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.
1,612 questions
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Fourier multiplier bound for disjoint union of intervals
Let $\mathcal I$ be finitely many disjoint closed intervals contained in $[0,1]$. For instance, they could come from a tiling of $[0,1]$ by intervals of the same length. Define a Fourier multiplier $...
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Poincaré–Wirtinger inequality with geometric median center?
Original post
Let $L>0$ and let $u \in H^1_{\mathrm{per}}(0,L;\mathbb{R}^n)$, where
$$ H^1_{\mathrm{per}}(0,L;\mathbb{R}^n)
:= \Bigl\{\,u\in H^1(0,L;\mathbb{R}^n)\;:\;\text{the traces satisfy }u(0)=...
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Writing the limit of integrals as an integral over a subspace of $\mathbb{R}^n$
Let us denote by $1_B(x)$ the characteristic function of a box $B$ inside $\mathbb{R}^n$ and define
$$
I_P = \int_{\mathbb{R}^n }1_B(x) \Phi_P( c_1 x_1 + \dotsb + c_n x_n ) dx
$$
where $\Phi_P (t) = P(...
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Sharp $L^{\infty}$ Bernstein inequality for bandlimited functions with 2D square symmetry
This is a follow-up to this question. Based on that question, the following sharp $L^{\infty}$ Bernstein inequality for bandlimited functions is known:
Suppose $f(x)$ is a smooth function $\mathbb{R}^...
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Properties of Sturm-Liouville eigenfunction expansion
There are plenty of classical results about Fourier series. For instance, if a periodic function $f$ is continuously differentiable, then its Fourier series converges absolutely and uniformly to $f$.
...
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Reference for $L^\infty$ decay of spherical harmonic coefficients for $C^{1,\alpha}$ functions
Let $f \in C^{1,\alpha}(\mathbb{S}^{d-1})$, $0<\alpha<1$, and consider its spherical harmonic expansion $f = \sum_{\ell=0}^{\infty} \sum_{k} a_{\ell k} Y_{\ell k}$.
A standard result states that ...
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Non-asymptotic Szego theorem for Toeplitz matrices
Assume $w > 0$ on $[-\pi,\pi]$ and $\log w\in L^1([-\pi,\pi])$.
Define the Fourier coefficients of $w$ by
$$
w_k=\frac{1}{2\pi}\int_{-\pi}^{\pi} w(\theta)\,e^{-ik\theta}\,d\theta,
\qquad k\in\...
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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 $...
- 747