# Tagged Questions

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### A proof regarding Fourier-Polynoms

I want to prove the following: Let $f:\mathbb{R}\rightarrow \mathbb{C}$ so that $f \big |_{[0,2\pi]}$ is integrable. Let $V$ be the vectorspace of all $2\pi$-periodic functions and $U \subset V$ be ...
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### Prove that $||f||_2 \le \sqrt{2 \pi} ||f || _{\infty}$

Let $||f||_2=\sqrt{\int_{-\pi}^{\pi} f^2(x) dx}$ $||f||_{\infty}=\sup \{ |f(x)| \mid x \in [-\pi,\pi]\}$. Suppose $f: \mathbb{R} \to \mathbb{R}$ an in the space of piecewise continuous functions ...
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### Uniform convergence of the Fourier Series using Bessel's inequality

Consider the Fourier series of $f$, $$\frac{a_0}{2} + \sum_{n=1}^{\infty} a_n \cos(nx) + b_n \sin(nx)$$ Let $$f_n(x)= a_n \cos(nx) + b_n \sin(nx)$$ Then to show that $f_n(x)$ is uniformly ...
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### Decay of Fourier coefficients sequence

If $f:\Bbb R\to \Bbb R$ is a $2\pi-$ periodic, $C^1$ function, then $k^2a_{k}(f)\to 0$ where $$a_{k}(f)=\frac {1}{\pi}\int_{-\pi}^{\pi}f(x)\cos kx dx$$ are the Fourier coefficients. I ask if this is ...
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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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### Showing that $\mathrm{P}(t,x) = \sum_{n\in\mathbb{Z}} \mathrm{G}_t(x-2\pi n)\in\mathbb{C}^\infty((0,\infty)\times\mathbb{R})$

Welcome everybody :) I need your help in answering the following question: Let $t > 0$ and $\mathrm{G}_t(x) = (2\pi t)^{-1/2}e^{-x^2/2t}$ Show that \mathrm{P}(t,x) = \sum_{n\in\mathbb{Z}} ...
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### State-of-art of the Discrete Fourier Transform

I would like to know what is the state-of-art in the research of the discrete Fourier transform. I have listed some questions to help answering, please add your own to make the list more ...
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### Is a Fourier Series a continuous function?

My question relates to the properties of the Fourier series of a function, $f: \mathbb{R} \to \mathbb{R}$. I know from an elementary course in differential equations (for engineers) that, for all ...