Questions tagged [fourier-analysis]
Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).
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Fourier transform and the acoustic tensor
Let $C$ the fourth-order tensor of elastic constants which can be seen as as a linear transformation from Sym into Sym (matrices). We denote the action of $C$ on a symmetric matrix $A$ as $C[A]$. ...
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Moderate Decrease [closed]
The exercise in question.
A function is of moderate decrease if $|f(x)|\leq \frac{C}{1+x^2}$ for some constant $C$. I can’t seem to find a way to show that $\hat{f}(ξ)$ is of moderate decrease.
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Fourier inversion theorem for compact groups
I am searching for a good reference for finding the fourier inversion theorem for compact (abelian, or non abelian) groups (we shall denote such a group by $G$). In particular, I would like to see a ...
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Decomposition of a Schwartz function by Lacey and Thiele
Suppose we have a Schwartz function $\varphi\colon \mathbb{R}\to \mathbb{R}$ supported in (0,1) such that $||\varphi||_{L^2}\leq 1$ satisfying that for all $\xi\in \mathbb{R}$,
\begin{align}
\sum_{l\...
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Finite Fourier series for square-like wave
I'm looking to find a finite Fourier series that satisfies these conditions:
It equals a square wave with wavelength $\lambda$ at the integers
There is a local maximum at every integer x value
Has at ...
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Quadrature rule for tensor product of fourier basis
For reasons not directly relevant to the question, I am constructing a function basis in $\mathbb{R}^3$ by taking the tensor product of the fourier basis (up to discretization).
I must now come up ...
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If all convolutions of a tempered distribution T with any compactly supported function are compactly supported, is T compactly supported?
What I mean is that we have $T \in \mathcal{S}'(\mathbb{R}^n)$ such that for all $f \in C^\infty_c (\mathbb{R}^n)$, we have that $\mathrm{supp}(T \ast f)$ is compact, and we're asking whether $\mathrm{...
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The Fourier coefficients are zero for all but finitely many integers $k$
For an appropriate function $g$ and for an integer $k$, the $k^{\text{th}}$ Fourier coefficient $\widehat{g}(k)$ of $g$ is defined as
$$ \widehat{g}(k):=\int_0^1g(x)e^{-2\pi ikx}dx. $$
$\textbf{...
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Can Young's inequality give a pointwise bound for a convolution?
I’m trying to understand a step in Appendix A.1 of
Bejenaru and Herr, The cubic Dirac equation: small initial data in $H^1(\mathbb{R}^3)$.
The paper proves global well-posedness for the cubic Dirac ...
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Why does this surplus-weighted Dirichlet sum reproduce the Riemann zeta function so closely?
I’ve been experimenting with a surplus-weighted Dirichlet series that seems to replicate the local behavior of the Riemann zeta function.
Below is the definition and a set of plots comparing it to $\...
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Question on Theorem on necessary and sufficient conditions for multiplier operator
In Grafakos' Fundamentals of Fourier Analysis, Theorem 2.8.5 states the following:
Let $u \in \mathcal{S}^\prime$ (tempered distributions) and $T(\phi) = \phi \ast u \in \mathcal{S}$. Then $T$ admits ...
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Can regularization (Abel/Borel) of a divergent derivative series tell us about $f'(x)$?
Question.
If $f(x) = \sum f_n(x)$ is a continuous function, and its formal derivative series $g(x) = \sum f_n'(x)$ diverges at a point $x_0$, but is summable (via Abel, Borel, etc.) by some ...
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Is there a name for the Gibbs' Phenomenon-like oscillations when plotting the Fourier Series of a Dirac Train?
Below is a plot of $\sum_{k=-20}^{k=20}e^{2\pi ix}$ plotted using desmos. I know when you have a differentiable function with jump discontinuities in it, you get oscillations near the discontinuity ...
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Plancherel's theorem and boundary terms
Consider the following functional of a real-valued vector field $\mathbf{u}(\mathbf{x}): \mathbb{R}^3 \to \mathbb{R}^3$:
$$ F[\mathbf{u}] = \int d^3x\, \frac{\partial u_i}{\partial x_j}\frac{\partial ...
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Computing the action of GL(2) on the Fourier-transformed principal series representation
Below is the setting,
Let $F$ be a non-Archimedean local field, $G=\text{GL}_2(F)$, and let $\chi_1, \chi_2:F^\times \to \mathbb{C}$ be characters. Define a character on the Borel subgroup $B$ of $G$ ...
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