# Tagged Questions

Harmonic analysis is the generalisation of Fourier analysis. Use this tag for analysis on locally compact groups (e.g. Pontryagin duality), eigenvalues of the Laplacian on compact manifolds or graphs, and the abstract study of Fourier transform on Euclidean spaces (singular integrals, Littlewood-...

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### Intuition behind the Riesz transform

Define the Riesz transform in singular integral form \begin{equation*} R_jf(x)=\lim_{\epsilon\to 0}\pi^{\frac{-(n+1)}{2}}\Gamma(\frac{n+1}{2})\int_{|y|>\epsilon}\frac{y_jf(x-y)}{|y|^{n+1}}dy. \...
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### When do closed subspaces of a Banach space fit together nicely?

Let $E$ be a Banach space, and let $F_1, F_2, F_3, ...$ be a sequence of closed subspaces of $E$ with $F_i\cap F_j=0$ whenever $i\neq j$. Denote by $\sum_n F_n$ the (not necessarily closed) subspace ...
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### Solve Intergrals Using Inverse Fourier Transform

a)Find f(x), the insverse fourier transform of F(ω) b) Does the fourier transform of f(x) equal to F(ω)? c)use your answers to calculate these Integrals: if I'm not mistaken the answer to a is: ...
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### Pontryagin dual of the unit circle

Define the unit circle as $\frac{\mathbb{R}}{2\pi\mathbb{Z}}.$ I know the Pontryagin dual (looking at properties of the Fourier transform on locally compact Abelian groups) is $\mathbb{Z}$ but why? ...
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### A proposition to prove the real interpolation of positive exponent

Let $p,A\in(0,\infty)$ $\|f\|_{L^{p,\infty}(X,\mu)} := \sup \{\lambda\mu(\{|f|\geq\lambda\})^{\frac{1}{p}}\}$ Show that the following are equivalent (1) $\|f\|_{L^{p,\infty}(X,\mu)}\leq C_pA$ for ...
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### Calculate $\mathscr{F}((1+t)^{-3})$

Let $$f(t)=\cases{\frac{1}{(1+t)^3}&t>0\\0&t<0}$$ Does: a.$\hat{f}$ is differentiable? b.$\hat{f}\in L^1(\mathbb{R})$? c.$\hat{f}\in L^2(\mathbb{R})$? Seems like we need to calculate ...
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### What are single layer and double layer potentials?

I have never encountered the terms "single layer potential" and "double layer potential" in my (under)graduate studies, although there was a firm development of partial differential equations and some ...