# Tagged Questions

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### Haar measure on $\mathbb{R} × \mathbb{T}$ and on dual $\mathbb{R} × \mathbb{T}$

I've solved this exercise somewhat.To complete it please help me Haar measure on G $=$ translation invariant on G $$μ(A)=μ(A+t)$$ if $G=\mathbb{R}$ then Haar measure on G is lebesgue measure. and ...
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### Using Harmonics to find a solution to a boundary value problem

Consider a boundary value problem with two given level sets of phi. One set is in the imaginary plane with center (1,i) and radius 1. This set has level set phi = 0. Another set is in the imaginary ...
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### Addition of two cosine waves with different periods

I was just wondering if anyone knows how to add two different cosine equations together with different periods to form one equation. Is there a way to do this and get a real answer or is it just all ...
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### Solving the equation $\int G(t) dt =\frac{\sin x}{x}$

I have to solve the equation $$\int_{\mathbb R} \frac{f(t)}{1+(x-t)^2} dt =\frac{\sin x}{x}.$$ I tried change of variables to make the $\frac{1}{1+(x-t)^2}$ part resemble $e^{h(x)}$ so I can use the ...
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### Show that two series are equal

In my harmonic analysis class I have to prove that for all $a>0$ the following equality holds: $$\sum_{n\in \mathbb{Z}}e^{-n^2\pi a}=\frac{1}{\sqrt{a}}\sum_{k\in \mathbb{Z}}e^{-k^2 \pi / a}.$$ I'd ...
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### Left regular representation of $L^1(G)$ for a locally compact group $G$

Let $G$ be a locally compact group (not discrete) and let $L$ be the left regular representation of $A = L^1(G)$ on itself i.e. $L: A \to \mathcal{B}(A)$ where $L(f): A \to A$, $L(f)(g) = f*g$. I want ...
Assuming the principle is stated as such: Let $U\subset\mathbb{R}^n$ be a bounded domain and $u$ harmonic in $U$ such that $\sup_{x\in U}u(x)\leq A$ for some $A\in\mathbb{R}$. Then either $\forall ... 3answers 239 views ### A Van der Corput style inequality for highly oscillatory integrals Suppose$f$has at least two continuous derivatives,$f'$is monotonically increasing, and$f' \geq \lambda$for some$\lambda > 0$. How might one find the upper bound$|\int_a^b ...
could anybody will help me to do this problems: Let $\mathcal D$ be the unit disk a Set $E\subseteq\partial\mathcal D$ has harmonic measure identically $0$ with respect to $\mathcal D$. What can you ...