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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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:
$$\mathscr{F}\{\psi\}(x)=-2\pi \gamma \delta(x)-\pi \...
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Density of zero modes
Let $(M,g)$ be a compact smooth Riemannian manifold with a smooth boundary. Let $\{(\lambda_k,\phi_k)\}_{k\in\mathbb N}$ be the spectral data on $(M,g)$, namely an orthonormal basis for $L^2(M)$ ...
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A simple bilinear estimate
Let $2\leq p,q <\infty$ and fix $0<\alpha<1$ such that
$\frac{1}{p}+\frac{1}{q}\leq 2-\alpha$.
Suppose that $f\in L^{p}([0,1])$ and $g\in L^{q}([0,1])$.
What is the optimal value of $t=t(\...
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Are the coefficients in the stationary phase approximation computed explicitly somewhere
In Stein's "Harmonic analysis" book, page 334, one can find
the asymptotic expansion
An instructive proof is given for the case $k=2$. It is clear enough to generalize to the cases $k\geq ...
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Fourier series but different waveform
Given a nondegenerate smooth simple closed convex curve $f: [0,2\pi]\to \mathbb C \setminus \{0\}$ with winding number (around origin) $1$, and $f$ have zero mean. Let $f_n: [0,2\pi]\to \mathbb C \...
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An asymmetric quadrilinear estimate
Fix $1<p<2$ and let $a_{i}=1-\frac{\theta_{i}}{p^{\prime}}$
where $\theta_{i}\in (0,1/2)$, $i=1,2,3,4$, and $p^{\prime}$ is the conjugate exponent of $p$. Note here that $0<a_{i}=1-\theta_{i}+...
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Is there a compactly supported differentiable function whose Fourier transform is not in L1?
In my MSE answer here, I discussed the example of compactly supported continuous function
$$g(x)=
\begin{cases}
\dfrac{\frac12 -x}{\log(x)},&0<x\leq1/2\\
0,&\text{otherwise}
\end{cases}$$
...
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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 ...
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Literature search for results on Bochner-Riesz means for real functions
MathOverflow community!
I'm delving into a specific area of Fourier analysis and came across an intriguing lemma (referenced as Lemma 4 of Chen & Chen Paper) stating that for a compact set U in $...
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Is there a Bochner-Riesz Theorem for Real Functions with Each Summand Being a Product of Two Real Functions?
I am exploring the Bochner-Riesz theorem and its implications on real functions. Specifically, I'm curious whether there exists a version of the Bochner-Riesz theorem where each summand in the Bochner-...
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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 \...
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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
(...
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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$, ...
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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 ...
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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 ...
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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_{...
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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-...
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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 $\...
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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 ...
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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} \...
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Relationship between Fourier inversion theorem and convergence of "nested" Fourier series representations of $f(x)$
$\DeclareMathOperator\erf{erf}\DeclareMathOperator\sech{sech}\DeclareMathOperator\sgn{sgn}\DeclareMathOperator\sinc{sinc}$This is a cross-post of a question I posted on MSE a couple of weeks ago which ...
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Sufficient condition for a probability distribution on $\mathbb Z_p$ to admit a square-root w.r.t convolution
Let $p \ge 2$ be a positive integer, and let $Q \in \mathcal P(\mathbb Z_p)$ be a probability distribution on $\mathbb Z_p$.
Question. What are necessary and sufficient conditions on $Q$ to ensure ...
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Fast algorithm for computing certain signal transformations
Let $f,g,h:\mathbb Z\to\mathbb C$ supported on $[-n,n]$. For $\tau\in \mathbb Z$, let $\operatorname{sh}_\tau f$ be the shift of $f$ by $\tau$ (i.e. $(\operatorname{sh}_\tau f)(t) = f(t-\tau)$). ...
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Equivalent Littlewood-Paley-type decompositions
The theory of Besov and Triebel-Lizorkin spaces usually proceeds by taking a dyadic decomposition of unity, i.e. some non-negative functions $\psi_0,\psi \in C_c^\infty(\mathbb{R})$ such that
\begin{...
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Sobolev regularity via Laplace spectrum
Fix a positive integer $n$ and let $\mu$ be the uniform measure on the sphere $\mathbb{S}^n$, with respect to its usual Riemannian metric $g$. Let $\nabla$ be the Laplacian on $(\mathbb{S}^n,g)$ and ...
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microlocalisation of a Lagrangian state
Let
$f : \mathbb{R} \to \mathbb{R}$ a smooth function of exponential decay at infinity (I think of something like a Gaussian) ;
$g$ a polynomial of $\mathbb{R}$.
I would like to have a quantitative ...
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On the I-method's energy increment calculation in a paper of Dodson
I am currently reading Dodson's Global Well-posedness for the Defocusing, Quintic Nonlinear Schrödinger Equation in One Dimension for Low Regularity Data article and I am trying to understand Theorem ...
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Prove J.L. Lions’s Lemma without using Fourier transform
When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states
Let $\Omega \subset \mathbb R^n$ be a ...
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Why for $\psi$ square-integrable function the zero mean condition is equivalent to $\hat{\psi}(0) = 0$?
I am studying the classical book "Ten Lectures on Wavelets" written by Ingrid Daubechies and I do not understand a specific point. I would appreciate it if someone could help me with ...
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The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions
Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...
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Multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
How to rigorously demonstrate that multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
I am utilizing the exponential type definition established by Elias Stein in the book 'Fourier ...
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Monotone Characteristic Function
Let $X$ be a continuous, symmetric random variable such that its characteristic function $\phi_X$ is real, symmetric and with $\lim_{t\to\infty}\phi_X(t)=0$.
What other properties must $X$ have in ...
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$L^1$ norm for a product of cosines
Let $k$ be an integer and consider the function
$$
f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t).
$$
I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to ...
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Fourier coefficients of the logarithm of a given function
Let $f$ be a $1$-periodic real function that I know is bounded away from zero:
$$
f(x) = \sum_{n = -\infty}^\infty c_n e^{2\pi i n x}
$$
Let me also assume that $f$ is analytic with Fourier ...
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Is there a way to solve this integral on the sphere explicitly?
Let $k_{j}\in {\mathbb{Z}}^{+}$ and $\,a_{j}\in \,]0,1[$, be such that
$k_{j}\,a_{j}<1$, $j=1,\cdots,n$. Let $f:\mathbb{R}^{n}\rightarrow [0,\infty[$ be defined by the integral
$$f(y):=\int_{\...
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A function whose derivatives belong to $BMO(\mathbb{R}^n)$
I am reading the paper "Bounded Mean Oscillation and Sobolev Spaces" by Robert
s. Strichartz, Indiana University Mathematics Journal , 1980, Vol. 29, pp.
539-558. In this paper he defines ...
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$l^2(L^p)$ Decoupling constant of congruent tubes
Demeter's book Fourier Restriction, Decoupling, and Applications give a principle that one cannot decouple in a direction where the manifold is flat. Which is the below proposition:
Proposition 9.5 ...
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Truncating the high degree part of a positive boolean function doesn't change the distance to positive functions too much
Given $\displaystyle n\in\mathbb{Z}^{+}$, suppose $\displaystyle f:\{-1,1\}^n\to[0,1], $ then $f$ has a Fourier expansion:
$\displaystyle f(x)=\sum_{S\subseteq[n]} \tilde{f}(S)x^S,$ where $\...
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Question regarding proof of Littlewood-Paley
I posted this question on Math.SE where I unfortunately received no answers even after a bounty. As such, I am putting it here, in hopes to receive a response.
For the proof of Theorem 6.1.6 in ...
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Almost everywhere convergence of the Fourier series of a continuous function
I want to know if there is a simple proof of the almost everywhere convergence of the Fourier series of a function in $C(\mathbb{T})$. I am aware of Carleson-Hunt Theorem. I was wondering if the proof ...
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Any references for generalised square functions?
In harmonic analysis, there is a big chunk of literature studying the square function $Sf=\|\{P_jf\}_{j=1}^\infty\|_{l^2}$, where $P_jf=(\psi_j\hat f)\check{}$ and $\{\psi_j\}$ is a partition of unity,...
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Eric T. Sawyer's proof of Fourier restriction conjecture
Some days ago Eric T. Sawyer uploaded a paper to arxiv claiming a proof of the Fourier restriction conjecture https://arxiv.org/pdf/2311.03145.pdf. If complete and correct this work will be a landmark ...
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Why do we need the concept of Fourier measurability with growth function $\mathcal F$?
I'm studying the book Higher Order Fourier Analysis by Terence Tao (https://terrytao.files.wordpress.com/2011/03/higher-book.pdf). There, it defines that a function $f:[N]\to\mathbb{C}$ has Fourier ...
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What are the ‘refinements’ in Christ's method of refinements?
I have gathered that Christ's ‘method of refinements’ originated in his paper Convolution, curvature, and combinatorics: a case study to prove the sharp range of $L^p$-improving estimates for the ...
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Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
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A fractional weighted Poincaré inequality
Does there exists a constant $C>0$ such that
$$ \int_{-1}^1 \lvert x\rvert\lvert\partial_x u\rvert^2 \,dx \geq C\, \lVert u\rVert^2_{H^{1/2}((-1,1))},$$
for all $u\in C^{\infty}_0((-1,1))$?
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2
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Theoretical/Practical Implications of DFT Eigenvectors
Discrete Fourier transform (DFT) has only four distinct eigenvalues: $±1$ and $±i$. For large matrices , each eigenvalue $λ$ yields a multidimensional eigenspace, allowing linear combinations of ...
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Periodicity in one Fourier variable
Let $f:[0,1]\times [0,1] \to \mathbb C$ be a double periodic function (periodic in both variables) that depends real-analytically on its argument.
We can thus write $f$ as $$ f(x) = \sum_{n \in \...
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Pre-positive definite functions?
A function $f(x,y)$ is positive definite if matrices $( f(x_i, x_j) )_{i, j \in F}$ are positive definite for all finite index sets $F$. This is frequently hard or impossible to check given some ...
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Blow up for certain classes of distributions
Let $\mathbb D$ be the open unit disc centered at the origin and let $u \in H^{-N}(\mathbb D)$ be a distribution for some natural number $N>0$. Suppose that $$u|_{\mathbb D\setminus \{0\}} \in C^{\...