# 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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### 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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### 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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### 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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### Find the spectrum of an operator related to Fourier series

As an exercise, I was told to find the spectrum of the bounded operator $K\in B(L^2[-\pi,\pi])$ defined by $$K\varphi (t)=t\int_{-\pi}^\pi\varphi (x)\cos (x)dx+\cos t\int_{-\pi}^\pi x\varphi(x)dx.$$ ...
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### Counting solutions by estimating Fourier coefficients

In W. T. Gower's essay The Two Cultures of Mathematics, he mentions the following as an example of a 'general principle' in combinatorics: "If one is counting solutions, inside a given set, to a ...