express two consecutive values of X(n) (example: X(1) and X(2) ) to obtain a recursive expression. What I'...
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2answers
263 views

Fourier transform of eigenvalue distribution of GUE matrices

I am interested in explicit expression or bounds for the Fourier transform (characteristic function) of the joint probability distribution of eigenvalues of random matrices $X\sim \mathrm{GUE} (d)$, ...
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86 views

Weak-type inequality for the partial Fourier sum operator

I'm studying harmonic analysis by myself. One of the online notes gives the following claim as a remark: For any $N \in \mathbb{Z}^{+}$, let's use $S_{N}$ to denote the partial ($N$ terms) Fourier sum ...
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1answer
88 views

Analogous form of Hardy-Littlewood maximal inequality (weak/strong type) on affine subspaces

I'm using some online notes (Professor Schlag, Yale University) to study harmonic analysis by myself. He introduced the following claim as an exercise: For any function $f \in L^{1}(\mathbb{R}^{d})$ ...
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68 views

Definition of a continuous Gabor frame

I am trying to understand the definition of a Gabor frame and would appreciate some clarification with terminology. Let us begin with the setup: Let $G$ be a locally compact abelian group, and let $\...
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Magnitude spectrum of a cascade of filter

Given is a input vector $x=[x_1 x_2 x_3] \in \mathbb{R}^{3N}$ with 3 consecutive sub-blocks $x_1,x_2,x_3 \in \mathbb{R}^{N}$, which goes through a cascade of filtering operations defined as Step 1 (1-...
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1answer
230 views

What corresponds to the operation of taking traces in of the Fourier transformation on a finite group?

I have a question about the Fourier transfomation on a finite non-comutative group. I hope that it is a known fact in the Representation Theory but I cannot find it written explicitly in textbooks. ...
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1answer
159 views

“Reversed” Bernstein Inequality

I'm studying harmonic analysis by myself, and I read some online notes that introduce the Bernstein inequality. One of them mention a reversed form of the Bernstein inequality, which is stated below: ...
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70 views

Decay conditions on the Fourier coefficients ensuring that a smooth function doesn't vanish on some interval

Let $f$ be a non-zero smooth $1$-periodic function. Is there a decay condition on its Fourier coefficients ensuring that it won't vanish identically on any interval? I mean a condition weaker than &...
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1answer
646 views

What does $O(N)$ mean in this article and how does it imply this lemma?

In this article the author proves the following lemma: LEMMA: $\forall N \in \Bbb N$, there exists $v=v_N$ with compact support so that $$[M_S(M_S v)^\delta(x)]^{1/\delta} \geq c_\delta NM_S v(x), \...
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1answer
76 views

Integrability of fractional heat kernel

In Estimates of fractional heat kernel, it was stated that $$ \partial_{x_j} p_t^{(n)}(x) = -\frac{x_j}{2 \pi} \, p_t^{(n+2)}(\tilde x) $$ where $x = (x_1, \ldots, x_n) \in \mathbb R^n$, $\tilde x = (...
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69 views

What are the necessary/sufficient conditions for a Fourier transform to have at least $k$ roots?

Let $f(x)$ be a symmetric function from $\mathbb{R}\to \mathbb{R}$, and $\hat f(k)$ be it's Fourier transform. What are the necessary and sufficient conditions for $\hat f(k)$ to have at least $n$ ...
4
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1answer
168 views

Vanishing of the product of a function and its own Fourier transform

I have found the following question to be surprisingly hard: Is there a non-zero $f\in L^1(\mathbb R)$ or $f\in L^2(\mathbb R)$ such that $$ f\cdot\hat f=0 \qquad \text{Lebesgue-almost everywhere}, $$ ...
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63 views

When can one bound the Hilbert transform on the torus in $L^1$?

If $f$ is a function on $\mathbb T$ then the conjugate $\tilde f$ of $f$ satisfies Zygmund's bound $$\int |\tilde f|\leq A \int |f|\log(e+|f|)+B.$$ I am curious if there are any conditions where one ...
4
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1answer
178 views

Smallest regular $m$-gon covering a regular $n$-gon

I start by stating the problem, which is already hinted in the title of the question. I do believe it is a research-level question. Let us fix a regular $n$-gon with area $1$. What is the smallest ...
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1answer
55 views

Variance of spectral density is related to the gradient of signal?

Define the frequency variance as: $$ \sigma^2 = \int^\infty_{-\infty}\omega^2 P(\omega) d\omega$$ Where $P(\omega)$ is the spectral density function, which is the same as normalized power. Therefore, $...
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1answer
131 views

Does Bochner's Theorem apply to Fourier coefficients?

Let $f $ be a periodic function and denote by $c_n$, for $n \in \mathbb{N}$, its Fourier coefficients, i.e. $$ c_n := \frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)e^{inx}\ dx. $$ It is well known that Bochner's ...
6
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1answer
251 views

Wiener Corollary in “An introduction to harmonic analysis” by Yitzhak Katznelson

I can't understand a lemma in "An introduction to harmonic analysis" by Yitzhak Katznelson which is stated as follows: Corollary. Let $\mu\in M(\mathbb T)$. Then $$\sum\limits_{\tau\in\...
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51 views

Context for this discrete Cauchy integral formula

Notation: I will use the following conventions for discrete Fourier transforms (DFT) and discrete time Fourier transforms (DTFT): $$\mathcal{D}_N[x_j](k) := \sum_{j=0}^{N-1} e^{-2\pi i j k} x_j$$ $$\...
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0answers
33 views

Function of several variables whose hessian is a Hankel matrix

First of all, let me apologize because I asked this question a few days ago on https://math.stackexchange.com, but I did not get any reply. I am studying a function $f:\mathbb{R}^n\rightarrow\mathbb{R}...
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1answer
92 views

How many Fourier coefficients of a sparse signal $f=\sum_{n=1}^Nc_n\delta_{t_n}$ are needed to determine $f$ uniquely?

Let $N \in \mathbb N$ and $c_n \in \mathbb C$, $t_n \in \mathbb R$ for $n=1, \dots, N$. Suppose that $f$ is a linear combination of dirac-deltas with locations $t_n$ and coefficients $c_n$, i.e. $$ f=\...
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51 views

Some density properties about Sobolev periodic spaces

Let $L>0$ fixed. Consider the space $$ \mathcal{P}:=\{f: \mathbb{R} \longrightarrow \mathbb{C} \; ; \; f \: \text{is infinitely differentiable and periodic with period}\: L\}. $$ For $r \in \mathbb{...
7
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1answer
291 views

What makes Gaussian distributions special? Local field version?

This question is inspired by the recent one about Gaussian measures over the reals: What makes Gaussian distributions special? I would be interested in a similar list of characterizations for the ...
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0answers
48 views

Does every compact abelian group contain a Kronecker set generating a dense subgroup?

Let $G$ be a compact metrizable abelian group with infinite exponent. Let $S^1 = \left\{z \in \mathbb{C} : |z| = 1 \right\}$. A set $K \subset G$ is a Kronecker set if, for every continuous function $...
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0answers
66 views

Smoothing property of a certain singular integral operator of non-convolution type

For simplicity, suppose that the dimension $d=2$, and let $g_s(x)$ be the Coulomb or Riesz potential defined by $$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0<...
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0answers
57 views

Factoring a complex function such that it is analytic in upper and lower plane

Consider this function $$\frac{k^{2}-\xi^{2}}{k^{2}+1}$$ which has singularities at $k=\pm i$, the strips where it is analytic are $$ -1<k^{\prime \prime}<0 \quad \text { or } \quad 0<k^{\...
0
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1answer
60 views

Can a Fourier transform be performed on irregularly sampled data with timestamps?

Normally, when I think of performing a Fourier transform, I imagine that my samples are spaced regularly in time (or space). If I have a set of samples that are spaced irregularly, but have accurate ...
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0answers
58 views

Second question on a real sequence

I am thinking again about the real sequence $\{a_n\}_{n\ge1}$ which decays faster than any algebraic speed, that is, $\lim_{n\to \infty}n^pa_n = 0$ for every integer $p$. Actually, $a_n$ can be ...
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0answers
23 views

Deriving periodical processes from a finite time series

Suppose we have a finite time series of real-world events measured at $(t_k), k \in \mathbb{N}$ with $(t_{k-1} < t_k)$. The content of the actual events is irrelevant. I would like an automated ...
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0answers
48 views

Converse to Hausdorff-Young (or Riesz-Thorin) for finite cyclic groups?

Let $v$ be a vector $v \in \mathbb{R}^p$, with non-negative entries and $p$ prime. The Hausdorff-Young inequality gives bounds of the form: $$\|\mathcal{F}v\|_a \le C_{a,b} \|v\|_b$$ where the ...
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0answers
101 views

Integration by parts formula for the spectral fractional Laplacian

Let $f,g:C^\infty_c(0,1)$. Is there a formula similar to $$ \int_0^1 (f_{xx})^2g \ dx = \int_0^1 \frac{1}{2} (f_x)^2 g_{xx} \ dx - \int_0^1 f_{xxx}f_x g \ dx $$ for the spectral fractional Laplacian ...
3
votes
1answer
145 views

Using Fourier series to prove $-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)^2 \eta_{xx}$

Let $u, \eta$ be smooth functions and $\eta$ compactly supported in $(0,1)$. Integrating by parts, we can easily prove $$-\int_0^1 u_{xxx}u_x \eta = \int_0^1 (u_{xx})^2\eta - \int_0^1 \frac{1}{2} (u_x)...
2
votes
0answers
147 views

Functions whose Fourier coefficients satisfy $ \sum_{k=1}^\infty |c_k| < 1 $?

Let $f:(0,1) \to \mathbb R$ be a function that can be written as $$f(x) = \sum_{k=1}^\infty c_k \phi_k(x),$$ where $\phi_k(x) = \cos(\pi k x)$. What is the minimal assumption required on $f$ to ...
1
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1answer
266 views

Deriving the functional equation for $\zeta(s)$ from summing the powers of the zeros required to count the integers

When counting the number of integers $n(x)$ below a certain non-integer number $x$, the following series could be used: $$n(x) = x-\frac12 + \sum_{n=1}^{\infty} \left(\frac{e^{x \mu_n}} {\mu_n}+\frac{...

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