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A Fourier series is a decomposition of a periodic function as a linear combination of sines and cosines, or complex exponentials.

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Here is a problem and answer from my notes, where I have issues understanding parts of the provided answer. Problem: Solve the P.D.E. $U_t=U_{xx}$ with the following initial and boundary conditions: $...
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This is a problem and answer from my notes: Solve heat equation for $l=\pi$ and with the initial and boundary conditions: $U(0,t)=U(\pi,t)=0;\;\;U(x,0)=u_0(\sin x+\sin 3x)$ The answer to the above ...
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Suppose we have a Schwartz function $\varphi\colon \mathbb{R}\to \mathbb{R}$ supported in (0,1) such that $||\varphi||_{L^2}\leq 1$ satisfying that for all $\xi\in \mathbb{R}$, \begin{align} \sum_{l\...
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I was reading stuff about discrete Fourier transform (DFT) and its inverse when I produced the following proof which seems to lead to a contradiction for me. I don't know where the error is. Could ...
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As mentioned, the problem is: Derive the Fourier series of a half-wave rectified cosine with period T=4p and amplitude A=1 as you can see in Desmos graph (orange color). The solution given in textbook ...
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For an appropriate function $g$ and for an integer $k$, the $k^{\text{th}}$ Fourier coefficient $\widehat{g}(k)$ of $g$ is defined as $$ \widehat{g}(k):=\int_0^1g(x)e^{-2\pi ikx}dx. $$ $\textbf{...
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I wanted a plot of: \begin{equation} f(x) = e^{-|x|} \end{equation} and I wanted to compare $f(x)$ to its Fourier series ($n = 1,3,20$): \begin{equation} F(x) = \frac{e^{\pi}-1}{\pi e^{\pi}} + \frac{2}...
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I’m working on a problem from Asmar’s PDE textbook (2nd ed, 2004). The question comes from §2.2 #13: In Exercises 5-16, the equation of a 2$\pi$-periodic function is given on an interval of length 2$\...
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Below is a plot of $\sum_{k=-20}^{k=20}e^{2\pi ix}$ plotted using desmos. I know when you have a differentiable function with jump discontinuities in it, you get oscillations near the discontinuity ...
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Consider the integral \begin{equation} \int_0^\infty \dfrac{dT}{T} \sum_{(m,n)\in\mathbb{Z}^2}e^{-\alpha Q(m,n) T} \end{equation} This integral is divergent both at $T\to0$ logarithmically and $T\to\...
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So obviously the Fourier series of $\sin(x)$ is just $\frac{1}{2i}\left(e^{ix} - e^{-ix}\right)$ almost by definition. However, I was wondering if there is a way to compute the Fourier coefficients of ...
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I am teaching Engineering Mathematics (II) in my school. I use the textbook [1]. In [1], the Fourier series is defined in DEFINITION 12.2.1: DEFINITION 12.2.1 Fourier Series The Fourier series of a ...
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Say I have a monotonically increasing function $f : [0,1] \to {\Bbb R}$. I only know the values of $f$ for a finite set of points $x_1, \dots, x_n$. Can I use a Fourier series to approximate the ...
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The solution to the Legendre differential equation $$ (1-x^2) \frac{\mathrm{d}^2y}{\mathrm{d}x^2} - 2x\frac{\mathrm{d}y}{\mathrm{d}x} + n(n+1) y = 0 $$ is a linear combinations of the Legendre ...
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$$\sum_{n=1}^∞ \frac{(-1)^n\sin(nx)}{n^3}$$ I'm brushing up on Fourier series. I was looking at deriving the Fourier series for: $$ f(x) = x^3 - \pi^2 x, x\in [-\pi,\pi] $$ I have an undergrad in ...
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8.45 corollary. Suppose that $f \in L^1(T)$ and $I$ is an open interval of length $\leq 1$. a. If $f$ agrees on $I$ with a function $g$ such that $\hat{g} \in l^1(\mathbb{Z})$, then $S_mf \to f$ ...
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I have a question when reading R. H. Shumway and D. S. Stoffer's Time Series Analysis and Its Application With R Examples, 5th edition. On page 181, section 4.1, it's said that Note that, if in (4.4),...
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Investigating the $$\cosh (\pi \sin (\pi r))$$ Looked at the graph and seems it is somehow related to some combination of $sin$ and some unknwon constants $$1.68 \pi (\sin (2 \pi r-1.68)+2.4)-2 \...
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Let $f\in C^{0,\alpha}_{even}(\mathbb{R}/2\pi\mathbb{Z})$ be an even function of period $2\pi$ and of class $C^{0,\alpha}$ with $0<\alpha<1$. In wikipedia it's mentioned that the partial sum of ...
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$$a_{m} = \frac{2}{R^{2}J^{2}_{n + 1}(\alpha_{mn})}\int_{0}^{R}xf(x)J_{n}(\alpha_{mn}x/R)dx \tag{1}$$ As an exercise I've been trying to write GNU Octave code that will take an arbitrary function that ...
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It should start from the very beginning deriving the Fourier series. I have tried a book by Elias M. Stein & Rami Shakarchi. It's a good book but they assume that reader has already been ...
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I'm trying to solve the following PDE using the Fourier series method: \begin{align} &\partial_t u(t, x) - t\partial^2 u(t, x) = 0 && x \in [0, \pi], \; t\in\mathbb{R}^+ \\ &u(t, 0) = ...
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Suppose that $f$ is a function on the torus $\mathbb T$ such that its Fourier series is absolutely convergent, i.e. $\sum_{n \in \mathbb Z} |\hat{f}(n)| < \infty$, where $\hat{f}(n)$ is the $n$-th ...
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I am trying to solve an exercise which requires to find the Fourier expansion for the function: $$ f(x) = \begin{cases} x^2 - 1 & -\pi \le x < 0 := A \\ x^2 + 1 & 0 \le x < \pi := B \end{...
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IMC 1995 Problem 9 Prove that every function of the form $$f(x)=\frac{a_{0}}{2}+\cos(x)+\sum_{n=2}^{N}a_{n}\cos(nx)$$ with $|a_{0}|<1$ has positive as well as negative values in the period $[0,2\pi)...
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I came across the following result on the heat equation when taking a second look at my calculus course : First the heat equation is : $$\frac{\partial F}{\partial t}=a^2 \frac{\partial^2 F}{\partial^...
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Inspired by this question. Let $\tilde{f}:[0,1)\to \mathbb{R}$ be a bounded, non-constant continuous function. Let $f$ be the $1$-periodic extension of $\tilde{f}$ to $\mathbb{R}$. Let $g:[0,1]\to\...
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Question: I am trying to evaluate the following integral \begin{equation} \int \mathrm dp_1\cdots\int \mathrm dp_N \delta(p_0-p_c) \prod^N_{k=1}\exp\left[-\frac{\beta}{2Nm}\left(p_k-i\frac{Nm}{\beta\...
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This question is possibly a duplicate, but I can't find any other ones which answer this. I've been learning about Fourier analysis, and I understand how Fourier series work. I know that the Fourier ...
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I'm investigating the following function: $$ f(x)=\sum_{n=-\infty}^\infty \frac{a}{\sqrt{2\pi}\sigma}\exp\left(-\frac{(x-na)^2}{2\sigma^2}\right) $$ Conceptually, this is a bunch of Gaussians added ...
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I came across a puzzling statement in Lanczos' "Discourse on Fourier Series". It arises in the context of integrating the Dirichlet Kernel: $$\Gamma_n(t) = \int_0^tC_n(\xi)\,d\xi+\text{const....
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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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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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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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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 $$\lim_{N\to\infty}\frac{1}{\pi} \int_{0}^{\pi}g(t)\cdot D_N(t)\,dt$...
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Consider the Fourier series of $g(x) = e^{-2\pi i ax}$ on $[0, 1]$ where $a$ is not an integer. Computing the complex Fourier coefficients gives $$c_n = \int_0^1e^{-2 \pi i ax}e^{-2 \pi i nx} \, dx = \...
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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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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 $$S(f,x) := \sum_{n \in \mathbb{Z}} \hat{f}(n)e^{...
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I have this task on mathematical analysis from my college teacher. I tried using Dini test but I am confused. How I can determine the convergence of the Fourier series of this function in points? The ...
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My attempt is as follows: if we multiply both sides of a Fourier-Bessel series (first kind, $\nu$th order) of the form, \begin{align*} f(x)&= \sum_{n=1}^{\infty} C_n \: J_\nu\left(\frac{\...
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I have just been graphing the following functions defined by a sort of 'Fourier series'. Fix an integer $m> 1$, and define $$f_m(x)=\sum_{n=1}^\infty \left(\exp\left(\frac{ix}{n^m}\right)-1\right)$$...
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Question Is the everywhere‑divergent series(Steinhaus) $$ \sum_{k=2}^{\infty} \frac{\sin\bigl(k(x-\ln\ln k)\bigr)}{\ln k} $$ a genuine Fourier series? H.Steinhaus: A divergent trigonometrical ...
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Cody mentions in his answer here that A handy formula when integrating a polynomial times cot or csc. It can be shown that: $\displaystyle\int_{a}^{b}p(x)\cot(x)dx=2\sum_{k=1}^{\infty}\int_{a}^{b}p(x)...
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The integral posted six years ago contains 10 solutions for proving that $$\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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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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Question Consider the two periodic signals $$ y(t)=B\sum_{n\in\mathbb Z}\operatorname{rect}\!\bigl(2t-n\bigr), > \qquad x(t)=A\sum_{n\in\mathbb > Z}\operatorname{rect}\!\Bigl(\tfrac{t}{2}-6n\...
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I studied Fourier analysis. The classes were like other mathematics classes where we made definitions, proved theorems, etc. However, it was much later, that I began to understand that these ...
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This semester I'm taking finite field theory and Fourier analysis. The finite field professor introduced the subject as "Clock arithmetic" while the Fourier professor introduced the topic as ...
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$$V=\sum_{n_x}\sum_{n_y}\frac{1}{n_xn_y}\frac{\sin(kn_x)\sin(kn_y)\sinh\left(k\sqrt{n_x^2+n_y^2}\right)}{\sinh(\pi\sqrt{n_x^2+n_y^2})}$$ Here, $n_x, n_y$ are odd integers and $k=\frac{\pi}{2}$ I ...
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I was asked to find the Fourier coefficient of :- $f(x)=\cos(x/2)$ Here is my attempt $$I:=\int \cos(x/2)e^{ax}dx= \frac{e^{ax}\cos(x/2)}{a}+\frac{1}{2a}\int \sin(x/2)e^{ax}dx $$ $$=\frac{e^{ax}\cos(x/...
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