Questions tagged [fourier-analysis]

Question 1

For a single-variable function $f(x)$, the Fourier sine series is: $$f(x)=\sum_{n=1}^\infty b_n\sin(\frac{n\pi}lx)\quad\text{where}\quad b_n=\frac2l\int_0^lf(x)\sin(\frac{n\pi}lx)dx$$ However, I ...

Question 2

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 $C[A]$. ...

Question 3

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.

Question 4

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 ...

Question 5

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\...

Question 6

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 ...

Question 7

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 ...

Question 8

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{...

Question 9

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{...

Question 10

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 ...

Question 11

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 $\...

Question 12

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 ...

Question 13

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 ...

Question 14

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 ...

Question 15

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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Questions tagged [fourier-analysis]

Why Fourier sine series of $F(x,t)$ is $\sum_{n=1}^{\infty}b_n(t)\sin(\frac{n\pi}l x)$ with $b_n(t)=\frac2l \int_{s=0}^l F(s,t)\sin(\frac{n\pi}ls)ds$?

Fourier transform and the acoustic tensor

Moderate Decrease [closed]

Fourier inversion theorem for compact groups

Decomposition of a Schwartz function by Lacey and Thiele

Finite Fourier series for square-like wave

Quadrature rule for tensor product of fourier basis

If all convolutions of a tempered distribution T with any compactly supported function are compactly supported, is T compactly supported?

The Fourier coefficients are zero for all but finitely many integers $k$

Can Young's inequality give a pointwise bound for a convolution?

Why does this surplus-weighted Dirichlet sum reproduce the Riemann zeta function so closely?

Question on Theorem on necessary and sufficient conditions for multiplier operator

Can regularization (Abel/Borel) of a divergent derivative series tell us about $f'(x)$?

Is there a name for the Gibbs' Phenomenon-like oscillations when plotting the Fourier Series of a Dirac Train?

Plancherel's theorem and boundary terms

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