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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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Fourier transform relation for spherical convolution

Let $f$ and $g$ be two functions defined over the 2d sphere $\mathbb{S}^2$. The convolution between $f$ and $g$ is defined as a function $f * g$ over the space $SO(3)$ of 3d rotations as $$(f*g)(R) = \...
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Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$. Schwartz space is dense in $A$ wrt $\|f\|:= \|\hat{f}\|_1+\|\hat{f'}\|_1$?

Let $A:=\{f\in C^1(\mathbb{R}): \hat{f}, \hat{f'} \in L^1(\mathbb{R})\}$, where $\hat{f}$ is the Fourier transform of $f$. Then is it true that Schwartz space $\mathcal{S}(\mathbb{R})$ is dense in $A$ ...
mathlover's user avatar
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Does $i\partial_tu = \Delta^2 u$ exhibit more or less dispersion than $i\partial_t u= \Delta u$?

Consider the initial-value problems in $d=1$ $$\begin{cases} i\partial_tu = \Delta^2 u \\ u(x,0)=u_0 \end{cases}$$ and $$\begin{cases} i\partial_t u= \Delta u \\ u(x,0)=u_0, \end{cases}$$ Solutions to ...
Diffusion's user avatar
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199 views

Computing a closed form representation for a Fourier series summation

I want to compute a closed form representation for the below given summation expession. $$g_{\lambda}(\boldsymbol{x}) = \sum\limits_{\boldsymbol{l}\in\mathbb{Z}^m} \frac{1}{1+\lambda\|\boldsymbol{l}\|...
Rajesh D's user avatar
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estimate a singular integral using a dyadic decomposition

Let $0<\alpha_{j}<1$, $j=1,\dots,d+1$. I am trying to estimate the following singular integral: $$I(y_{1},\dots,y_{d},z):=\int_{\substack{ x\in[0,1]^{d}\\ 1/2<|x|<1}} \frac{d x_{1} \dots d ...
Medo's user avatar
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1 answer
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How to show such result for generalized $ O(|x|^{-1/2}) $ function?

Assuming that $ \chi\in C_c^{\infty}([-2,2]) $ is a cutoff function such that $\text{supp }\chi\subset[-2,2]$, $\chi\equiv 1 $ in $ [-1,1] $, and $ 0\leq\chi\leq 1 $, suppose that $ f\in C^{\infty}(\...
Luis Yanka Annalisc's user avatar
4 votes
1 answer
396 views

Reference or proof of a theorem of L. Fejér on summability of Fourier series

In the article "Ensembles exceptionnels" by A. Beurling, the author cites the following theorem of Fejér: Suppose that a $2\pi$ periodic function $ f $, Lebesgue integrable in $(0,2\pi)$ ...
an_ordinary_mathematician's user avatar
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Orthogonal space of polynomials

Let $f \colon [0,+\infty) \to \mathbb R$ be a continuous function. Assume that for any non-negative integer $n$, the function $f(t) t^n$ in integrable in $(0,+\infty)$ and $$ \int_0^{+\infty} f(t) t^n ...
henrysupercool's user avatar
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Maximal function on mixed $L^{p}$

Consider $ f_{j,k}$ to be a function in $L^{p}(l^{q}(l^{2}))$, that is $$ \Vert f_{j,k} \Vert^{p}_{L^{p}(l^{q}(l^{2}))} = \int_{\mathbb{R}^{n}} \left( \sum_{k} \big[ \sum_{j} \vert f_{j,k}(x) \vert^{2}...
User091099's user avatar
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1 answer
103 views

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 ...
Mohammad A's user avatar
2 votes
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Construction of an analytic function whose Fourier transformation has compact support [closed]

Is there a non-constant real analytic function $f$ on $\mathbb{R^2}$ satisfying the following properties? $f$ vanishes on $x$-axis and $y$-axis; the Fourier transformation $\hat{f}$ of $f$ has a ...
adobereader's user avatar
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Rate of convergence of mollified distributions in Besov spaces with negative regularity

Given a standard mollifier $\rho_\delta$ and a distribution $ u \in B^\alpha_{ p, p}$ with $\alpha<0$, $p \in [1, \infty]$ and $B^\alpha_{p,p}$ is a not-homogeneous Besov space, I'm trying to prove ...
Marco's user avatar
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Relation between Fourier series and Schur polynomials

Asked initially at MSE. I would like to know how to express the Fourier series of a symmetric function, $f(\theta_1,...,\theta_N)$, in terms of Schur polynomials $s_\lambda(x_1,...,x_N)$ in the ...
thedude's user avatar
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Fourier series of function with zero of infinite order?

It is easy to give examples of positive periodic functions with zeros of order $n$ where $n$ is a natural number and whose Fourier series expansion is explicit (powers of cosine will do). I wonder if ...
António Borges Santos's user avatar
4 votes
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154 views

Measurability of $L^{p}(L^{q})$ integrable functions

Let $ F: \mathbb{R}^n \times (0,\infty) \to \mathbb{R}$ be a function with the property that $ \int_{\mathbb{R}^n} \big[ \int_0^\infty |F(x,r) |^q \, dr \big]^{p/q} \, dx < \infty $ In addition we ...
User091819's user avatar
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Vectors of complex exponentials span $\mathbf{C}^N$

Let $Q = [0,1]\times [0,1]$. Let $1\leq k < \infty$ and $\{(x_l,\xi_l)\}_{l=1}^{k}\subseteq Q$ be such that $(x_i,\xi_i)\ne (x_j, \xi_j)$ for $i\ne j$. Additionally, for $1\leq l \leq k$, let $n_l\...
Doofenshmert's user avatar
1 vote
1 answer
106 views

approximating differentiable functions with double trigonometric polynomials

Let $Q = [0,1]^2$. For sake of notation, let $$ f^{(i,j)}(x,\xi) = \frac{\partial^{i+j}}{\partial x^i \partial \xi^j}f(x,\xi). $$ Fix some non-negative integer $k$. Moreover let $f\in C^k(Q)$ if $$ \|...
Doofenshmert's user avatar
1 vote
1 answer
99 views

An asymptotic integral with complex phase

Suppose that $D\subset \mathbb R^2$ is the closed unit disk and that $f\in C^{\infty}(D)$. Assume that for all $\lambda \in (1,\infty)$ there holds $$ \left|\int_D f(x^1,x^2)\, e^{\lambda (x^1+ix^2)}\,...
Ali's user avatar
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On weighted Fourier transforms

Suppose that $f\in L^{\infty}((0,1))$ and that there exists $c_1,c_2>0$ such that $$ \left|\int_0^1 e^{i \xi x} e^{-|\xi|^{-1}x}f(x)\,dx \right| \leq c_1 e^{-c_2|\xi|} \quad \forall\, |\xi|>1.$$ ...
Ali's user avatar
  • 4,103
7 votes
3 answers
535 views

Real-world examples of unweighted directed graphs

Social Networks, Internet Traffic, Citation Networks, Transportation Networks, and Biological Networks exemplify real-world instances of unweighted directed graphs. Within these domains, the Graph ...
ABB's user avatar
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2 votes
0 answers
188 views

Finding specific coefficients of product of high-dimensional Fourier series faster than FFT

I need a fast algorithm to perform a specific Fourier-type computation in my physics research. Suppose I have the following two Fourier series in three dimensions $$ a(t_1,t_2,t_3)=\sum_{j=-n}^{n}\...
groupoid's user avatar
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0 answers
47 views

Equating two Fourier Series with different periods

For some $\tau\in(0,1)$, let $f : (-\infty,0]\times [0,\tau] \rightarrow \mathbb{C}$ and $g:[0,\infty)\times[0,1]\rightarrow\mathbb{C}$ with $$f(0,t)=g(0,t)\text{ for }t\in [0,\tau]$$ be two smooth ...
Martin's user avatar
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165 views

Construct a continuous function $f(x)$ periodic with period $2\pi$ such that the Fourier series of $f(x)$ is divergent at $x = 0$

The following question was asked on Math Stack Exchange by me 15 days ago. I used a bounty, but still no response. So I am posting the question here. Here is the link of the question here. problem ...
Martin.s's user avatar
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112 views

Radial Fourier transform vs Hankel transform

I was wondering about the relationship between the Hankel transform and the Fourier transform for radial functions. Let $f : \mathbb{R}^d \to \mathbb{R}$ be a (sufficiently regular) radial function. ...
Chris Jones's user avatar
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32 views

Reference request: injectivity of CWT, density of dilations and translations in $L^p$

Recently, I encountered the notion of Continuous Wavelet Transform (CWT), and I find it very intriguing (for a reference, see the wiki). I believe it offers a different and more general perspective on ...
Zhang Yuhan's user avatar
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63 views

Multiplication with dilations of nonzero measurable function is injective

Denote $f_s(x):=f(sx)$ as the dilation of a function $f$. I want to know whether the following statement is true: Suppose $f$ and $g$ are measurable functions on $\mathbb{R}$, and $f$ is not almost ...
Zhang Yuhan's user avatar
1 vote
1 answer
105 views

A bilinear estimate with a simple one-dimensional oscillatory integral kernel

Let $f\in L^{p}(\mathbb{R})$, $1\leq p\leq 2$. I am trying to show that $$\int_{\mathbb{R}}\int_{\mathbb{R}} \,K(y,z)\, \frac{f(y)f(z)}{y^{\frac{1}{2\,p^{\prime}}}\,z^{\frac{1}{2\,p^{\prime}}}}\,dy\,...
Medo's user avatar
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7 votes
1 answer
287 views

High dimensional Lusin conjecture

Lusin, in 1913, while considering the properties of Hilbert's transform, conjectured that every function in $L^2[-\pi, \pi]$ has an a.e. convergent Fourier series. Kolmogorov, in 1923, gave an example ...
Tomas's user avatar
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6 votes
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135 views

Estimating the Hausdorff dimension of the discontinuity set of a function

Suppose $\sum_{\xi \in \mathbb{Z}^d}{a_{\xi}}e^{i\langle x, \xi\rangle }$ converges spherically pointwise to $0$ for all $x \in \mathbb{T}^d$, i.e. $\lim_{R \to \infty} \sum_{|\xi| < R}{a_{\xi}}e^{...
fwd's user avatar
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10 votes
5 answers
898 views

Proving Poincaré's inequality for Boolean functions over the hypercube without Fourier analysis

$\DeclareMathOperator\Inf{Inf}\DeclareMathOperator\unif{unif}$I have been attempting to find a non-Fourier-analytic proof of Poincaré's inequality for Boolean functions over the hypercube. Let's call ...
Mathews Boban's user avatar
3 votes
1 answer
152 views

When is a solution $P(f'(x)) = Q(f(x))$ periodic or double periodic?

$\newcommand{\cl}{\operatorname{cl}}\newcommand{\sl}{\operatorname{sl}}\newcommand{\cm}{\operatorname{cm}}\newcommand{\sm}{\operatorname{cm}}$Consider the differential equation $$P(f '(x)) = Q(f(x))$$ ...
mick's user avatar
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0 answers
117 views

Counterexamples in Laplace transforms

Of all the examples I know of (bilateral) Laplace transforms $F$ defined on their maximal vertical strips $V_{a,b}=\{ z \in \mathbf{C} : a < \operatorname{Re}(z) < b \}$ with $-\infty < a <...
proofromthebook's user avatar
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0 answers
67 views

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 \...
Grandes Jorasses's user avatar
1 vote
0 answers
55 views

Density of zero modes

Let $(M,g)$ be a compact smooth Riemannian manifold with a smooth boundary. Let $\{(\lambda_k,\phi_k)\}_{k\in\mathbb N}$ be the spectral data on $(M,g)$, namely an orthonormal basis for $L^2(M)$ ...
Ali's user avatar
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0 votes
1 answer
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A simple bilinear estimate

Let $2\leq p,q <\infty$ and fix $0<\alpha<1$ such that $\frac{1}{p}+\frac{1}{q}\leq 2-\alpha$. Suppose that $f\in L^{p}([0,1])$ and $g\in L^{q}([0,1])$. What is the optimal value of $t=t(\...
Medo's user avatar
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2 votes
1 answer
144 views

Are the coefficients in the stationary phase approximation computed explicitly somewhere

In Stein's "Harmonic analysis" book, page 334, one can find the asymptotic expansion An instructive proof is given for the case $k=2$. It is clear enough to generalize to the cases $k\geq ...
Medo's user avatar
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2 votes
1 answer
299 views

Fourier series but different waveform

Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. Let $f_n: [0,2\pi]\to \mathbb C \...
Zhang Yuhan's user avatar
5 votes
1 answer
220 views

An asymmetric quadrilinear estimate

Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$ where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+...
Medo's user avatar
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2 votes
2 answers
193 views

Is there a compactly supported differentiable function whose Fourier transform is not in L1?

In my MSE answer here, I discussed the example of compactly supported continuous function $$g(x)= \begin{cases} \dfrac{\frac12 -x}{\log(x)},&0<x\leq1/2\\ 0,&\text{otherwise} \end{cases}$$ ...
D.R.'s user avatar
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-2 votes
1 answer
261 views

Predicting the peak "amplitude" of a damped sine wave in the frequency spectrum with FFT

In one line: Given an exponentially decaying sine wave $x(t)$, how can we predict the amplitude of the resulting peak in frequency spectrum using discrete Fourier transform. In nuclear magnetic ...
AChem's user avatar
  • 801
1 vote
0 answers
40 views

Looking at a frequency reassignment rule as a Möbius transform

Suppose we have some Schwartz function $h$. Denote its Fourier transform $\widehat{h}$. Let $\xi_0$, $a$, $\Delta$ be positive and fixed. I have a function $\Omega: \mathbb{R}\times \mathbb{R}^+ \to \...
mathim1881's user avatar
1 vote
1 answer
148 views

To find a $2\pi$-periodic function with a property

I recently came across the following question in my research, and I don't know how to proceed this problem. Question: How to find a function $g(x)$ such that it satisfies (1) $2\pi$ periodic (2) odd (...
tony's user avatar
  • 329
3 votes
1 answer
143 views

Rescaling Fourier coefficients of a continuous function by a bounded sequence

This question stems out of: which sequences $(a_n)_{n\in\mathbb{Z}}$ of complex numbers have the property that if there exists a continuous function $f$ on the circle with Fourier coefficients $b_n$, ...
Logan Hyslop's user avatar
1 vote
0 answers
64 views

The Discrete Fourier Transform (DFT) decomposes any signal into four orthogonal signal components [closed]

Let $F=(w^{kl})_{k,l=0}^{n-1}$ be the discrete Fourier matrix of size $n$ where $w=\exp\left(-\frac{2\pi i}{n}\right)$. It is a well-known that $F_n^4 = I_n$ where $I_n$ represents the identity ...
ABB's user avatar
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0 votes
0 answers
92 views

Fourier integral operators and parametrix

Consider the classical wave equation in a bounded domain in $\mathbb{R}^n$, assuming vanishing of the function and its normal derivative on the boundary. Question: Is there an expression for the ...
0x11111's user avatar
  • 493
1 vote
0 answers
51 views

Best approximation rates of various classes of functions by truncated Fourier series

Let $f\in C([-1,1]^d)$ have periodic boundary, $N$ be a positive integer, and let $S_N(f)$ be the best approximation of $f$ by its truncated Fourier expansion truncated approximation $$ S_N(f):=\sum_{...
Math_Newbie's user avatar
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0 answers
84 views

Why does $\omega$ belong to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-2 + n\varepsilon^3}$ different $\theta$?

In this paper, there is the following claim (Pg. 1850): If $1 - \eta(w) \ne 0$, then $|\omega| \ge \rho$. In that case, $\omega$ belongs to $D^{\varepsilon^3}\theta^\ast$ for $\lesssim \rho^{-n} D^{n-...
stoic-santiago's user avatar
2 votes
1 answer
117 views

Singular Integrals and $L^1$

Let us consider in one dimension the Fourier multiplier $\vert D\vert$ and the derivative $iD$. Both are well-defined on the Schwartz space $\mathscr S(\mathbb R)$ with the derivative sending $\...
Bazin's user avatar
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2 votes
0 answers
99 views

Fourier multiplier on $L^1$

On the Wikipedia page, one can read that an iff condition for L1 boundedness of the Fourier multiplier m(D) is that $$ \hat m\quad\text{ is a Borel measure with finite total mass. } $$ There is no ...
Bazin's user avatar
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3 votes
0 answers
111 views

How are distributions and divergent series summations related?

When we do Fourier analysis, we don't always get convergent series. A classic example comes from considering the Sawtooth function. It has Fourier Coefficients $$s(x) = \frac{1}{2} + \sum_{n \neq 0} \...
Caleb Briggs's user avatar
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