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.

Filter by
Sorted by
Tagged with
0 votes
0 answers
26 views

The norm and the Fourier coefficient if the inner product $\langle f(x), f(x)\rangle$ is negative.

For a real function $f_m(x)$ orthogonal with respect to $w(x)$ where $w(x)>0$. We have inner product $$ \langle f_m,f_n\rangle_{w}=\int_{x_0}^{x_0 + T}{w(x)f_m(x)f_n(x)dx} $$ The norm is $||f_m||_{...
user avatar
1 vote
2 answers
30 views

Doing transformations on trignometric functions

I have a function $$f(x)=\sqrt{1-\cos(x)}$$ with the fundamental period $2\pi$. But I can also write this as $$\sqrt{2} \sin(x/2)$$ whose fundamental period is $4\pi$. Why has the fundamental period ...
user avatar
  • 2,557
1 vote
0 answers
22 views

Infinite series Sum of zeroth order Bessel Functions of first kind

I am trying to find the upper bound of $$ \sum_{n \geq 1} J_0(an) J_0(bn) \sin(cn) \sin(dn) $$ where $J_0(x)$ is the zeroth order Bessel function of first kind, and $a,b \geq 0, \textit{ and } c,d \in ...
user avatar
  • 21
0 votes
0 answers
21 views

How to graph the periodic function of a fourier series?

I have been given this function $$G(x)=\left\{\begin{array}{rcl} 2x,&& \mathrm{if}\ 0\le x\le 1,\\ 2,&& \mathrm{if}\ 1\le x\le 3,\\ 8-2y,&& \mathrm{if}\ 3\le x\le 4 \end{array}\...
user avatar
1 vote
0 answers
21 views

Is there a hyperbolic variant of the sine-cosine fourier series?

The sine-cosine form of the fourier series is given by: $$ s_{\scriptscriptstyle N}(x) = \frac{a_0}{2} + \sum_{n=1}^N \left(a_n \cos\left(\tfrac{2\pi}{P} nx \right) + b_n \sin\left(\tfrac{2\pi}{P} nx \...
user avatar
0 votes
0 answers
22 views

Error in solution of 3-45 in Solution Manual of Signals and Systems Oppenheim 2nd edition

Hello I was solving the question 3-45 of the book Signals and Systems 2nd edition and I assume there is an error in the solution manual, kindly if someone confirm that am I right? or the solution ...
user avatar
0 votes
0 answers
10 views

Discrete fourier series from impulse train

Im struggling to derive the DFS from a periodic impulse train: let x(t) be a periodic signal over $NT_s$. As such, its sampled, impulse train version is: $x_p(t) = \sum_{n\in Z}x(nT_s)\delta(t-nT_s) = ...
user avatar
  • 45
0 votes
0 answers
17 views

About the Dirichlet conditions and the Fourier series of the sine function

One of the Dirichlet conditions, at least in the texts I've used, is: "$f$ must have a finite number of maxima and minima." This is not true for the sine function as its derivative is the ...
user avatar
  • 379
0 votes
0 answers
32 views

Upper bound for sum of waves

Let the function $f(t)$ be defined as a linear combination of sinusoidal waves, with different amplitudes, frequencies and phases, i.e. $$f(t) = \sum_{i=1}^n a_i \sin \left( \omega_i t + \varphi_i \...
user avatar
  • 563
1 vote
1 answer
31 views

What is the value of the sum of $e^{i(-ak^2 + bk)}$ when $k$ goes through every integer? [closed]

I'm doing a physics question sheet and I stumbled upon a series of the kind $$ \sum_{k \in \mathbb{Z}} e^{i(-ak^{2}+bk)} $$ Where $a$ and $b$ are real numbers with $a>0$. I have no idea how to do ...
user avatar
4 votes
0 answers
134 views

Expressing $ \sum_{n=1}^{\infty}\frac{\mu(n)}{n^{2}} \left \lfloor x^{1/n}-1 \right \rfloor$ in terms of the nontrivial zeros of $\zeta(s)$

Let $\left \lfloor \cdot \right \rfloor$ be the floor function. Is there a way to express the function $A(x)$ given by : $$A(x)=\sum_{n=1}^{\infty}\frac{\mu(n)}{n^{2}} \left \lfloor x^{1/n}-1 \right \...
user avatar
0 votes
0 answers
20 views

Frequency peak always appearing at half of Nyquist frequency in Fourier transform

When taking the FT of a signal I always get a sharp peak at exactly half the Nyquist frequency. My signal is shown here: and its FT here: The Nyquist frequency is 36.7 KHz. As can been seen in the ...
user avatar
  • 1
0 votes
0 answers
34 views

Show that $\underset{N\to\infty}{\lim}\int_{-\pi}^\pi [g(t)\cos(\frac{t}{2})]\sin(Nt)dt=0$, where $g(t)=\frac{f(x-t)-f(x)}{\sin(t/2)}$.

Problem: $\underset{N\to\infty}{\lim}\int_{-\pi}^\pi [g(t)\cos(\frac{t}{2})]\sin(Nt)dt=0$, where $g(t)=\frac{f(x-t)-f(x)}{\sin(t/2)}$. The Problem arises from Walter Rudin's PMA: For the texts ...
user avatar
0 votes
0 answers
26 views

Finding solutions to $ \sin\left(n \pi \frac{T_p}{T}\right) - \sin\left(n \pi \frac{T_0}{T}\right) = \sin\left( \frac{{n \pi}}{2}\right) $

I'm trying to figure out how I can find a solution(s) to the equation below. $$ \sin\left(n \pi \frac{T_p}{T}\right) - \sin\left(n \pi \frac{T_0}{T}\right) = \sin\left( \frac{{n \pi}}{2}\right) $$ ...
user avatar
-2 votes
0 answers
45 views

Meaning of Fourier series expansion

I tried to solve this question related to Meaning of Fourier series expansion (foundation) in Mathematics for ultrasonic signal processing Question Calculate $\sin(\theta)\cos(\theta)$ utilizing the ...
user avatar
  • 1
0 votes
1 answer
35 views

Finding the complex term of a Fourier series [closed]

Expand $$f(x) = \begin{cases} 0 & -\pi < x < 0 \\ 1 & 0 < x < \pi\end{cases}$$ My doubt is regarding step number seven how does $0$ comes when $n$ is even. Any explanation will be ...
user avatar
1 vote
1 answer
24 views

Poisson Summation at Half Integers

In the proof of Poisson Summation, for a Schwarz function $f$, you define $$ F(x)=\sum_{n\in\mathbb{Z}}f(x+n) $$ and you show that $$ F(x)=\sum_{n\in\mathbb{Z}}\hat{f}(n)e^{2\pi inx} $$ Then plugging ...
user avatar
0 votes
1 answer
37 views

Proof that $\sum_{k = 0}^{N-1} \cos \left(\frac{2 \pi}{N} \left( k + \frac{1}{2} \right) \right) = 0$

I have come across the expression $$ \sum_{k = 0}^{N - 1} \cos \left( \frac{2 \pi}{N}\left(k + \frac{1}{2}\right)\right) $$ and I happened to notice that it equals $0$ for all integer values of $N >...
user avatar
  • 35
0 votes
0 answers
45 views

Fourier series expansion of a $L^2$ function.

For a function $f \in L^2(\mathbb{T})$ (where $\mathbb{T}$ denotes the unit circle) I know that it can be expressed as $f(z) = \sum_{j = -\infty}^{\infty}f_j z^j$. The Fourier coefficients are given ...
user avatar
-3 votes
1 answer
50 views

Fourier series of $f(x) = \cos(x)$

The Fourier series of $f(x) = \cos(x)$ on $|x| < \pi/2$ and $f(x) = 0$ otherwise. Since $\cos(x) = \cos(-x)$, we have $b_n=0$. Then I computed 2 different $a_0$'s, thinking the latter one is the ...
user avatar
  • 531
0 votes
2 answers
38 views

Fourier series of $f(x) = 1$ on the interval $\pi/2 < |x| < \pi$

I am trying to calculate the Fourier series of $f(x) = 1$ on the interval $\pi/2 < |x| < \pi$ and $f(x) = 0$ otherwise. $f(x) = 1$ is an even function. Therefore, $b_n = 0$. I am troubling how ...
user avatar
  • 531
0 votes
0 answers
29 views

Can one show that $\sum_{k=10}^{50} a_k \cos(k \theta)$ has at least four zeroes on $[0,2\pi]$ for $a_k \in \mathbb{R}$?

This is a complex analysis puzzle that seems tricky. How can one show that for $a_k \in \mathbb{R}$, $\sum_{k=10}^{50} a_k \cos(k \theta)$ has at least four zeroes on $[0,2\pi]$? A hint is to consider ...
user avatar
  • 279
0 votes
2 answers
34 views

$f(x) = x$ on the interval $|x| < \pi/2$

I am trying to calculate the Fourier series of: $f(x) = x$ on the interval $|x| < \pi/2$. First, I observed that $f(x) =x$ is odd and thus $f(x) * \cos(nx)$ is odd. Hence, $a_n = 0$. $b_n = 2/\pi \...
user avatar
  • 531
2 votes
1 answer
79 views

Fourier series of $\sqrt[3]{\sin x}$

So I've done some experiments with how to add distortion to audio, and one of the methods I'm proposing is to take the cube root of an audio signal as a way to add overdrive. As the waveform that you'...
user avatar
  • 1,013
0 votes
0 answers
12 views

Use of sine-cosine orthogonality rules to derive expressions for the Fourier coefficients

I am trying to follow a proof from my textbook for the equation which gives the value of the $a_r$ coefficients of the Fourier series: $$\int^{x_0 + L}_{x_0} f(x) \cos (\frac{2 \pi r x}{L}) dx = \frac{...
user avatar
  • 145
0 votes
0 answers
18 views

Complex fourier expansion of $\cosh(t-1)$ in a given period,

So I attempted this question 'An even function 𝑓(𝑡) is periodic with period $𝑇 = 2$ and $𝑓(𝑡) = \cosh(𝑡 − 1)$ for $0 ≤ 𝑡 ≤ 1$. Find a complex Fourier series representation for 𝑓(𝑡).' my ...
user avatar
1 vote
1 answer
84 views

Which even function $f(x)$ satisfies $\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)~dx=\frac{1}{n}$?

Question: Which even function $f(x)$ satisfies $$\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\cos(nx)~dx=\frac{1}{n}? $$ Alternatively, which even function $f(x)$ satisfies $$\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)\...
user avatar
1 vote
1 answer
58 views

solving $u_t = u_{xx} +2u_x +u+1$ [closed]

I have this problem : $$u_t = u_{xx} +2u_x +u+1$$ with boundaries of $$0<x<1$$ $$ 0<t$$ and we also have $$ u(x,0) = x$$ $$u_x (0,t) = sin t$$ $$ u(1,t) + u_x(1,t) =2$$ I need to solve ...
user avatar
  • 147
1 vote
0 answers
29 views

Bound on amount of minima of finite Fourier sum

Consider a finite Fourier sum of the form $$f(\theta) = \sum_{i=1}^n r_i \cos(m_i \theta) \,,$$ where $n \geq 1$ is an integer, the $r_i$ are positive real numbers and the $m_i$ are integers. Is there ...
user avatar
  • 109
0 votes
2 answers
32 views

Fourier series of a non-periodic function possible?

I have read online that Fourier series are only applicable to periodic functions. Ive included the following example in my work however it is not a periodic function, does this mean that the following ...
user avatar
-1 votes
0 answers
43 views

The best approximation for $L^2$ is the partial sum of a fourier series?

Looking over my notes, I've found the following lemma Define $f_n$ as the $n^{th}$ partial sum of the Fourier series of $f$ and let $g_n$ be an arbitrary trigonometric approximation to $f$. Consider $...
user avatar
1 vote
1 answer
39 views

Find the fourier coefficient of given function

I'ven been reading Dimitris Koukoulopouls' analytic number theory book. I'm stuck at p.63, the following part. Let $f : \mathbb{R} \to \mathbb{C}$ be twice-continuously differentiable function of $\...
user avatar
  • 1,057
0 votes
0 answers
41 views

The first part of the proof of Szegő's theorem, using Fourier series. Why $\hat f(0)=e^{\frac{\hat g(0)}{2}}, \hat{1/f}(0)=e^{-\frac{\hat g(0)}{2}}$?

I'm reading Szegő's theorem in P$189$ of "Fourier series and Integrals" by Dym, Mc kean. I have difficulty in understanding the first part of this. The proof begin from P$190$. Let $\Delta \...
user avatar
  • 2,251
0 votes
0 answers
10 views

Fourier transform on a lattice

I am trying to prove an identity that ensures consistency of the discrete Fourier transform. Namely, to show that $\sum_{k}^{ }︁e^{ikn}=Nδ_{n0}=\left\{N,\ n...N;\ 0\ -otherwise\ \right\}$ $\sum_{k}^{ }...
user avatar
1 vote
1 answer
18 views

Fourier decomposition by solving an infinite order linear ODE

I am sorry for not being rigorous in asking this question as I lack knowledge in several of the concerned areas. Under some conditions, the exponential of the scaled derivative operator $D$ is the ...
user avatar
  • 11
1 vote
0 answers
29 views

Inequality about Fourier series

I am doing a question below Let $f\in C^{1},~2\pi~periodic~and~f(x_{0})=0~for~some~x_{0}\in~[-\pi,\pi]~and$ $$\frac{1}{2\pi}\int_{\pi}^{\pi}|f(x)|^{2}dx=1$$ then show $$(\sum n^{2}|\hat{f}(n)|^{2})\...
user avatar
  • 111
4 votes
0 answers
86 views

$f$ continuous with summable Fourier coefficients $\implies$ Fourier series converges to $f$?

Suppose that $f$ is continuous and has (absolute) summable Fourier coefficients: $$ \sum_{k \in \mathbb{Z}}\vert c_k \vert < \infty. $$ Is it then true that the Fourier series for $f$ converges to ...
user avatar
  • 691
0 votes
0 answers
22 views

Why does Fourier transform gives the correct amplitudes?

For instance we would like to get the Fourier transform of $A\cdot cos(2\pi fx)$. At some point, when we find the frequency $f$, we arrive to the following integral. $$\int_{-T}^{T}A\cdot cos^2(2\pi ...
user avatar
0 votes
0 answers
21 views

Fourier Transform of a periodic function and its Fourier series

Function working on: $$f(t) = \sum_{k = -\infty}^{\infty}\delta(t - KT_{s})$$ I found the Fourier transform via the following 2 method: 1.Directly finding Fourier Transform: $$F(e^{j\omega}) = \int_{-\...
user avatar
0 votes
0 answers
36 views

Matching Coefficients of Fourier Series with Separation of Variables Solution for Discontinuous Boundary Conditions: 2D Slab Conduction

I am trying to find the temperature profile in a 2D domain with steady heat conduction. The non-dimensional domain is shown below. Domain dimensions, coordinate system, boundary conditions, and ...
user avatar
4 votes
1 answer
73 views

Multi-dimensional Dirichlet-Dini criterion for Fourier series

Let $\mathbb I^d$ be the $d$-dimensional unit cube, and $f\in L^1(\mathbb I^d)$. Further let $x\in\mathbb I^d$ and assume that (some representative of) $f$ is differentiable at $x=(x_1,\dotsc, x_d)$ (...
user avatar
  • 1,257
2 votes
2 answers
74 views

Finding Fourier coefficients $a_n$ of $\ln|\sin(\frac{x}{2})|$ [duplicate]

$\ln|\sin(\frac{x}{2})| = -\ln2 + \sum_{n=1}^{\infty} \frac{\cos{nx}}{n}$ where $x\neq2k\pi$ for any integer k. I'd like to find $a_n$ and here is my stuck. \begin{align*} \frac{1}{\pi}\int_{-\pi}^{\...
user avatar
  • 21
-1 votes
0 answers
17 views

The first five terms to the Fourier series.

question Hi, I try to find that the first five terms to the Fourier series. Could you help me?
user avatar
3 votes
0 answers
33 views

Closed formula for variation of Fourier series of Bernoulli polynomials

The Fourier series for the periodic Bernoulli polynomials $$ \sum_{k \in \mathbb{Z}-\{0\}} \frac{e^{2\pi ikx}}{k^n} = - \frac{(2\pi i)^n}{n!} P_n(x), \hspace{0.5cm} n \geq 1 $$ is well known. I am ...
user avatar
  • 825
1 vote
1 answer
111 views

Evaluating a limit of a Fourier series

I have a Fourier series representation of a solution to the heat equation, given by $\displaystyle u(t,x) = \\ \displaystyle\sin \omega t + \sum_{n = 1}^{\infty}{\frac{4( - 1)^{n}}{(2n - 1)\pi}\omega\...
user avatar
-4 votes
1 answer
43 views

Plotting Square Wave Motion with a Period of $2\pi$ [closed]

How to Plot a square wave motion with a period of $2\pi$ for the interval $-2\pi +2 \pi$? $x(t) = \begin{cases} +3, & \text{0 $<t\leq\frac{\pi}{2}$ } \\ \\ -3, & \text{$\frac{-\pi}{2} <...
user avatar
  • 1
0 votes
0 answers
15 views

Fourier coefficients restriction

I have a function $p(t)=sin(t/k)$ in the interval $0<t<2\pi k$ and $p(t)=0$ for $2\pi k<t<T$ And i have to figure out what restrictions the fourier coefficients have. I understand that ...
user avatar
0 votes
0 answers
21 views

Determine where function is discontinuous by its fourier coefficients

The fourier coefficients of ${f} \in C_{PW}^0$ are given by $\widehat f(n)=(-i)^n(\frac{3i}{4n}+\frac{1}{2\pi n^2}) $ for $n \ne 0$. Find where $f$ is discontinuous and what is its jump there? it's ...
user avatar
  • 31
0 votes
1 answer
63 views

An easy version of Riemann-Lebesgue Lemma on $L^2([-\pi,\pi])$ [closed]

Question Let $\left\{{\varepsilon}_{k}\right\}$ be an orthonormal set (it may be complete or not) in a Hilbert space $H$. Explain why for any $x \in H^{ \perp}$ $$ \lim _{k \rightarrow \infty}\left\...
user avatar
3 votes
1 answer
54 views

Representation of $\sum_{k=1}^{\infty} \frac{f^{(k)}(a)}{k!} k^{-p} $ in terms of another series

I face a problem when I'm dealing with one of the transforms and the problem is as follows: Suppose that we have $$ f(a+z) = \sum_{k=0}^{\infty} \frac{f^{(k)}(a)}{k!}z^{k} \tag{1} $$ $$ f\left(a+e^{\...
user avatar

1
2 3 4 5
104