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.

learn more… | top users | synonyms

2
votes
0answers
31 views

Fourier series using Bessel function

so Im stuck on the following problem; Use the identity $\exp(ix\sin\theta) = \sum\limits_{k=-\infty}^\infty J_k(x)\exp(ik\theta)$ to find the Fourier series of $\cos(\theta + 4\sin\theta)$, where ...
1
vote
0answers
16 views

Expansion of function in polar coordinates

I'd like to expand a function in polar coordinates to something that splits radius and angle $f(r,\theta)=\sum_i A_i(r)B_i(\theta)$ I've found some hints on the internet by the name of polar Fourier ...
1
vote
1answer
36 views

Find Fourier Coefficients

I am asked to find the coefficients for $f(t)=\sin^{2}(5t)$ $$Period =\frac{\pi}{5}$$ so I wrote $$a_n\cdot\sin(\frac{n\pi{t}}{\frac{\pi}{10}})=\sin^{2}(5t)$$ $$a_n\cdot\sin(10n{t})=\sin^{2}(5t)$$ ...
0
votes
0answers
18 views

An upper bound for the Fourier coefficient of the “infinite cake” function

Consider a function $x_{s_n} (t) = s_n$ for $t\in[-\frac{s_n}{T_0}, \frac{s_n}{T_0}]$ and $x_{s_n} (t) = 0$ for $t$ everywhere else, with period $2T_0$. Now let $s_n=\frac{1}{n^2}$, and define the ...
0
votes
1answer
30 views

Confused about Fourier series?

From linear algebra we know that if a set of vectors form a basis for a space, their is a unique linear combination of the basis to form any vector in that space. I'm assuming this extends to scalar ...
1
vote
1answer
15 views

Computing the Fourier series of $\lvert x\rvert$

I am getting very confused when trying to compute the Fourier series of $f(x) = \lvert x\rvert$, $x \in [-1/2,1/2]$. Normally I have no trouble with this because it is mindlessly integrating to get ...
0
votes
1answer
18 views

Relation between fourier coefficients of $f\in \mathcal{C}^1[-\pi, \pi]$ and $f'$

I'm given $f\in \mathcal{C}^1[-\pi, \pi]$ with $f(-\pi)=f(\pi)$. It's fourier coefficients are given by: $$\gamma_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-int}f(t)dt,\ n\in \mathbb{Z}$$ And now I'm ...
1
vote
0answers
22 views

Does this reasoning about fourier analysis make sense?

I'm asked to show that there cannot be $\alpha_1,\alpha_2,...\in\mathbb{C}$ s.t. $$\lim_{N\to\infty}\int_{-\pi}^{\pi}|e^{it}-\sum_{k=1}^{N}a_k\sin(kt)|^2dt=0$$ Here is my attempt: Assume there are ...
1
vote
1answer
45 views

Finding limit under integral [closed]

Evaluate if $f \in C [-\pi,\pi]$ $$\lim_{n\to\infty} \int_{-\pi}^{\pi}f(t)\cos(nt)dt$$ and $$\lim_{n\to\infty}\int_{-\pi}^{\pi} f(t)\cos^2(nt)dt$$
2
votes
3answers
74 views

Can a non-periodic function have a Fourier series?

Consider two periodic functions. Assume their sum is not periodic. The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents ...
0
votes
0answers
25 views

When has the Fourier transform for some values equal values?

Definition We take a function $F : \mathbb T^n \rightarrow \mathbb R$ that is even ( $F(x)=F(-x)$) and continuous (hence bounded), where $\mathbb T^n$ is the $n$-dimensional Torus. Now we define the ...
0
votes
0answers
17 views

How can I solve this differential equation with fourier series?

Find a formal solution $u(x; y)$ by using Fourier series. (Hint: In two dimensions the basis functions have one of the forms $\sin(ax) \sin(by)$, $\sin(ax) \cos(by)$ and $\cos(ax) \cos(by)$, with ...
0
votes
0answers
33 views

Closed form of a series with sinh

Is there a simple form for following function (where $a$ and $b$ are constants)? Can it be simplified to a simple form if $a>>b$? $$ u(x) = \sum _{n=0}^{\infty } \frac{ \, (-1)^n ...
0
votes
0answers
27 views

Fourier series question - represent $x$ as a series of $\cos$

I was asked to represent $f(x)=x$ in $(0,\pi)$ as a sum of $\cos$ functions, using fourier series. I couldn't solve it on my own, but here is what the teacher did, and I don't fully understand why ...
2
votes
2answers
30 views

Discrete fourier transfomation and harmonics

I have a very simple question that I would like to understand. If you have a DFT of a function: $$ X_k \stackrel{\mathrm{def}}{=}\sum_{n=0}^{N-1}x_n\cdot e^{-i2\pi kn/N},\qquad k\in\mathbb{Z} $$ Did ...
1
vote
1answer
34 views

How to find $ \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^2}$?

Let $f$ be a $2\pi$-periodic function whose restriction on $[-\pi, \pi]$ is $f(x)_{[-\pi, \pi]} = |x|$ It is easy to see that its fourier series converges uniformily to $f$ and is $$f(x) = \frac \pi2 ...
0
votes
0answers
14 views

Characterstic Functions and Recovery

Assume that I have a pdf, call it $f$, that is supported on $[0,2]$. Let $\varphi(t)$ be the corresponding characteristic function, which is known to me. Is there some common method to recover the pdf ...
0
votes
0answers
24 views

Fourier Series: even extension and Parseval Identity

I'm trying to solve this exercise but I have some problems, because I haven't seen an exercise of this type before. $f(x)= \pi -x$ in $[0, \pi]$ Let's consider the even extension of f(x) in ...
7
votes
0answers
57 views

Example of continuous function whose Fourier series doesn't converge on an uncountable dense set.

According to a well-known theorem (Theorem 5.12 in Rudin's Real and Complex Analysis), there is a dense $G_\delta$ set of continuous periodic functions $f:\mathbb{R}\to\mathbb{C}$ such that the ...
1
vote
1answer
22 views

Fourier series: $\lim_{n\to\pm\infty} n^p \hat{f}(n) = 0$

Let $f:\mathbb{R}\to\mathbb{C}$, $f\in C^\infty$ (differentiable infinitely many times) and periodic,$T=2\pi$. Prove that for every $p>0$: $$ \lim_{n\to\pm\infty} n^p \hat{f}(n) = 0$$ So I ...
2
votes
1answer
14 views

Fourier series: Show that $f$ is a trigonometric polynomial

Let $N\in\mathbb{N}$ and $f_m:\mathbb{R}\to\mathbb{R}$, continuous functions and periodic, $T=2\pi$. Let's assume that $f_m \to f$ uniformly and for all $m\ge 1$: $$\left| \hat{f_m}(n)\right| \le ...
0
votes
1answer
21 views

Even or odd function. Fourrier coefficients

This is probably a very easy question, but I can't find the answer to it.. I'm working on Fourier coefficients and whether or not the integrals become zero. As far as i'm concerned this integral ...
4
votes
1answer
48 views

Fourier coefficients intuition?

I just learned about Fourier series, and this is how I interpreted them: The complex exponentials form a basis for all periodic functions, and the Fourier series essentially decompose the function ...
0
votes
2answers
71 views

Proof of the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series

I have tried and failed to prove the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series which is as follows (and is described here: ...
5
votes
1answer
128 views

Double sum and zeta function

This is a personal research that came to an end , since the results were not those which were being anticipated. I was unable to come up with a solution therefore I post the topic here: Prove (it ...
2
votes
1answer
27 views

fft phase plot of pure sine function, why so messy?

I am plotting the phase plot of $sin(2*pi*60*x)$ in the frequency domain. Ideally, we should only see two peaks. How come this is not the case in matlab? ...
0
votes
0answers
9 views

Phase difference of two signal of different frequency

Currently, I have two signals, the main components of both signals are 60Hz, but both also have weaker response at 180Hz + small amount of noise. As shown in the photo below, I want to find the phase ...
0
votes
1answer
27 views

How are sinusoids and roots of unity related to each other?

The discrete Fourier transform (DFT) is often teached as being a transform that decomposes a given signal or sequence of numbers into sinusoids with frequencies $\large\frac{k}{N}$ where $k \in [0, ...
0
votes
1answer
18 views

Sum involving the “distance to the nearest integer function”

I want to prove that if $||x||$ is the distance between $x$ and the nearest integer to $x$, $\{\alpha_1,\ldots, \alpha_N\}$ are points in $\mathbb{R}$/$\mathbb{Z}$ and we define $$S(y) = ...
15
votes
5answers
2k views

Why do Fourier Series work?

I would like to have an intuitive understanding of Fourier Series. I mean, I know the formulas: $$ f(t) =\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(n\pi tL)+\sum_{n=1}^\infty b_n \sin(n\pi tL) $$ And ...
0
votes
1answer
27 views

General Fourier coefficients and smoothness

Suppose $f\in L^2([0,1],\lambda)$. Are there assumptions on the smoothness of $f$ which translate into the particular behavior of Frourier coefficients. Namely, I have arbitrary complete orthonormal ...
1
vote
1answer
40 views

Fourier sine series of $f = \cos x$

Let $f:(0,\pi) \to \mathbb{R}$ defined by $x \mapsto \cos x $ Show that the Fourier sine series of (odd extension) is given by $$\sum\limits_{n=2}^\infty \frac{2n(1+(-1)^n)}{\pi(n^2-1)}$$ So far, ...
2
votes
1answer
138 views

Derivation of fourier series equation

No matter where I search, every time if there's an article about Fourier series derivation, the first step made by author is to present the following formula: $$f(x) = \frac{a_0}{2}+\sum_{n=1}^\infty ...
0
votes
0answers
20 views

Fourier series - different equations

There are two very popular forms of Fourier series equation. $$f(x) = \frac{a_0}{2}+\sum_{n=1}^\infty \left(a_n \cos(\frac{2\pi}{T}nx) + b_n \sin(\frac{2\pi}{T}nx)\right)$$ and $$f(x) = ...
2
votes
0answers
34 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
0
votes
1answer
50 views

Help with proof of Poisson summation formula

I am trying to understand a proof of the Poisson summation formula and I cannot understand a vital part of it which the author seems to think is obvious, but is not obvious to me. If anyone can fill ...
2
votes
0answers
8 views

inserting absolute value in Hilbert transform and a discrete version of Hilbert transform

It is well known that the Hilbert transform $H(f)(x)=p.v. \int\frac{f(x-y)}{y}dy$ is bounded on $L^p(\mathbb{R})$ for $p\in(1,\infty)$. I want to consider some variants of $H$. 1) What happens if we ...
1
vote
1answer
28 views

Heat flow in 1D bar fourier series problem

I am stuck on this problem: The temperature $T$ in a one-dimensional bar whose sides are perfectly insulated obeys the heat flow equation $$ \frac{\partial T}{\partial t} = \kappa ...
1
vote
1answer
30 views

How to prove these Fourier-series identities?

The first series is : $$\sum_{n=1}^{\infty }(-1)^n\frac{4}{(n\pi )^2}\{(\cos(A(n\pi) )-\cos(B(n\pi ))\left. \right \}=(A^2-B^2)$$ Where $A$, $B$ are positive real numbers less than $1$. I need a ...
1
vote
1answer
42 views

Fourier Series and differential equation with epsilon

Happy New Year! I am stuck for days on expressing the solution of a differential equation using Fourier series. The question is: Consider the equation: $$\ddot{x}+x+\epsilon\left(\alpha ...
0
votes
0answers
16 views

Is the spherical harmonic representation of a 2D field independent of grid?

What I am currently unable to understand is whether the spherical harmonic representation of a 2D field is in any way tied to the nature of the grid on which decomposition/composition is performed. I ...
0
votes
1answer
19 views

Fourier Series Reduced Form: Phase Angle and Spectra

Im very confused regarding how to determine the angle on the reduced or harmonic form representation of the Fourier series. Some books state the following: $$f(t)=F_0+\sum_{n=1}^\infty |F_n ...
0
votes
1answer
30 views

Is there a trigonometric Fourier transform formula?

I wonder if one can express the Fourier transform in the trigonometric approach like, say, in the case of the Fourier series, where we can write it as: $Sf(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left ...
0
votes
1answer
56 views

What is the Fourier Series of a piecewise constant wave?

I am looking for the Fourier Series of this function: This is a winding function method for calculation of rotor inductances. The distance between each stator slot (each segment) is $10$ degrees or ...
2
votes
0answers
30 views

Norm Inequality (Vinogradov Notation)

I'm going through a proof of differentiability of fourier series on the d-dimensional torus and while proving the following inequality: $$ ...
2
votes
3answers
69 views

Does the phrase “orthogonal” mean the same thing when used in the terms “orthogonal function” and “orthogonal vector”?

I was reading about Fourier series when I came across the term "orthogonal" in relation to functions. http://tutorial.math.lamar.edu/Classes/DE/PeriodicOrthogonal.aspx#BVPFourier_Orthog_Ex2 I've ...
0
votes
0answers
21 views

Why is $\|f-s_n(f)\|_2=\inf_{T\in\mathcal{T}_n}\|f-T\|_2$

I am working through some examples in my book in the section on Fourier Series. Why is $\|f-s_n(f)\|_2=\inf_{T\in\mathcal{T}_n}\|f-T\|_2$? where $f$ is a continuous $2\pi$ periodic function, $T$ is ...
2
votes
1answer
65 views

Sum of $\sum_{n=1}^{\infty }\frac{1}{\pi n }\sin ^k\left(\frac{2\pi n}{k}\right)$

We have: $$S_k=\sum_{n=1}^{\infty }\frac{1}{\pi n }\sin ^k\left(\frac{2\pi n}{k}\right)$$ where $k$ is an odd number greater than $1$. I was able to find the sum of the series when $k=3,5$ as ...
0
votes
2answers
28 views

How does the Fourier transform get you the frequency amplitude

I understand that the Fourier transforn gets you the function which gives the amplitude of each frequency. But I don't understand how that is possible by multiplying it by an exponential. How is that ...
1
vote
1answer
60 views

wave equation on a square domain

I'm stuck on the following problem. Let $u(x, y, t)$ denote a solution to the linear wave equation $k^2(u_{xx}+u_{yy}) = u_{tt}$ with $k = 2$ on a square domain with corners at (0, 0), (0, 1), ...