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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Writing a function $f : [-\pi,\pi) \to \mathbb{R}$ as $\sum c_k e^{ikx}$ where $c_k$ is to be found

I have a function on $[-\pi, \pi)$ defined as: $$ f(x) = \begin{cases} -1 & \mbox{if} \;x \in [-\pi,0) \\ 1 & \mbox{if} \;x \in [0,\pi) \\ \end{cases} $$ And I have to write it in the form ...
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2answers
31 views

'Obtain' the Fourier transform

If $g(t) = e^{-a|t|}$ and a is a real positive constant, obtain the fourier transform. I'm a bit unsure what this is asking. I can write out the expression for the fourier transform. Should I stop ...
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7 views

Using the Fourier Series in Variational Optimization Problems

Say I have a functional $L(f)$ which takes as input the function $f:\mathbb{R}\to\mathbb{R}$, and I want to find the function that optimizes $L$. Unfortunately, there's no way to define a functional ...
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1answer
26 views

Odd and Even Fourier Series Extension of $f(x)=x$ on $[0,\pi]$

I'm confused on finding the odd and even extensions of $f(x) = x$ on $[0,\pi]$. I know the general forms and how to find the co-efficients, but for the sin series, $f(0)$ =/= $f(\pi)$, so then I only ...
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1answer
53 views

Challenge in trignometry and integration [on hold]

Can anyone prove how the two equations are equal? Thanks $$=\frac1\pi \int_0^{2\pi} f(x) \left\{\frac12+\sum_{n=1}^N \cos [n(t-x)] \right\} \, dx$$ $$=\frac1{2\pi} \int_0^{2\pi} f(x) ...
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31 views

Can a sum of trigonometric functions equal a constant for all inputs?

Let $r_1,...,r_n$ and $\phi_1,...\phi_n$ be real numbers. Consider the following sum: $S=\sum\limits_{k=1}^{n}r_k\sin(\phi_k+k\alpha)$ Suppose $S$ is constant for all $\alpha \in R$. Does it ...
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Show that $ \lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}e^{ik^2}=0$

TL;DR : The question is how do I show that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{ik^2}=0$ ? More generaly the question would be : given an increasing sequence of integers ...
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1answer
23 views

Apply Periodic Boundary to PDE (Fourier Transform)

Use Fourier Transform to solve the BVP: \begin{cases} u_t + a u_x - b u_{xx} = 0, & \mbox{for } x \in [-1,1] \\ u(x,0) = f(x) \\ u(x+2,t) = u(x,t) \end{cases} I solved the problem (attached); ...
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17 views

Agile method to find Fourier coefficients [on hold]

Is there a way to calculate Fourier serie coefficients which doesn't pass through an integral?
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642 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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Calculating Fast Fourier Transform from given set of data

I am trying to calculate the Fast Fourier Transform numerically from the given data : Given: f0 f1 f2 f3 f4 f5 f6 f7 1 2 3 4 4 3 2 1 I have to find the ...
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28 views

Fourier sine and cosine: reconstruction depends on 'noise data' outside signal

I am working in strain analysis. Strain in a mechanical testing machine is captured by strain gages. Signals are like the slim line in the graph below showing strain versus time. The data are of the ...
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31 views

How to approach solving this Fourier series [closed]

$$f(x):=\frac{1}{e^{2+\cos x}-1}$$ Source. Hi. I need to find Fourier series for this function. This is even function so Fourier coefficient $b_n$ is 0. Basically I need to solve this integral ...
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2answers
21 views

Harmonic Motion - Fourier Approximation What does this mean below?

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ...
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13 views

Sawtooth wave as a sum of sines

Wikipedia gives the equation for a sawtooth waveform composed as a sum of sines as: $$ x_\mathrm{sawtooth}(t) = \frac{A}{2}-\frac {A}{\pi}\sum_{k=1}^{\infty}\frac {\sin (2\pi kft)}{k} $$ Where $A$ ...
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1answer
5k views

Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft

Rebuilding original signal from frequencies, amplitude, and phase obtained after doing an fft. Greetings I'm trying to rebuild a signal from the frequency, amplitude, and phase obtained after I do ...
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2answers
28 views

Fourier series and convolution

Let $f$ and $g$ be $2\pi$-periodic, piece-wise smooth functions having Fourier series $f(x)=\sum_n\alpha_ne^{inx}$ and $g(x)=\sum_n\beta_ne^{inx}$, and define the convolution of $f$ and $g$ to be ...
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1answer
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Is the complex form of the Fourier series of a real function supposed to be real?

The question said to plot the $2\pi$ periodic extension of $f(x)=e^{-x/3}$, and find the complex form of the Fourier series for $f$. My work: ...
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17 views

What is an angle in fractional fourier transforms?

I would like to know the geometrical interpretation of an angle in fractional Fourier transforms. Is this a rotation of time-frequency plane or rotation of the signal?
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1answer
32 views

Fourier cosine series giving nonsense answer

I'm currently trying to find the cosine Fourier series of $f(x) = \left | \sin \frac{\pi n }{L} x\right |$ on the interval $0 < x < L$. I first started by calculating the first term of the ...
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39 views

Solution of boundary value problem using Fourier series

I want to solve the following PDE using Fourier series. $u(x,y): \Omega \to \mathbb{R}$, $\Omega=(0,\pi)\times (0,2\pi)$ $u-3u_{xx}-u_{yy}= 3\sin(2x)-\sin(5x)$ $u_{xx}$ and $u_{yy}$ are second ...
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1answer
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Problem calcualting Fourier coeff. of tent function.

Consider the tent-function on $[-\pi,\pi]$ depending on some $\delta$. I.e $(1-\frac{\mid x \mid}{\delta})$, $x$ is zero when larger then $\delta$ When I compute ...
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Example of multidimensional Fourier transform

Please, take the function, for example $\sin(xy)+\cos(xz)$ from dimension 3 to $R$, and give me a multidimensional Fourier transform for it. I'll be also thankful for general multidimensional Fourier ...
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1answer
27 views

Fourier Coefficient

I have to compute the coefficient $b_3$ of the odd Fourier Series associated with the function $y=2-x$ in the interval $(0,1)$, period $2$. By using the formula $$ b_k = \frac{1}{T}\int_{-T}^{T} ...
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29 views

Moving limits inside inside an infinite sum, a special case.

I have come over a problem where I have found that this expression is most likely equal to a square wave with period 4 and phase shift 1. $$ f(t) = \lim_{n \rightarrow \infty}\sum_{k=1}^n ...
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1answer
15 views

Compute the Fourier Series of a trig function

I want to compute the Fourier series for the following function $$ g_n(\theta) = -2nK_{n}(\theta)\sin(n\theta)$$ where $K_n(\theta)$ is the Fejer Kernel. I tried to compute the Fourier coefficients ...
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1answer
44 views

fourier transform of $f(x) = x^2+\frac{1}{1+2x^4}$

I really have no thought on this. I can't seem to use residue thm., nor could I find a inverse transform for it. by some Fourier Calculator I know it's solvable but how?
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20 views

Resolvent of the operator

Consider the Laplace operator defined on the biggest possible subset of$L^{2}(R^{2})$: $T= - \partial^{2}_{x} -\partial^{2}_{y}+x^{2}+y^{2}+ 2.i(x \frac{\partial}{\partial ...
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Fourier Polynomials: standardly used term?

When teaching Fourier series to students, I realized that one of my references (only one or two I know that does this) calls the $n$-th partial sum of the Fourier series of an $L^2$ function $f$, the ...
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“Fourier” subsets of a complete basis

If we have some complete basis where the basis functions have a finite bandwidth in fourier space, and we are interested in reproducing a function with a finite bandwidth, we know that there is some ...
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24 views

How to transform an even function into an odd one?

Expand $$ x(t) =\begin{cases} t,& 0 < t < \pi/2 \\ \pi - t,& \pi / 2 < t < \pi \end{cases}$$ in Fourier sine series. First, $x(t)$ needs an horizontal translation into an ...
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Fourier series - Why does $\hat f(0) \ne 0$?

Let $f\in C^1$, $2\pi$-periodic, and let's assume $\int_{-\pi}^\pi |f'|^2 \le 1$. Prove: $$\sum_{n\in\mathbb{Z}} |\hat f(n)|^2 \le \frac{1}{2\pi}$$ There's a $c\in\mathbb{C}$ such that: ...
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Understanding Fourier transform example in Matlab

I'm studying about Fourier series and transform and I get confused with the following Matlab example of Fourier transformation: ...
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89 views

Show that $f_n\to f$ uniformly on $\mathbb{R}$

Let $$P_n(x) = \frac{n}{1+n^2x^2}$$. First, I had to prove that $$\int_{-\infty}^\infty P_n(x)\ dx = \pi$$ And that for any $\delta > 0$: $$\lim_{n\to\infty} \int_\delta^\infty P_n(x)\ dx = ...
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2answers
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Why does convolution of delta function commute (test distribution perspective)?

If I understand correctly, for test functions $f$ we define $$ \langle\delta, f\rangle = f(0)$$ and convolution is defined as $$ \langle g * T, f\rangle = \langle T, g^- * f\rangle,$$ where $f$ ...
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1answer
98 views

Fourier series of $\sin(x)$

I know that this series has been calculated here for more then one time but I need help with a specific thing. We define $f$ as an even function with period $2 \pi$ by $f(x)=\sin (x) $ where $0 \leq ...
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2answers
51 views

Show that $f(x)\equiv 0$.

Let $f:[0,2\pi]\to\mathbb{R}$, which is $2\pi$ periodic and continuous. It is given that for every $n\in\mathbb{Z}$:$$\int_0^{2\pi} f(x)e^{i\left(n+\frac{1}{2}\right)x} = 0.$$ Show that $f(x)\equiv ...
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Characterize a set of functions

While computing matrix elements of the evolution operator in Quantum Field Theory for the harmonic oscillator using the path integral formalism, I came across the assumption that all physically ...
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37 views

Good reference for Fourier Analysis

Would you please indicate a good reference about Fourier analysis (Fourier series, convergence theorems: pointwise, uniform convergence, $L^2-$convergence...etc)? It should concern the organisation of ...
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Finding the zeroth Fourier coefficient using limit

The $\text{n:th}$ Fourier coefficient (for the $\cos(nx)$ part) is defined by $$a_n = \frac{1}{\pi} \int_{-\pi}^{\pi} f(\theta) \cos(n\theta)d\theta.$$ Inserting $n=0$, we get $$a_0 = \frac{1}{\pi} ...
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1answer
37 views

Which way does the Fourier Transform go?

This might be a silly question, but I'm really confused by the way Fourier Transform was taught in my algorithms class, and everything else I found on the internet. The way we defined FT is first ...
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1answer
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Double integral calculation and fourier transform

I try to find the following $$\int_{\mathbb{R}}^{}\int_{\mathbb{R}}^{} e^{-y(x+z)-(x^2+z^2)} dxdz$$ and I change variables $x=r\cos(\theta)$ and $z=r\sin(\theta)$ and the integral becomes: ...
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Find a function $u(x,t)$ satisfying some initial conditions for a vibrating string of length $\pi$.

Solve the following problem for a vibrating string of length $π$: Find a function $u(x, t), 0 ≤ x ≤ π, t ≥ 0$, satisfying $∂^2u/dt^2 = ∂^2u/dx^2, 0 < x < π, t > 0$ the boundary conditions ...
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Applying Fourier transform to equation

I would like to know why we divide the $(i*ω)^2$ to the equation. When I asked my supervisor, he said I need to learn Fourier Transform: more specifically I need to understand relationship between ...
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19 views

Matrix representation of nonlinear functions

Let $\tau : [0,1]\rightarrow [0,1] $ be a continuous invertible map. Then the 'extension of $\tau$ to the space of square integrable real valued functions on $[0,1]$ is defined by the linear operator ...
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Coefficients of Fourier-Bessel series for a Neumann condition

What is the expression for coefficients of Fourier-Bessel series for a Neumann condition? I know what it is for Dirichlet condition. $\frac{\partial f}{\partial x} = 0$
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21 views

Expanding a formula

We have the below formula $$\int_{-\pi}^{\pi}s_n^2 (x) dx =\int_{-\pi}^{\pi}\left[\frac{a_0}{2}+\sum _{k=1}^n a_k \cos (k x)+b_k \sin (k x)\right]^2 dx,$$ using the aforementioned formula, how (from ...
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367 views

Can we express all doubly periodic functions as one of doubly periodic function?

Singly Periodic Functions $e^{x},\cos(x),\sin(x),\tan(x), .. etc.$ Euler's identity is $$e^{i\alpha}=\cos(\alpha)+i\sin(\alpha)$$ $$e^{-i\alpha}=\cos(\alpha)-i\sin(\alpha)$$ Thus, we can express ...
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2answers
59 views

Evaluating the Fourier coefficients of $abs(x)$

Let's get started: $$\hat f(n) = \frac{1}{2\pi}\int_0^{2\pi} |x|e^{-inx} dx$$ since $|x|$ is an even function: $$= \frac{1}{\pi}\int_0^{\pi} xe^{-inx} dx$$ Integration by parts yields: ...
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Proving Inverse DFT

I have trouble understanding the proof I was provided of the IDFT, here is what I have: $$ \nu_n = \frac{n}{\Delta N} \\ x(t) = \int_{-\infty}^{\infty}X(\nu)e^{i2\pi\nu_n t}d\nu \\ $$ the next step I ...