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.

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Green's function of Harmonic Oscillator using Fourier modes

First off, I know this is similar to an already answered question concerning the Green's function of a harmonic oscillator. I wanted to ask a question there in the comments, but couldn't due to ...
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1answer
24 views

Fourier series-odd and even functions

f+ is the even part of the function and f- is the odd part. I'm not able to understand how it is that they got the values of modulus of x and x for the even and odd parts of the function ...
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4answers
152 views

A Fourier Analysis Question I am stuck at

If $f,g\in C[-\pi,\pi]$,and $f,g$ are $2\pi$ periodic, prove that $$\lim_{n\to\infty}\dfrac{1}{2\pi}\int_{-\pi}^\pi f(t)g(nt)\mathbb dt=\big(\dfrac{1}{2\pi}\int_{-\pi}^\pi f(t)\mathbb ...
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0answers
23 views

How can I make the mean of samples be approximately equal to the mean of actual continuous signal?

Suppose there is signal f(t) that is continuous and periodic. It is known that this f is T-periodic. (but it's not necessarily a single cosine f(t).( I'd like to make the mean of samples be ...
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2answers
33 views

I don't understand the relation.

$$e^ix - 1 =e^{ix/2}* 2i * sin({x/2}) $$ I don't understand why that is true, but I do know the relation $$sin (x) = \frac {e^{ix} - e^{-ix}}{2i}$$ However I don't see where the 1 came from
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1answer
22 views

Frequency scaling property for Fourier series

For Fourier transform, there is an equation connecting time-scaling with frequency-scaling. (By scaling, I mean multiplying by constant for time or frequency) Is there such a relation for Fourier ...
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5answers
100 views

Can anyone suggest a book on Fourier Analysis containing many good problems

I am taking a basic course in Fourier Analysis in my undergrad Analysis class and I know the theory and related theorems. However, this is a relatively new zone for me and I would like a book that ...
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0answers
11 views

Fourier Amplitude Sensitivity Test (FAST)

I am new in the domain of sensitivity analysis, I am trying to investigate the global sensitivity analysis method FAST (Fourier Amplitude SEnsitivity testing). I read alot about this subject, starting ...
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2answers
75 views

Solving a PDE by Fourier Series

I want to solve the following PDE: $$\begin{cases} u_t=u_{xx}+1\\ u_x(0,t)=0, \quad u(1,t)=0\\ u(x,0)=\cos\left(\frac{\pi}{2}x\right) \end{cases}$$ using a Fourier series. The thing that is throwing ...
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1answer
44 views

Find the Fourier series representation of $f(t)=\sin(3\pi t)$

Find the Fourier series representation of $$f(x)=\sin(3\pi t)\qquad \text{for }-1\leq t\leq1$$ When I calculate the coefficients, I always get $0$. Why is that? Is the series indeed zero?
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1answer
68 views

Number of zeros of a periodic function

Let's consider a periodic real function of a real variable $f(x)$. If the function is analytical and it is not the zero function, can one infer that the number of zeros in one period $[x,x+P)$ is ...
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22 views

Is there anything similar to DTFT for Fourier series?

So if sampling condition is met well, with aperiodic signals we have discrete-time Fourier transform (DTFT) that allows us to get frequency-domain data that resemble continuous-time fourier transform. ...
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2answers
42 views

Fourier integral problem?

Show that $$ \int_0^{\infty} \frac{\sin \pi \omega \sin x\omega}{1-\omega^2}d\omega= \begin{cases} \frac{\pi}{2}\sin x,&\mbox{ if } 0\leq x\leq\pi\\ \quad\\ 0,&\mbox{ if } x\geq\pi ...
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1answer
40 views

What is a window function with positive spectrum?

I need a real, symmetric window function $x(t) = x(-t)$ whose Fourier transform $\hat{x}(\omega)$ (also real and symmetric) is non-negative $\hat{x}(\omega) \ge 0$ for all $\omega$. The function does ...
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1answer
23 views

Is the DTFT of a sampled Gaussian a positive function?

I have an infinite sequence $x_{n}$ for $n \in \mathcal{Z}$ which is a sampled Gaussian function $x_{n} = \exp(-n^2/a)$ with a > 0. I need to check whether its DTFT $x(\theta) = \sum_{n \in ...
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1answer
35 views

Find the Fourier Coefficients that minimize the error [duplicate]

I know that the coefficients that minimize the expression are the ones that make it's derivative 0. I have also expanded the whole expression and taken it's derivative, but still I can't figure out ...
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1answer
39 views

differentiation and integration of Fourier series.

If I have the fourier series of $|x|$ for $-l < x < l$ and I make it periodic with period $2l$ I get a cos series: $$ \frac{l}{2} ...
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0answers
9 views

Result obtained on deletion of finite number of Fourier Coefficients

I want to know the answer to the following question. If a finite (but fixed) number of Fourier coefficients (of any choice) of a Fourier series are made $0$, then will the new series be a Fourier ...
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2answers
24 views

DTFT and its convergence

In the textbook "signals and systems", by prof. Simon Haykin, it says:   If $x[n]$ is not absolutely summable, but does satisfy square summable, then it can be shown that the following equation ...
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2answers
26 views

Find complex Fourier coefficients of $f(-x), f^*(x)$

For $f(-x)$ i have tried to replace the $k$ with $k'=-k$ but still i can't find any relationship between the coefficients. What could be a better way to approach this problem?
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1answer
31 views

Find the coefficients of the Fourier series that minimise the error.

I am having a little trouble understanding what I have to actually do here. What does differentiate with respect to bn? I thinks after differentiation I must use some calculus theorem about extreme ...
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0answers
44 views

Why do sines and cosines form a basis, and can be considered a vector space?

Many times I've seen that Fourier series are justified because we are thinking that the set of all functions of the form $sin(ax)$ and $cos(ax)$ form a vector space. A function can therefore be ...
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1answer
33 views

Fourier series solution of the heat equation on $-2<x<2$

I have to solve the following boundary value problem: $u_t=u_{xx}$, $u(t,-2)=u(t,2)=0$ and $u(0,x)=f(x)$. I tried to solve the problem using the method of separation of variables. So assume ...
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1answer
17 views

Finding the value of a series using a known Fourier series

We are given the function $$f(x)=\begin{cases}1&\text{for }-\dfrac{\pi}{2}<x<\dfrac{\pi}{2}\\ 0&\text{for }\dfrac{\pi}{2}<x<\dfrac{3\pi}{2}\end{cases}$$ which I have found to have ...
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1answer
52 views

Fourier transform and splitting frequency range into 4 channels

I have code example that divides audio frequency into 6 channels. It uses Fast Fourier Transform (FFT). Algorithm process the frequency range using 6 capture[x] samples based on the range of n between ...
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1answer
54 views

Prove that $\{\sin x, \sin 2x, … , \sin nx\}$ is a linearly independent set

Prove that $\{\sin x, \sin 2x, ... , \sin nx\}$ is linearly independent. The short solution that I do not understand is as follow: For p and q are positive integer, we have $$ ...
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2answers
239 views

Why can the equality sign be used for Fourier series expansion of a discontinuous function?

Many of the Fourier series problems I deal with right now are with discontinuous functions. Many times the integrals involved have to be separated because there are discontinuities. However this is ...
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1answer
42 views

Fourier series convergence question from big Rudin.

I am working on some problems from the 3rd edition of Rudin's "Real and Complex Analysis" and I'm stumped on proving the following part from question #19 of chapter 5. Suppose $\lambda_n/\log n \to ...
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0answers
19 views

Exponential form of Fourier Series,

For a function of period $2L$ the exponential form of the fourier series is defined above. Why however is $|x|<L$ as opposed to $|x| \leq L$?
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1answer
25 views

Is it possible to use a fourier series to make a sin wave with a wave length that is not in the fourier series?

This may seem backwards since a fourier series isn't typically used this way but I'm trying to prove whether or not the sum of sin and cos waves could produce a sin wave with a wave length that is not ...
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1answer
49 views

Fourier Series Expansion, error in coefficients?

After reworking the problem many times I keep getting the same (incorrect?) answer. So the problem as stated is Find the Fourier expansion of : $$ f(x) = \begin{cases} x &\text{ if }0 ...
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1answer
29 views

fourier series sketching (by hand)

I calculated the Fourier Series representation of $f (x) = 1 − |x|$ on $−1 ≤ x ≤ 1$ and now I am asked to sketch the graph of the series on $−3 ≤ x ≤ 3$ by hand. How do I do this? I read through my ...
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1answer
57 views

Solve $\int_{0}^{2\pi} f(t) \sin ^2 (t-\theta) dt = g(\theta)$ for unknown function $f$

Let $g(\theta)$ be a known real-valued function with domain $[0, 2\pi]$. Given that: $$\int_{0}^{2\pi} f(t) \sin ^2 (t-\theta) dt = g(\theta)$$ How would I solve for the unknown real-valued function ...
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1answer
22 views

Quick Fourier Series Question about Cn Integration

If I am given a function $$ f(x) = \left\{ \begin{array}{ll} 2 & \quad x \in (0,6) \\0 & \quad x\in(0,-6) \end{array} \right. $$ $I=(-6,6)$ and I want to ...
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1answer
88 views

Bessel's Inequality simplification

Let $f:[-\pi,\pi] \to \mathbb{R}$ be a piecewise smooth function with $\int_{-\pi}^{\pi}f(x)dx = 0$. Does anyone have ideas on how to apply Bessel's inequality to show that $\int_{-\pi}^{\pi} ...
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0answers
34 views

Fourier Series and Uniform Convergence

This question is an extension of Fourier series simplification I know wish to show $$(\frac{1}{\pi})\int_{-\pi}^{\pi}f^2(x) dx = a_{0}^{2}/2 + \sum_{n=1}^{\infty} (a_{n}^{2} + b_{n}^{2})$$ But, we no ...
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2answers
37 views

Using Complex Fourier Series to Find Real Coefficients

I am about to go insane with this problem, so I really hope some kind, kind soul out there can help me. I am trying to find the complex Fourier series of the following function and interval, and then ...
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3answers
80 views

Fourier series simplification

I want to show that $$\frac{1}{\pi} \int_{-\pi}^{\pi} f(x)g(x)dx = \frac{a_0\alpha_0}{2} + \sum_{n=1}^{\infty} (a_n\alpha_n + b_n\beta_n)$$ where $f,g: [-\pi,\pi] \to \mathbb{R}$ are integral ...
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1answer
75 views

Series of exponential function

I had a thought today and I've tried to see if it is a thing. I'm certain it is a thing, I just don't know how to search for it. We have the Taylor series which is a summation of monomials: ...
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1answer
20 views

Why do we write the first term of the Fourier cosine series as $c_{0}/2$ instead of simply $c_0$?

The Fourier cosine series of some function $f(x)$ defined over the interval $[0, L]$is written as: $$f(x) = \sum_{k = 0}^{\infty} c_k\cos(\frac{k\pi}{L} x)$$ Where $c_k$ can be determined by the ...
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1answer
73 views

Writing a partial sum of Fourier series as an integral

Show that the partial sum in equation (3) may be written as:$$f_N(x)=\frac{2}{\pi}\int_{0}^x \frac{\sin(2Nt)}{\sin(t)}\,dt$$ Can someone please explain me how to show these 2 are equal? The first ...
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2answers
45 views

Fourier series question

I am just a beginner in Fourier series.How should I get start to tackle this question and show the partial sum has extrema? I have no clue to this question. Any help would be highly appreciated.
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1answer
73 views

Gibbs Phenomenon and Fourier Series

a) Show the partial sum $$S = \frac{4}{\pi} \sum_{n=1}^N \frac{\sin((2n-1)t)}{2n-1}$$ which may also be written as $$ \frac{2}{\pi}\int_0^x\frac{\sin(2Nt)}{\sin(t)}dt$$ has extrema at $x= ...
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0answers
3 views

Estimating certain singular discrete sums

I want to estimate sums of the following form: $S^d(\alpha,\beta,l):= \sum_{k \in \mathbb{Z}^d, k \notin \{0,l\}} \frac{1}{|k|^\alpha} \cdot \frac{1}{|k-l|^\beta}$, where $l \in \mathbb{Z}^d$ and ...
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0answers
25 views

Double Fourier series for inhomogeneous BC

So the task is, that the following 2D eigenvalue problem on a unit square is given. \begin{equation} -\nabla^2M(x,y)=\lambda M(x,y),\quad 0<x<1,0<y<1\\ M(x,y)=0\quad \text{on the boundary ...
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1answer
16 views

Discrete Fourier Transform real f_j's

Could you help me show that if $$\hat{f}(k)=\frac{1}{N}\sum\limits_{j=0}^{N-1}f_j \exp\left(-i\frac{2\pi jk}{N}\right)$$ (k=0,1,...,N-1) is the Discrete Fourier Transform of $f_0, f_1,\ldots, ...
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0answers
22 views

Function approximation by various means

I know several ways to approximate a function: Taylor series, Fourier series, or polynomials, like e.g. Legendre polynomials. Is the only difference between those various methods the speed at which ...
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1answer
52 views

Partial Sum Fourier Series

Show that the partial sum $$f_N(x)=\frac{4}{π}\sum^N_{n=1}\frac{\sin((2n-1)x)}{2n-1}$$ may be written as $$f_N(x)=\frac{2}{π}\int_0^x\frac{\sin(2Nt)}{\sin(t)}\,dt$$ The original question is 'Sketch ...
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1answer
90 views

Fourier sine series expansion

The function $f(x)$ is defined as $$f(x)=1\qquad0<x<\pi$$ Sketch the odd extension and show that the Fourier sine series expansion is ...
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0answers
54 views

Proving Gibbs phenomenon using Dirichlet kernel

I am working on a problem$^{(1)}$ on using Dirichlet kernel to prove Gibbs phenomenon. It is a long proof broken down into 7 steps, and on each step I have to answer some questions. Long story short, ...