1
vote
0answers
26 views

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 ...
0
votes
0answers
29 views

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 ...
0
votes
1answer
52 views

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 ...
0
votes
1answer
58 views

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 ...
2
votes
1answer
57 views

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 ...
0
votes
1answer
51 views

Confustion with Fourier Transform of harmonic functions

Note: this is homework. Say we assume a time-varying momentum is of the form $p(t) = \Re \{p(w) e^{-iwt} \}$ Now we would like to know the solution of the following equation, in frequency space: $ ...
0
votes
1answer
123 views

Harmonic mean of absolute value squared discrete Fourier transform

Let $X[k] \in \mathbb{C}$ be the discrete Fourier transform of $x[n] \in \mathbb{R}$, where $k,n = 0,1,2,...,N-1$. Parseval's theorem relates the arithmetic mean (AM) of absolute value squared ...
1
vote
1answer
155 views

Another aspect of Heisenberg uncertainty principle

In fourier transformation theory, we have the Heisenberg uncertainty principle, i.e. Suppose $\phi$ is a function in [Schwarz space][1] which satisfies the normalizing condition ...
1
vote
1answer
51 views

$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq \int_{-\infty}^{\infty}|f|^2 dx$?

Suppose complex function $f$ in the Schwartz Space, its definition see http://en.wikipedia.org/wiki/Schwartz_space how can we argue that $$\int_{-\infty}^{\infty}f '\bar{f}'+x^2 f\bar{f}dx\geq ...
1
vote
1answer
26 views

$(y^2+2iy)^{m-\frac{1}{2}}=y^{m-\frac{1}{2}}(2i)^{m-\frac{1}{2}}+O(y^{m+\frac{1}{2}})(0\leq y \leq 1)?$

This question comes from stein's book Introduction to fourier analysis on euclidean space,page 159. when $m > 1/2$, ...
1
vote
1answer
220 views

Fourier transform of a function is square integrable

Is there a result stating that if a function $f:\mathbb{R} \rightarrow \mathbb{R}$ is square integrable and decays at infinity, then its Fourier transform is also square integrable?
1
vote
0answers
25 views

$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$

Here $i$ is complex number, $n$ is positive integer. Show that $$i^{-k}\pi^{n/2}\Gamma(\frac{k}{2})/\Gamma(\frac{n+k}{2})\approx k^{-n/2}~~(k\rightarrow \infty)$$ This question appears from Stein's ...
8
votes
1answer
483 views

How to show convolution of an $L^p$ function and a Schwartz function is a Schwartz function

We have the Schwartz space $\mathcal{S}$ of $C^\infty(\mathbb{R^n})$ functions $h$ such that $(1+|x|^m)|\partial^\alpha h(x)|$ is bounded for all $m \in \mathbb{N_0}$ and all multi-indices $\alpha$. ...
2
votes
1answer
92 views

Proof for Fourier transform in $L^2$

This question makes me really confused: Let $f$ and $g$ two functions in $L^2$. Show that: $$\int \widehat f\cdot gdx= \int f\cdot\widehat gdx,$$ where $\widehat f$ is the Fourier transform ...
7
votes
3answers
335 views

Nonexistence of a cyclic vector for a representation on $\ell^2(\mathbb{Z})$

Let $S$ be the bilateral shift on $\ell^2(\mathbb{Z})$ and let $T = S + S^*$. I want to show that there is no cyclic vector for the representation of $T$ on $\ell^2(\mathbb{Z})$ i.e. $\forall x\in ...
3
votes
1answer
261 views

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 ...
1
vote
0answers
284 views

Proving the maximum principle for harmonic real valued functions in $\mathbb{R}^n$

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 ...
8
votes
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 ...
1
vote
0answers
179 views

Harmonic measure

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 ...