# 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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### Calculation of Fourier transform

How to calculate the Fourier transform of $f(x)=x$. I know using the formula $f(\varepsilon)=\int_xe^{-ix\varepsilon}x \, dx$. But I have problem calculating this complex integral.
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### Asymptotics of the Fourier transform of a non-analytic function

I have to find the asymptotic behavior of the Fourier transform of a function that is non analytical, but has a cusp. Given $$f(p)=\frac{1}{|p|^\sigma+1}$$ with $\sigma \in \mathbb{R}$, $\sigma >0$,...
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### FourerIntegrals

Trying to understand a proof about Fourer Intergrals in my book. I can't understand how they got formula (3). As you can see delta W goes to $0$ then $L$ goes to infinty. Doesn't that mean that we get ...
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### Different Alternate Representations of Functions

Could someone please point out a source with detailed steps / other pointers for different alternate representations of functions? For example, I know of two such ways 1) Taylor Series Expansion 2)...
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### Fourier Transform Properties Established

I have a function, f, in $L_1(R)$. I need to establish the following fourier transforms and I don't know how to do so. Can anyone guide me through one of these that would then help me get the other ...
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### Can the phase of a function be extracted from only its absolute value and its Fourier transform's absolute value?

If for a function $f(x)$ only its absolute value $|f(x)|$ and the absolute value $|\tilde f(k)|$ of its Fourier transform $\tilde f(k)=N\int f(x)e^{-ikx} dx$ is known, can $f(x) = |f(x)|e^{i\phi(x)}$ ...
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### On the frequency of primes.

Condsider the ("2D") sequence $(mn)_{m,n>1}$ (with $m,n \in \mathbb{N}$). It contains all natural numbers (in various multiplicities) except the primes and the number $1$. Now we construct a ...
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### Bounds for the Fourier transform of characteristic functions on $\mathbb{Z}/N\mathbb{Z}$ supported on large sets

Suppose $A \subseteq \mathbb{Z}_N := \mathbb{Z}/N\mathbb{Z}$ with $|A| \geq N/2$. Let $$\hat{A}(h) := \sum_{a \in A} e_N(ha),$$ where $e_N(x) := e^{2\pi i x/N}$. Clearly $|\hat{A}(h)| \leq |A|$ for ...
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### Spectrum of a convolution operator

Let $T$ be the operator from $L^2(\mathbb R^n)$ to $L^2(\mathbb R^n)$ that is given by $Tf := f * g$ where $g$ is in $L^2$. How do I now find that the spectrum of $T$ is equal to the essential range ...
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### Fourier series of two functions composition with small parameter

A function $f(x)$ has fourier series: $$f(x) = \sum_{n=-\infty}^{\infty}f_n e^{in\omega_x x}$$ where $x \in [x_1, x_2]$, $\omega_x = 2\pi/(x_2 - x_1)$. Now let's say $x = t + \varepsilon g(t)$ ...
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### Where were my mistakes when I've combined $\eta(s)=\left(1-2^{1-s}\right)\zeta(s)$ and the Fourier series for the fractional part?

Let $s=x+it$ the complex variable, thus we are denoting $\Re s=x$. Combining the identity $$\eta(s)=\left(1-2^{1-s}\right)\zeta(s),$$ that holds for $0<x<1$, where $\zeta(s)$ is the Riemann Zeta ...
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### Getting the DFT of irregularly spaced points

I am trying to estimate the discrete Fourier transform of a discrete surface, $x:\{1,\dots,N\}\times \{1,\dots,N\} \to\mathbf{R}$, given a sparse set of samples on the grid. If we had all the ...
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