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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2answers
46 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
11 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
30 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
14 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 ...
-1
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0answers
10 views

Fourier transform for $F_C(\frac{1}{1+x^2})$ and $F(\frac{1}{1+x^2})$? [on hold]

What is Fourier transform for $$F_C(\frac{1}{1+x^2})$$ and $$F(\frac{1}{1+x^2})$$? Help: Use the duality theorem Duality Theorem: If $ \ phi (w) $ Fourier transform $ f (x) $ , then $ f (w) ...
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5answers
64 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
7 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 ...
3
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2answers
52 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 ...
-1
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1answer
27 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?
2
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1answer
57 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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0answers
18 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. ...
2
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2answers
37 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 ...
0
votes
1answer
32 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 ...
2
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1answer
19 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 ...
1
vote
1answer
24 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 ...
0
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0answers
19 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
8 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 ...
0
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2answers
16 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 ...
0
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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?
1
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1answer
27 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
19 views

Fourier coefficients [closed]

Could you please tell me what is a general way to approach these kind of problems and maybe solve one of the above examples just for clarification?
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0answers
40 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 ...
0
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1answer
30 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 ...
0
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1answer
16 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
47 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 ...
3
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1answer
50 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 $$ ...
2
votes
2answers
222 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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0answers
27 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 ...
0
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0answers
16 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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0answers
70 views

Fourier theorem proof

How do you prove the completeness of the trigonometric system? More precisely, how do you show that the trigonometric system ($\cos(nx)$, $\sin(nx)$, $\forall\; n\in \mathbb{N}_0$) provides a basis ...
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1answer
24 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
37 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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0answers
6 views

Relation between DFT and fourier transform on tempered distributions

Since tempered distributions can well capture the idea of discrete data, via sums of Dirac deltas, I wonder if there is a way to think of DFT as just the restriction of the analytic fourier transform ...
1
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1answer
23 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
53 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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0answers
27 views

Fourier problem with method of separation of variables

I'm trying to solve the following set of equations, using the method of separation of variables commonly found in textbooks on Fourier analysis (here's a sample) $u_{tt} = 3u_{xx}$ $u(0,t) = 0, ...
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1answer
21 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 ...
2
votes
1answer
83 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} ...
1
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0answers
28 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
33 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 ...
2
votes
3answers
72 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 ...
1
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1answer
56 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 ...
2
votes
1answer
70 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
35 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
65 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
20 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 ...