# Tagged Questions

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### Estimate of Projection Operator on two-torus

Let $\Lambda$ be a lattice, $\mathbb{T}=\mathbb{R}^2/\Lambda$ be a flat torus and $\Delta$ be the Laplace-Beltrami operator. There is any reference where the norm of the projection operator ...
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### Construct a Fourier series that diverges almost everywhere.

Andrey N. Kolmogorov was one of the greatest mathematicians and polymaths of the 20th century. One of his first achievements was to construct a Fourier series that diverges almost everywhere. How ...
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### an “alternate derivation” of Poisson summation formula and discrete Fourier transformation

Inspired by this post, I am trying to do a derivation of a Poisson summation formula. My starting point is this: $$\frac{1}{2\pi} \int^{\infty}_{-\infty} e^{i k x} dx=\delta(k)$$ I simply wish ...
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### Find Fourier series coefficients of $f(x)$.

$T=2$ $$f(x) = \begin{cases} 1, & \text{-\frac12\le x \le\frac12} \\[2ex] |2x|, & \text{\frac12 < x \le1\frac12} \\ \end{cases}$$ The image: I found that $a_0=\frac12$. Since ...
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### Real part of a holomorphic function is bounded by polynomial then the holomorphic function is a polynomial [duplicate]

Let $u$ be a harmonic function on $\mathbb{R}^2\cong \mathbb{C}$ such that $Ref= u$ where $f$ is an entire function. If $|u(z)|\leq |z|^n$ for any $z\in\mathbb{C}$, then $f$ is a polynomial of degree ...
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### Contradiction between $a_0$ and $a_k$ for Fourier Series

I need to calculate the Fourier Series for the function $f(x) = |x| \; f:[-\pi,\pi] \to \mathbb{R}$ When calculating $a_k = {1 \over \pi} \int_{-\pi}^{\pi} f(x) \cos{(kx)} dx \; (k \in \mathbb{N_0})$ ...
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### Uniqueness of Fourier Coefficients

I'm reading through Stein & Shakarchi's book on Fourier Analysis on my own, and have a question about the proof of the following theorem: Suppose that $f$ is an integrable function on the circle ...
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### Infinite trigonometric series, find the constant C_n

Hi this is my first post :) I am not sure how to do part b. You get the infinite series of $\displaystyle c_n\cdot \sin(\frac{n\cdot \pi\cdot x}{L})$ from $n=1$ to infinity And this is equal to ...
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### The property of positive fourier series. [duplicate]

This is the problem in the book 'Classical and multilinear harmonic analysis, volume 1' Let $f(x)=\sum_{n=0}^{N}[a_{n}\cos{2\pi nx}+b_{n}\sin{2\pi nx}]$ be a nonnegative function defiend on $[0,1]$. ...
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### How to use Parseval' s( Plancherel' s) identity?

Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put, $F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt, \ (n=1,2,...).$ Fix $\alpha \in (0, \infty)$ and we define $H_{n}(x)$ ...
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### Why should we use the Fourier Transform?

I'm a CS/Math double major, and during my study (and reading sources out of my own interest) I've had some encounters with the Fourier Transform. I understand the theory behind Fourier series, and ...
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### spectral structure of sinusoidal model

let us consider following code ...
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### Example of a false proof when a Fourier series is not unique?

I am attempting to come up with an example to illustrate why one should care that a function has a unique Fourier series expansion. Inspired by the fact that one can rearrange terms in a ...
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### Diagonalization of circulant matrices

Why does the following hold?: $A$ circulant matrix iff it has a representation of the form $F^{-1}DF$ where $D$ is a diagonal matrix and $F$ is a discrete Fourier transformation. I get that $F^{-1}DF$ ...
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### Show that for $0<t<1$, $\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$

Show that for $0<t<1$, $$\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$$ So I derived the following Fourier series: ...
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### Do the Fourier series of a function-valued Hermitian matrix converge?

Let $\mathbf{A}(t):\mathbb{R}\rightarrow\mathbb{C}^{n\times n}$ be a rank deficient periodic function-valued positive semi-definite Hermitian matrix. The entries $a_{ij}(t)$ of $\mathbf{A}(t)$ are ...
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### Convergence of the series $\sum_{\xi\in\mathbb Z^n} e^{2\pi ix\cdot \xi} a(x, \xi)\hat{f}(\xi)$?

I need some help with the following problem: let $a:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb C$ be a smooth function and suppose there are constantes $C_{\alpha, \beta}$ and $M(\alpha, \beta)$ ...
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### Finding real part of fourier series

I have encountered the following problem in one of my textbooks but I'm not really getting anywhere: Let $f$ be complex-valued and piecewise continuous on the interval $[-\pi,\pi]$. Find the complex ...
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### Does $\int_{0}^{\pi/2n}\left|\frac{\sin(2n+1)t}{\sin t}\right|\text{d}t\leq \pi$ hold for $n \geq 2$?

Today I am trying to prove an integral inequality: $$\frac{1}{\pi}\int_{0}^{\pi/2}\left|\frac{\sin(2n+1)t}{\sin t}\right|\text{d}t<\frac{2+\ln n }{2}$$ where $n\geq 2$ and $n \in \Bbb{N}$. First, ...
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### Show a Fourier series converges uniformly

I need to show that the Fourier Series of |x| in the interval $(-\pi, \pi)$ converges uniformly to |x| in $[-\pi, \pi]$. I know that |x| = $\frac{\pi}{2}$ + ...
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### Fourier-Series of a part-wise defined function?

I have a function f given as $$f(x) = \begin{cases} ax&\text{ if }\quad-\pi \leq x \leq 0\\ bx&\text{ if }\quad 0<x\leq\pi \end{cases}$$ I am supposed to develop the fourier series of ...
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### Poisson summation formula and Schwartz functions

I am reading a proof of the Poisson summation formula which states that (with my version of the Fourier transform - I think they sometimes vary by a constant factor) for $f$ a Schwartz function on ...
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### A function and its Fourier transform cannot both be compactly supported

I am stuck on the following problem from Stein and Shakarchi's third book. I can't figure out how to use the hint productively. Once I know $f$ is a trigonometric polynomial, I see how to finish the ...
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### Fourier Coefficients of periodic function

Consider a Function $f\in L^2(\mathbb{T})$. Is there any lower bound for the decay of the Fourier coefficients $$\hat{f}(n)=\frac{1}{2\pi}\int_{-\pi}^{\pi} f(t) e^{-int} dt$$ known? There are a lot ...
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### what's the difference between “convergent” and “reconstruct-able”?

I am reading this book: http://www.abdn.ac.uk/~mth192/html/maths-music.html There is a sentence on page 54: "However, the question of convergence of the Fourier series is not the same as the question ...
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### Interval type for Fourier Analysis on $L^2( [-\pi,\pi) )$

Why is it that in many texts the natural domain of choice is $L^2 ( [- \pi, \pi) )$ as opposed to $L^2 ( [- \pi, \pi] )$? I would to think of the space $L^2 ( [- \pi, \pi] )$ as the completion of ...