# Tagged Questions

Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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### Show Fejer kernel on the real line is good, without using trignometric integrals.

This is from page 163 of Stein's Fourier Analysis. Fejer kernel on the real line is defined by $$\mathcal{F}_R(t) = R\left(\frac{\sin(\pi t R)}{\pi t R}\right)^2$$ When $t=0$, $\mathcal{F}_R(t)=R$...
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### One to one map $f$ equal to its power series

Across a difficult exercise sheet I encountered this exercise : Let $f$ be a continuous map from $\bar D$ the closed unit disk (in $\mathbb{C}$) to $\mathbb{C}$. We suppose that $f$ is one to one ...
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### Proof of Dirichlet L-function Euler Product formula (from Fourier Analysis by Stein)

On page 260 of Stein and Shakarchi's "Fourier Analysis," there's a proof of the Dirichlet product formula: $\sum_{n}\frac{\chi(n)}{n^s}=\Pi_{p}\frac{1}{1-\chi(p)p^{-s}}$ where $s>1$, $\chi$ is a ...
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### Help with an Inverse Fourier transform

Can anybody please guide me how to compute the inverse Fourier Transform of: $$f(k) = \frac{1}{1+k^2} \frac{\pi}{4}(\rm{sgn}(1-k) + \rm{sgn}(1+k))$$
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### $p$-adic Fourier transforms and orthogonality relations

In $\mathbb{C}$, we have the following orthogonality relation $$\int_{0}^{1} e^{2\pi i (m-n)x} dx = \begin{cases} 1 & \mbox{ if } m = n;\\ 0 & \mbox{ otherwise.} \end{cases}$$ Do we have ...
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### Fourier Tranform of $\frac {1}{(1+k \sin(t))^{3}}$

I'm stuck with some Fourier transforms that I'm not being able to solve and Mathematica is not helping: $\frac{1}{(1+k \sin(t))^{3}}$ (and the same one for sinh), where $k$ is a constant. Any ...
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### Windowing effect and Fourier Transform

I understand how windowing effect helps to improve side lobes, of transformed signal in fourier spectrum. One way to explain this, is by pointing out that the sampled signal is considered periodic, ...
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### Characters with values on the $p$-adic complex field $\mathbb{C}_p$?

Characters $\psi : G \to \mathbb{C}$ from abelian groups $G$ to the complex field $\mathbb{C}$ are well-known and appear all over. Is there an analogue for the $p$-adic complex numbers $\mathbb{C}_p$, ...
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### Fourier transform of $f(x+h)$?

Show that $f(x+h)\to \hat{f}(w)e^{2\pi i h w}$ Let $g(y) = f(x+h)$, then $\hat{g}(w) = \int_{-\infty}^\infty g(y) e^{-2\pi i y w} dy = \int_{-\infty}^\infty f(x+h) e^{-2\pi i (x+h) w} dx$, then I ...
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### Fourier Transform pdes

I have an exam next week and I was hoping someone might be able to help me out with this question. Show that the Fourier transform of the function $f(t+a)$ is $e^{iwa}\hat{f}(w)$ . There is a list of ...
Consider $f(t) = \frac{\pi - t}{2}$, $t \in [0, 2\pi]$ The complex fourier coefficients are $c_n = \frac{1}{2\pi}\int_0^{2\pi}\frac{\pi - t}{2}e^{-int}dt$ Which turns out to be $-\frac{i}{2n}$ if im ...