Questions tagged [fourier-series]
A Fourier series is a decomposition of a periodic function as a linear combination of sines and cosines, or complex exponentials.
5,815 questions
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Decomposing functions into an analytic component
I came across a puzzling statement in Lanczos' "Discourse on Fourier Series". It arises in the context of integrating the Dirichlet Kernel:
$$\Gamma_n(t) = \int_0^tC_n(\xi)\,d\xi+\text{const....
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Modern References to Fourier series in PDEs
I've been looking for reference about solving PDE with Fourier Series. I have a lot of references about Harmonic Analysis like "Fourier Analysis" by Javier Duoandikoetxea and Classical ...
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Convergence of Fourier series under scaling at the origin
I’m working on the following problem:
Let $f$ and $g$ be $2\pi$-periodic integrable functions such that in some neighborhood of $0$ one has
$$
g(x) = f(a x)
$$
for a fixed constant $a\neq 0$. Prove ...
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Show that $\sum_{k=0}^{4}\sqrt{1-c^2\sin^2\left(x+2k\pi/5\right)}$ is decreasing on $[0, \pi/10]$
I need help proving that this function is decreasing $(\frac{dv}{dx}\leq0)$ over the interval $x\in[0,\frac{\pi}{10}]$.
$$ v(x,5)=\sum_{k=0}^{4}\sqrt{1-c^2\sin^2\left(x+\frac{2k\pi}{5}\right)}$$
for $...
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Dirichlet's kernel and Dirichlet integral
Let $g(t)$ be an integrable function on $[0, \pi]$ such that $g(0) = 0$ and $g(t)$ is continuous at $t = 0$.
Question: If the limit
$$\lim_{N\to\infty}\frac{1}{\pi} \int_{0}^{\pi}g(t)\cdot D_N(t)\,dt$...
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The Derivation of a Well-Known Sum
Consider the Fourier series of $g(x) = e^{-2\pi i ax}$ on $[0, 1]$ where $a$ is not an integer.
Computing the complex Fourier coefficients gives
$$c_n = \int_0^1e^{-2 \pi i ax}e^{-2 \pi i nx} \, dx = \...
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What are the coefficients of the spectral form of the fourier series of a complex-valued function?
Say I have a function $f(x)$ define on $x\in[-\pi,\pi]$ where $f$ itself is complex and the argument $x$ is real. Then the trigonometric form of Fourier series of $f(x)$ is:
\begin{equation}
f(x) = \...
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Question about equivalent forms of Fourier Series
Let $f \in L^1(T)$, where $T$ is the interval $[-\pi, \pi]$ with $-\pi \sim \pi$ (identified). Then we can define the Fourier series of $f$ at $x$ by $$S(f,x) := \sum_{n \in \mathbb{Z}} \hat{f}(n)e^{...
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At which points of $(-π, π)$ the Fourier series of function $f(x) = \sqrt[10]{|x(x + 1)|}$ converges to the value f$(x)$?
I have this task on mathematical analysis from my college teacher.
I tried using Dini test but I am confused. How I can determine the convergence of the Fourier series of this function in points?
The ...
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How do I correctly find the Fourier-Bessel coefficient, $C_n$?
My attempt is as follows: if we multiply both sides of a Fourier-Bessel series (first kind, $\nu$th order) of the form,
\begin{align*}
f(x)&= \sum_{n=1}^{\infty} C_n \: J_\nu\left(\frac{\...
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Are these functions well-known?
I have just been graphing the following functions defined by a sort of 'Fourier series'. Fix an integer $m> 1$, and define
$$f_m(x)=\sum_{n=1}^\infty \left(\exp\left(\frac{ix}{n^m}\right)-1\right)$$...
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Is the everywhere‑divergent series $\sum_{k=2}^{\infty} \frac{\sin\bigl(k(x-\ln\ln k)\bigr)}{\ln k}$ a genuine Fourier series?
Question
Is the everywhere‑divergent series(Steinhaus)
$$
\sum_{k=2}^{\infty} \frac{\sin\bigl(k(x-\ln\ln k)\bigr)}{\ln k}
$$
a genuine Fourier series?
H.Steinhaus: A divergent trigonometrical ...
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Distributional Fourier expansion of $\cot$ or something else?
Cody mentions in his answer here that
A handy formula when integrating a polynomial times cot or csc.
It can be shown that:
$\displaystyle\int_{a}^{b}p(x)\cot(x)dx=2\sum_{k=1}^{\infty}\int_{a}^{b}p(x)...
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Need help to generalise $\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x$.
The integral posted six years ago contains 10 solutions for proving that
$$\int_0^\infty\left(\arctan \frac1x\right)^2 \mathrm d x = \pi\ln 2,$$
I then want to go further with the general case
$$I_n=\...
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Transformation rule under modulation for real Fourier series
The transformation rule for modulation in time (= shift in frequency) for complex Fourier series is easy to prove and can be found, for instance, on Wikipedia: see the last row in this table. Deriving ...
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