# Tagged Questions

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### Inequality on $L_1$ norms of tirgonometric polynomials generated with a smooth function

Let $\varphi\in C_0^\infty(\mathbb R)$ and for $n\ge1$ $$f_n(x)=\sum_{k=-\infty}^\infty \varphi(k/n)e^{i k x}.$$ I seem to remember that there is an inequality $\|f_n\|_{L_1(\mathbb T)}\le C$, where ...
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### Further studies on Fourier Series and Integrals.

If you had to choose two books from the following list, which pair would you chose, and why? If you haven't read any, would you pick any pair among the list based on the author of the book? I am ...
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### Complex form of Fourier Series

So, the last part of the university syllabus in the chapter of Fourier Series is: ...
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### approximate $[0, 1]$ continuous function with 2d basis.

everyone. I've been thinking of this problem when reading papers about Fourier series. I think I can state my question as follows: in the interval $[0, 1]$, I want to approximate an unknown ...
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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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### Inverting a discrete linear transformation

Consider the transformation from the set $\{a_n\}_{n=0}^N$ to the set $\{p_j\}_{j=0}^N$: $$p_j = \sum_{n = 0}^Na_n\phi_n(x_j)$$ where $\{\phi_n(x)\}_{n=0}^N$ is a set of basis functions (linearly ...
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### Proof of $\frac{1}{e^{\pi}+1}+\frac{3}{e^{3\pi}+1}+\frac{5}{e^{5\pi}+1}+\ldots=\frac{1}{24}$

I would like to prove that $\displaystyle\sum_{\substack{n=1\\n\text{ odd}}}^{\infty}\frac{n}{e^{n\pi}+1}=\frac1{24}$. I found a solution by myself 10 hours after I posted it, here it is: ...
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### Fourier Analysis

I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function? Can someone provide me simple information ...
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### Known facts about a function

In my work I have met the function on the unit circle whose Fourier coefficients are $$c_n=\frac{1}{|n|}\prod (d_k+1)$$ if $n=\pm\prod p_k^{d_k}$ is the decomposition of the integer $n$ into the ...
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### An elegant non-technical account on the work of Joseph Fourier.

It would seem difficult for a naive person to understand the beauty of work done by Fourier. So as far as I know, one can use the Fourier transforms, analysis and series to apply them for heat ...
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### Does number theory have any role in the proof of convergence of Fourier series for certain functions?

Does number theory have any role in the proof of convergence of Fourier series for certain functions? I vaguely remember reading in a book on signal processing, way back, that the proof (original ...
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### reference for Fourier series for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$

I am in need of a good reference which has a complete treatment (with all the convergence proofs) for Fourier series representation for periodic functions of the form $f : \mathbb{R} \to \mathbb{C}$. ...
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### reference request for proof of Gibbs phenomenon at jump discontinuities

Please suggest a reference for a proof of Gibbs phenomenon at jump discontinuities of a function.
Pointwise convergence of Fourier series of a piecewise continuous (and Lipschitz continuous everywhere) function.I basically want to understand a proof for convergence of a Fourier series of $f(x)$ to ...