# 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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### $p$-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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### A funtion and its fourier transformation cannot both be compactly supported unless f=0

Problem : Suppose that $f$ is continuous on $\mathbb{R}$. Show that $f$ and $\hat f$ cannot both be compactly supported unless $f=0$. Hint : Assume $f$ is supported in [0,1/2]. Expand $f$ in a ...
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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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### Characterizing functions with controlled Fourier coefficiens

It's a well known fact that an infinite dimensional Banach space $E$ is not locally compact. One may consider, at which point, is this property lost, i.e. what kind of compact sets $K \subset E$ exist....
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### Bit operations to count longest string of 1s in a binary number - connections to FFT?

I found this rather applied question on another forum. How to calculate size of largest string of consecutive 1s in a binary number. However the other forum had neither much of a focus on applied ...
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### trigonometric series

It is known that the eigenvalues of Sturm liouville problem: $$u''(x)+\lambda u(x)=0 \\ u(0)=u'(\pi)=0$$ are $\sin\left(\left(\frac{1}{2}+n\right)x\right)$ for $n=0,1...$ If for example we expand ...
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### Why is Fourier Analysis effective for studying uniform distributions

On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim: When it comes to the primes, we find that we do not have a good ...
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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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