Question

If $f \in L^1[-\pi, \pi]$ is odd and $f(x + \pi) = f(x)$ for $x \in \mathbb{R}$, then $\beta_{2k - 1} = 0, \forall{k} \in \mathbb{N}$

score 73 · Answer 1 · 2026-04-18 10:13:05

73

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If $f \in L^1[-\pi, \pi]$ is odd and $f(x + \pi) = f(x)$ for $x \in \mathbb{R}$, then $\beta_{2k - 1} = 0, \forall{k} \in \mathbb{N}$

Published on 18 Apr 2026 - 10:13

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202

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216

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37

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86

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List Question

If $f \in L^1[-\pi, \pi]$ is odd and $f(x + \pi) = f(x)$ for $x \in \mathbb{R}$, then $\beta_{2k - 1} = 0, \forall{k} \in \mathbb{N}$

Classical Full Fourier Series of f(x) converges uniformly to f(x)

Fourier coefficients of a triangle function

Solving Viscous Burgers using spectral method

Expanding a function in a Fourier Series

Show that $\sup_{x \in \mathbb{R}}|\sigma_nf(x) - f(x)| \leq C\frac{\log n}{n}$, for $f$ $2\pi$-periodic and Lipschitz.

Periodic Foricing Terms

Why does specifying an interval for a function make the function odd or even?

Heat and Wave equation - Green's function versus Fourier series?

Is square wave is an odd function if there is a dc shift?

Bounding Fourier coefficients using translates of a function

How to prove this function $f(x):= \sum_{k=1}^\infty \frac{1}{k^k} \exp(2\pi \, i\,k\,x)$ is entire?

Prerequisites for Fourier Series/Self-Study?

Index of a derivative operator on a circle

Fourier Series of a piecewise-linear function

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