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 series approximation error decay under different norms
Given a BV function $f:(0,1)^m\to\mathbb{R}$, how does the error term $f-S_N(f)$ in $L^1$ norm, $L^2$ norm and total variation $TV(f-S_N(f))$ decay/grow with $N$?
$L^1$ norm = $O(N^{-1})$
$L^2$ norm = ...
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What are the current best known results for linear dispersive Strichartz estimates on the circle $\mathbb{T}$?
I am currently trying to understand Strichartz estimates for linear disperive equations on the circle:
$$\begin{cases}
i \frac{\partial u}{\partial t}= \Phi(\sqrt{-\partial^2_x})u\:,\: &\text{ in ...
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Fourier transform and acoustic tensor
This is a repost from math stack since I have not received any answer
Let $C$ the fourth-order tensor of elastic constants which can be seen as as a linear transformation from Sym into Sym (matrices). ...
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Lower bounds for $\|f*g\|_1$ with mean-zero Lipschitz functions on $[0,1]$
Let $f,g \in L^{1}([0,1])$ satisfy
$$
\|f\|_{1}=\|g\|_{1}=1, \qquad \int_{0}^{1} f(x)\,dx=\int_{0}^{1} g(x)\,dx=0,
$$
and assume
$$
f \in \mathrm{Lip}_{L_f}, \qquad g \in \mathrm{Lip}_{L_g}.
$$
...
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Why does the Fourier transform of $μ(n)/n$ look like this?
Out of curiosity (and in relation to this MSE question), I computed numerically (an approximation to) the Fourier transform of $\mu(n)/n$ where $\mu$ is the Möbius function, viꝫ. $f\colon t \mapsto \...
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Rank of tensor product of irreducible representations over finite symmetric group
Let $G$ be a finite symmetric group, and let $\widehat G$ denote the set of equivalence classes of irreducible unitary representations of $G$.
For any non-trivial $\rho\in \widehat G$, we know that $\...
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Ergodicity in the Wiener-Wintner Ergodic Theorem [cross-post from MSE]
I'm studying the Wiener-Wintner (and related) ergodic theorems, and I've been running into a bit of confussion when passing the result from ergodic systems to non-ergodic ones. In most of the ...
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Sum in tensor product of irreducible representations on $S_n$
Let $G$ be a finite symmetric group, and let $\widehat G$ denote the set of (equivalence classes of) irreducible unitary representations of $G$.
For any non-trivial $\rho\in \widehat G$, we know that $...
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$L^2$-functions orthogonal to their own Fourier transform
It is well-known that, besides the standard Gaussian $e^{-|x|^2/2}$, there are many interesting functions which are eigenfunctions of the Fourier transform, for example the Hermite functions.
Mainly ...
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Traces mixing tensor products of Fourier coefficients on finite symmetric groups
Let $G$ be a finite (symmetric) group, and let $\widehat G$ denote the set of (equivalence classes of) irreducible unitary representations of $G$. Let $f:G\to \{0,1\}$ be a Boolean function on $G$ ...
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Pointwise estimate for a multilinear operator
Let $0<\alpha<1$, and let $$f=\frac{\chi_{[0,1]}}{x^{a}},\quad 0<a<1,$$
$$g=\chi_{[0,1]},$$
$$h(x)=\frac{\chi_{[0,1]}}{|x-1|^{b}},\qquad 0<b<1.$$
I have a multlinear operator on $L^{...
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Weak L2 norm in proof of Carleson's theorem
I am reading the paper of Michael Lacey called "Carleson's theorem: proof, complements, variations" 1, on Carleson's theorem in Fourier analysis. Under Lemma 2.18 on page 10 it says: "...
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Bounding the largest Fourier coefficient of $f$ minus a class function on symmetric group $S_n$
Let $G=S_n$ be a symmetric group. A class function on $G$ is any $g:G\to\mathbb C$ that is constant on conjugacy classes, i.e. $g(xy)=g(yx)$ for any $x,y\in G$. Let $\mathcal C$ denotes the set of ...
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Space of interpolating functions with constraints on interpolation
Disclaimer:
I am a first year mathematics student who is trying to write an applied math paper, so my question might seem trivial.
Definitions:
Let $N \in 2 \mathbb{N}$ and $u \in \mathbb{R}^N $ be a ...
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English translation of van der Corput's 1939 proof for three-term progressions in primes
I've seen van der Corput's paper "Über Summen von Primzahlen und Primzahlquadraten" [Mathematische Annalen 116 (1939), 1–50] referenced here and there. It proves that there are infinitely ...
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