Questions on Fourier series, the expansion of a function in terms of basis functions that satisfy an orthogonality relation. Usually, complex exponentials/sines and cosines are used.

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Let $a_{n}\downarrow 0$ and if series $\sum a_{n}\sin nx$ is a Fourier series of function $f\in L^{1}$ then $\sum \frac{a_{n}}{n}<+\infty.$

I want to show that if $\displaystyle\sum a_{n}\sin nx (a_{n}\downarrow 0)$ is a Fourier series of $f\in L^{1}$ then $\displaystyle \sum \frac{a_{n}}{n}<+\infty.$ I know i have to use some property ...
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Deriving fourier series using complex numbers - introduction

So this is the follow up thread to the one I asked before but you don't need to read the other one for this to make sense. If you want to, read PZZ's answer: link to the thread. So I know that there ...
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Is fourier series of a function with $e^{j\theta}$ replaced with a complex variable $z$ holomorphic on the unit disc?

Consider any continuous $2\pi$ periodic function (of bounded variation) $f : \mathbb{R} \to \mathbb{R}$ and its fourier series given as $f(\theta) = \frac{a_o}{2} + \sum\limits_{n = 1}^{\infty} ...
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Proving that the fourier coefficients for a pretty smooth function are pretty small

Let $f:[0,2\pi] \rightarrow \mathbb{R}$ be $C^k$ for some $k >0$. Prove that $|\widehat{f}(n)|n|^k|$ is bounded above by some constant independent of $n$. To do this, we've been ...