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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1answer
35 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
22 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
31 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
33 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
22 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
28 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
43 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
32 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
49 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
52 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
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2answers
230 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
39 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
48 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
25 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
54 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
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 ...
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1answer
86 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
31 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
34 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
75 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
66 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
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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
42 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
67 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
23 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 ...
0
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1answer
47 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
85 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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52 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, ...
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0answers
33 views

Does the sum $\sum_{n=1}^{\infty}{a_nb_n}$ converge(fourier series coefficients)?

Let $f\in H(0,2\pi)$, with inner product $<f,g>=\int_0^{2\pi}{f(t)g(t)dt}$ $S_f=a_0 + \sum_{n=1}^{\infty}{a_ncos(nx)}+\sum_{n=1}^{\infty}{b_nsin(nx)}$, is the fourier series for f. Where ...
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2answers
58 views

Fourier Series Representation $e^{ax}$

a) Compute the full Fourier series representation of $f(x) = e^{ax}, −π ≤ x < π.$ b) By using the result of a) or otherwise determine the full Fourier series expansion for the function ...
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0answers
29 views

Do three valued basis vector elements lead to the fastest discrete Fourier transforms?

When sin() and cos() are approximated to 1, 0 and -1 in the basis vectors in a real or discrete Fourier transform the basis vectors have a lot of elements of zero or in common leading to an algorithm ...
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0answers
9 views

Find distortion exponent from Fourier fitting

I'm facing this problem in my master thesis: we are measuring the signal from a sensor which is, physically, a $\sin^2$ (or $\cos^2$). Some non idealities distort the signal by introducing an exponent ...
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1answer
25 views

Why are discrete-time Fourier series and discrete Fourier transform only defined on integer $k$?

In ordinary Fourier series/transform of a continuous signal $f(t)$, fourier frequencies $\omega$ of series/transforms can be any of $\mathbb{C}$, not just $\mathbb{Z}$. But why is it the case that ...
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2answers
27 views

Test question regarding convergence of Fourier series

I'm preparing for a test and I have no clue how I should solve the following question. Let $f:\Bbb{R}\to\Bbb{R}$ be $2\pi \text{-periodic}$ function such that $f(0)=1$ and $$\forall ...
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2answers
41 views

How to prove this simple fact without using distribution theory?

Suppose function $f(x) $ is normalized to unity, i.e., $$ \int dx |f(x)|^2 =1 . $$ Now consider the Fourier transform of $f$, i.e., $$ F(k) = \int d x f(x) e^{-i k x} . $$ Here we assume that $f $ ...
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0answers
23 views

DPE problem invlolving Fourier transforms / partial eq.

Don't even know where to start with this question! would really appreciate some guidance.
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1answer
25 views

Fourier series coefficients which do not approach to zero

I want to know whether there are a finite number of coefficients in a Fourier series of a periodic function (with period $P$), whose magnitude are above a certain threshold. Those coefficients can can ...
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1answer
45 views

Fourier Series estimation

I know that the Fourier coefficient of $t\mapsto \frac{1}{\sqrt{\vert t\vert}}$ are given by some Fresnel integral, and behave like $O(n^\frac{-1}{2})$. Reciprocally, if I get a Fourier Series whose ...
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2answers
65 views

Prove that $\int_{-\infty}^\infty P_n(x) \, dx = \pi /n$

Let $P_n(x) = \frac{n}{1+n^2x^2}$. Prove that for every $n\in\mathbb{N}$ $$\int_{-\infty}^\infty P_n(x) \, dx = \frac{\pi}{n}$$ And for every $\delta > 0$: $$\lim_{n\to\infty} ...
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1answer
26 views

Fourier series of function $f(x)=0$ if $0 < x \leq L/2$ and $f(x)=1$ if $L/2 < x \leq L$

I am attempting to work through a very simple problem. Determine the Fourier series expansion for: $$ f(x) = \begin{cases} 0 & 0 \leq x \leq L/2 \\ 1 & L/2 < x \leq L\end{cases}$$ I ...
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1answer
14 views

Show that for every real-valued $L^2$ function $u$ on $S^1$ there is a real-valued $v$ in the same space such that $u + iv\in \widetilde{\mathbf H}^2$

For a homework exercise ($1.8$ in the book An Introduction to Operators on the Hardy-Hilbert Space) I am asked to show Let $u$ be a real-valued function in $L^2(S^1)$. Show that there exists a ...