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

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99 views

How did Fourier series lead to the development of rigorous analysis?

Once I've heard that the studies of Fourier series have lead to rigorous definitions of such concepts as function, convergence, integral, limit. And also that Cantor's study of Fourier series led him ...
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1answer
37 views

Fourier Series Coefficient Question

In calculating the Fourier Coefficients a0, an, bn: Why are the an and bn coefficients integrated over 2 times the inverse of the period, 2(1/T) while the a0 coefficient is integrated only over one ...
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1answer
292 views

Showing a series is not the fourier series of a riemann integrable function.

I want to show that the series $\sum_1^\infty \frac{sin(nx)}{\sqrt{n}}$ is not the Fourier series of a Riemann integrable function on $[-\pi,\pi]$. I was going to do this by showing that the partial ...
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1answer
42 views

Fourier series representation of $\sin^4 x$

I tried solving for fourier coefficients of Fourier series for the multiples of fundamental frequency $\omega_0=2$. So $F_n=\int_0^{\pi} \sin^4 x \, e^{-i2nx} dx$. And my calculator says answer should ...
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1answer
249 views

Weighted sum of cosines

Consider $$f(x) = \sum_{k=1}^\infty \cos(kx) k^\alpha.$$ The first question is: does this have a name (Mathematica gives it as a sum of polylogs of complex arguments, but this seems unnatural). Also, ...
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1answer
44 views

Fejer's theorem with Riemann integrable function

If $f$ is integrable and $f(x+), f(x-)$ exists for some $x$, then $$ \lim_{N \rightarrow \infty} {\frac{1}{{2\pi }}\int_{ - \pi }^\pi {f\left( {x - t} \right){K_N}\left( t \right)dt} } = ...
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1answer
32 views

Express as a complex Fourier series

My function is $f(x)= \dfrac{1}{1-2e^{ix}} + \dfrac{1}{1-2e^{-ix}} $, which has been periodically extended by $2\pi$. I found $C_0$ to be $\pi$. I'm having trouble expressing $C_n$. All I have is ...
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1answer
54 views

Setting up my Fourier series for $B_n$

Related but not necessary to know: here Looking at the temperature distribution in an infinitely long cylinder of metal with insulated sides and initial temperature distribution $f(x)= ...
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2answers
49 views

Theorem of the convergence of the series of fourier! [duplicate]

During the demonstration of the theorem of the convergence of the series of fourier, my teacher wrote :$$ \frac{1}{2}+ \sum_{k=1}^{n} \cos(ky)=\frac{\sin((n+\frac{1}{2})y)}{2\sin(\frac{y}{2})} $$ he ...
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1answer
168 views

Sufficient Condition for the convergence of Fourier Series

I'm studying real analysis and I know about derivative, Riemann integral, sequence and series, basic concepts. I'm having trouble understanding the sufficient conditions for a Fourier series of a ...
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1answer
48 views

Is it always the case that lower frequencies contribute the most in a Fourier series?

Is it always the case that lower frequencies contribute the most in a Fourier series? Or to put it in other words, in the equation: $$f(t)=a_0+\sum^\infty_{m=1} a_m\cos \left(\frac{2\pi mt}{T}\right) ...
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1answer
59 views

How to find the coefficients in the Fourier series solution of a 1-D heat equation?

I am trying to use Fourier's method to solve a problem. $u(x,t) = \sum \limits_{n=1}^\infty B_ne^{-(n\pi C / L)^2 t}\sin\left(\frac{n\pi x}{L}\right), B_n=\frac2L\int_0^L \sin\left(\frac{n\pi ...
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0answers
81 views

Complex Fourier Series. I Might Neeed Some Help On This Problem

The Problem: If $f(x) $ is a real funciton, rewrite the integral: $$ \frac{1}{2L} \int_{-L}^{L} |f(x)|^{2} \, dx$$ in terms of the usual Fourier Coefficients, $A_n$ and $B_n$ The attempt: Recall ...
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1answer
69 views

Rewriting partial differential equation

I have some trouble rewriting a partial differential equation, more specifically the heat equation in one dimension: $ \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + f(x,t)\\ $ ...
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1answer
37 views

I might need some help on this Complex Fourier Series Problem

Here is the problem: Use the complex Fourier Series on $[-L,L] $ with complex coefficients to find a representation of $\frac{1}{2L} \int_{-L}^{L} |f(x)|^{2} dx$ Here is my attempt: The ...
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1answer
147 views

Can piecewise $C^{1}$ on $[a,b]$ imply Lipschitz continuity

I saw a statement that if $f$ is continuous,$2\pi$-periodic function which is $C^{1}$ piecewisely on $[-\pi,\pi]$, then its Fourier series converges uniformly to $f$ on $[-\pi,\pi]$. I was wondering ...
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1answer
4k views

How to plot fourier series in matlab

For homework (with no prior experience in matlab, guh.) I'm asked to do the following: Plot the (2N + 1)-term approximation $$\sum\limits_{k=-N}^N{a_ke^{jk\omega_0t}}$$ where $a_k = ...
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1answer
36 views

Why is matlab giving me a single answer when I divide by a vector?

I'm attempting to do a stem plot of $\frac{sin(k2D\pi)}{k\pi}$ in matlab. Following is my procedure: ...
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1answer
40 views

Can you help me with this Complex Fourier Series Problem?

Find the Complex Fourier Series of $F(x) = \cos(2x) + \sin(x)$ on the interval $[-\pi, \pi]$ Here is my attempt: The complex Fourier Series is in the form $\cos(2x) +\sin(x) = \sum_{n= ...
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1answer
31 views

Fourier cosine series for a interval $[0, l]$

It is asked to find the Fourier Cosine Series for the function defined by $$f(x) = \cos \frac{\pi x}{l}, x \in [0, l/2]$$ $$f(x) = 0, (l/2, l]$$ I thought it should be $$\frac{a_o}{2} + \sum a_n ...
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2answers
73 views

If $f \in L^2(\mathbb T)$ then $S_n(f) \to f$ in $L^2$ sense.

Theorem: If $f \in L^2(\mathbb T)$, then $S_n(f) \to f$ in $L^2(\mathbb T)$ sense. Proof: Let $f \in L^2(\mathbb T)$, then by definition $\|f\|_2^2 = \frac{1}{2\pi} \int_0^{2\pi} \vert f(x) \vert^2 ...
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2answers
40 views

Terms in Fourier Series

Can any one explain why? $$\int_0^\pi \sin(nx)\sin(mx)\,dx=\begin{cases}0,&n\not=m,\\ {\pi\over 2},&n=m,\end{cases}$$ and $$\int_0^\pi \cos(nx)\cos(mx)\,dx=\begin{cases} 0, &n\not=m,\\ ...
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63 views

heat equation with fourier series

Original PDE $$T_t=\alpha T_{xx}$$ I need to solve this equation numerically and analytically and compared them. I've already done the numerical part. But I need to solve it analytically now. Given ...
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2answers
241 views

problem on a function being identically zero

Let $f:[0,\pi] \to \mathbb{R}$ be a continuous function such that $f(0)=0$. If $$ \int_0^\pi f(t)\cos nt\, dt = 0 $$ for all $n \in \mathbb{N} \cup \{0 \}$, is $f$ identically zero?
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0answers
35 views

Can the Fourier Series be made “ Shorter ”?

I have tried to give only the intuitive part of my question and haven't included many specific details. Please help me frame it more precisely. I have inluded the symbol (*) where I need more details. ...
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1answer
60 views

Are fourier series of a periodic function expanded on different intervals equivalent

I was given an assignment by my instructor where i had to write the function $$ f(t) = \begin{cases} 1-t & 0\leq t < 1 \\ t-1 & 1 \leq t < 2 \end{cases}\\ f(t + 2) = f(t) $$ as a ...
5
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1answer
115 views

Importance of groups $(\mathbb R,+)$ and $(\mathbb Z,+)$ for Fourier series

I have heard that the groups $(\mathbb R,+)$ and $(\mathbb Z,+)$ are the most important groups for Fourier series. Why is this the case? Supposedly, it has something to do with the fact that for any ...
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2answers
40 views

Fourier Series Proof

I have concerns about this problem Let $f_e(x)$ and $f_o(x)$ represent general continuous even and odd functions on $[-L,L]$. Prove that $\int_{-L}^{L} f_e(x) dx$ = 2 $\int_{0}^{L} f_e(x) dx$ ...
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0answers
47 views

Bound on the integral of a differentiable function against sine functions

Consider a function $f: [0, L] \to \Bbb R$ and $k$ natural number. Suppose that $f', f'', ...$ $f^{n-1}$ are continuous and that $f^{k}$ is absolutely integrable. Show $$ \left| \int_{0}^{L}{f(x)\sin ...
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1answer
686 views

Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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5answers
45 views

Proving a function $f(x + T)=k\;f(x)$ satisfies $f(x)=a^x g(x)$ for periodical $g$

I need to prove the following: If a function $\,f$ satisfies $$f(x+T)=k\;f(x), \forall x \in \mathbb R$$ for some $k \in \mathbb N$ and $T > 0$, prove that $\,f$ can be written as ...
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1answer
30 views

Fourier coefficient one-period function

Define a function $f(x) =(2\cos(\pi x))^{10} $$f\in L^{1}$ so it's one-period. I would like to calculate the Fourier coefficient $\hat{f}(2)$. So we get $\displaystyle\hat{f}(n)=\int_{0}^{1}e^{-2\pi ...
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1answer
95 views

Fourier Sine Series and Cosine Series

This is the Fourier Series representation for a periodic function with period 2p, given in my lecture note. $\dfrac{a_0}{2} + \sum_{n=1}^{\infty}(a_n cos(\dfrac{n\pi t}{p})+b_nsin(\dfrac{n\pi ...
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1answer
49 views

evaluate arithmetic sum by using fourier series

Hi I've been trying for 40 minutes to evaluate the sum of the following arithmetic series with no luck. $\sum_{n=1}^\infty \frac{sin(2k)}{k}$ I've tried to make this into a fourier series by ...
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0answers
53 views

Show that $x(\pi - x)= \frac{\pi^2}{6}-\sum_{k=1}^{\infty} \frac{\cos(2kx)}{k^2}$

Show that $$x(\pi - x)= \frac{\pi^2}{6}-\sum_{k=1}^{\infty} \frac{\cos(2kx)}{k^2}$$ for $ 0<x<\pi$ My idea: I've defined the periodic function $$f(x) = 0 \text{ if } x \in [- \pi, 0) \text{ ...
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0answers
89 views

Fourier Cosine Series question

If I have even piecewise periodic function ($T=6$) $$x(t)=\begin{cases} 0 &-3\leq t \leq-2  \\ 2+t &-2\leq t \leq-1 \\ 1 &-1\leq t \leq 1 \\ -t+2 &1\leq t \leq 2 \\ 0 &2 \leq ...
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0answers
66 views

Discrete Fourier Transform by hand

I have an assignment where I'm given the DFT of a sequence $x[n]$ as $X[k]=\{4,3,2,1,0,1,2,3\}$ and also $$y[n] = \left\{ \begin{array}[cc] xx[n/2] & \text{if n is even} \\ 0 & ...
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1answer
83 views

Calculating fourier series

I've a fourier series with a period = $2\pi$ that is even. f(t) = \begin{cases} 0 \text{, when: } 0<t<\pi-2 \\ \pi \text{, when: } \pi-2<t<\pi \end{cases} The functions trigonometric ...
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3answers
260 views

Finding the fourier series of floor function

Find the fourier series for $f(x)=\cases{x-[x]\quad x\in\mathbb{R\setminus Z} \\ \frac 1 2\quad x\in\mathbb{Z}}$ on $[-\pi,\pi]$ and its values for $x=1.5,3,5$. In order to find the series I need ...
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1answer
161 views

Why coefficients of Fourier series are countable, though the initial periodic function is described with an uncountable set of points

Coefficients in the Fourier series for any periodic square-integrable function $f(x)$ form a countable (though infinite) set, i.e., they have cardinality $\aleph_0$. As far as Fourier exponents form a ...
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2answers
71 views

Fractional Sobolev space $H^{1/2}(-\pi,\pi)$

Let $H^{1/2}(-\pi,\pi)$ be the space of $L^2$ functions whose Fourier series coefficients $\{c_n\}_n$ satisfy the summability constraint $\sum_n |n| |c_n|^2 < \infty$. Are functions in ...
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1answer
2k views

Finding the Fourier series of a piecewise function

I'm s little confused about Fourier series of functions that are piecewise. Here’s an example of such a function: $$f(x) = \begin{cases} x & -\frac\pi2 < x < \frac\pi2 \\[5pt] \pi - x & ...
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2answers
53 views

Derivative of Fourier series

Let function $f(t)$ is represented by Fourier series, $$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$ where $a$ and $b$ are lower and upper boundary, ...
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1answer
69 views

Using the Fourier Series of $f(t)=(t-\frac{1}{2})^{2}$ to deduce the sum $\sum_{n=1}^{\infty }\frac{1}{n^{2}}$?

So this is a question in one of the previous tests: My approach (if you want just skip to step 3.):$$$$ 1. Formulation of the problem and calculating the constant term of the series $a_o$ I ...
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2answers
18 views

Doubt in Fourier Series

When we solve the equation $$\frac 2{\pi}\int_{0}^{\pi}k\sin(nx)dx;$$ after integrating it, we get $\frac {2k}{n\pi}(1-\cos n\pi)$. Why is $\cos n\pi=(-1)^n$?
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0answers
49 views

Functions f(x) equal to Taylor series vs Fourier series vs Bessel series

(I had trouble phrasing the question below due partially to the fact that Bessel functions $J_{\alpha}(x)$ and $U_{\alpha}(x)$ are defined for any complex $\alpha$, so below I tried to express an ...
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1answer
163 views

Fourier series for a non-periodic function

My textbook says that: 'If we which to find the Fourier series of a non-periodic function only within a fixed range then we must continue the function outside the range so as to make it periodic.' ...
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1answer
89 views

complex fourier series with odd function

Consider the periodic and hybrid function defined as $$f(t)=x, 0\le x \le 1$$ and $$f(t)=1$$ $$1\le x\le 2$$ Attempt: I need to calculate Cn $$C_n=\frac{1}{2}\int_0^1 xe^{-in\pi ...
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2answers
65 views

Fourier Transform of mix partial derivative

I know FT{$\frac{\partial u}{\partial x}$} = (ik)FT{u}. Give a function $U(x,y)$. Is the following true? FT{ $\frac{\partial^2 U}{\partial y \partial x}$} = FT{$\frac{\partial U}{\partial y}$} ...
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2answers
40 views

Convergence of Fourier Series in $L^1(\mathbb{T})$

Suppose $f \in L^1(\mathbb{T})$ and the sequence of partial sums of its Fourier series converges (in $L^1(\mathbb{T})$) to $g$. How can I prove $f=g$?