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Compute Fourier transforms of homogeneous functions of the form, $$ \frac{1}{|x|^{n+d}}P_d(x) $$ where $P_d$ is a homogenous harmonic polynomial of degree $d$ in $n+1$ variables.

$L^2(\mathbb{S}^{d-1})$ is embedded in $H^{-s}(\mathbb{R}^d)$ with $s>\frac{1}{2}$, which means for $f\in L^2(\mathbb{S}^{d-1})$, the following holds: $$\DeclareMathOperator{\Dm}{\operatorname{d}\!}...

Given a rational point $p\in S^1$ and a continuous function $f:S^1\rightarrow \mathbb C$, we say that $f$ is an $\epsilon$-bump around $p$ (for some $\epsilon>0$) if $f(p)=1,|f|_{\infty}\leq 1+\...

I am trying to solve a nonlinear 3spring-2mass system under harmonic loading by using Fourier series expansion of states of the differential equation. The system is just basically two masses, two ...

Assume that we are given a sequence $\mathbf x := X_1,\dots,X_n \in \mathbb N^n$ for some $n \in \mathbb N$. I am interested in calculating the frequency of occurrence of some fixed sequence $\mathbf ...