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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Find Fourier Coefficients

I am asked to find the coefficients for $f(t)=\sin^{2}(5t)$ $$Period =\frac{\pi}{5}$$ so I wrote $$a_n\cdot\sin(\frac{n\pi{t}}{\frac{\pi}{10}})=\sin^{2}(5t)$$ $$a_n\cdot\sin(10n{t})=\sin^{2}(5t)$$ ...
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17 views

An upper bound for the Fourier coefficient of the “infinite cake” function

Consider a function $x_{s_n} (t) = s_n$ for $t\in[-\frac{s_n}{T_0}, \frac{s_n}{T_0}]$ and $x_{s_n} (t) = 0$ for $t$ everywhere else, with period $2T_0$. Now let $s_n=\frac{1}{n^2}$, and define the ...
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1answer
24 views

Confused about Fourier series?

From linear algebra we know that if a set of vectors form a basis for a space, their is a unique linear combination of the basis to form any vector in that space. I'm assuming this extends to scalar ...
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15 views

Computing the Fourier series of $\lvert x\rvert$

I am getting very confused when trying to compute the Fourier series of $f(x) = \lvert x\rvert$, $x \in [-1/2,1/2]$. Normally I have no trouble with this because it is mindlessly integrating to get ...
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16 views

Relation between fourier coefficients of $f\in \mathcal{C}^1[-\pi, \pi]$ and $f'$

I'm given $f\in \mathcal{C}^1[-\pi, \pi]$ with $f(-\pi)=f(\pi)$. It's fourier coefficients are given by: $$\gamma_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{-int}f(t)dt,\ n\in \mathbb{Z}$$ And now I'm ...
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Does this reasoning about fourier analysis make sense?

I'm asked to show that there cannot be $\alpha_1,\alpha_2,...\in\mathbb{C}$ s.t. $$\lim_{N\to\infty}\int_{-\pi}^{\pi}|e^{it}-\sum_{k=1}^{N}a_k\sin(kt)|^2dt=0$$ Here is my attempt: Assume there are ...
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45 views

Finding limit under integral [on hold]

Evaluate if $f \in C [-\pi,\pi]$ $$\lim_{n\to\infty} \int_{-\pi}^{\pi}f(t)\cos(nt)dt$$ and $$\lim_{n\to\infty}\int_{-\pi}^{\pi} f(t)\cos^2(nt)dt$$
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72 views

Can a non-periodic function have a Fourier series?

Consider two periodic functions. Assume their sum is not periodic. The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents ...
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21 views

When has the Fourier transform for some values equal values?

Definition We take a function $F : \mathbb T^n \rightarrow \mathbb R$ that is even ( $F(x)=F(-x)$) and continuous (hence bounded), where $\mathbb T^n$ is the $n$-dimensional Torus. Now we define the ...
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16 views

How can I solve this differential equation with fourier series?

Find a formal solution $u(x; y)$ by using Fourier series. (Hint: In two dimensions the basis functions have one of the forms $\sin(ax) \sin(by)$, $\sin(ax) \cos(by)$ and $\cos(ax) \cos(by)$, with ...
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33 views

Closed form of a series with sinh

Is there a simple form for following function (where $a$ and $b$ are constants)? Can it be simplified to a simple form if $a>>b$? $$ u(x) = \sum _{n=0}^{\infty } \frac{ \, (-1)^n ...
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24 views

Fourier series question - represent $x$ as a series of $\cos$

I was asked to represent $f(x)=x$ in $(0,\pi)$ as a sum of $\cos$ functions, using fourier series. I couldn't solve it on my own, but here is what the teacher did, and I don't fully understand why ...
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2answers
30 views

Discrete fourier transfomation and harmonics

I have a very simple question that I would like to understand. If you have a DFT of a function: $$ X_k \stackrel{\mathrm{def}}{=}\sum_{n=0}^{N-1}x_n\cdot e^{-i2\pi kn/N},\qquad k\in\mathbb{Z} $$ Did ...
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34 views

How to find $ \sum_{k=0}^\infty \frac{(-1)^k}{(2k+1)^2}$?

Let $f$ be a $2\pi$-periodic function whose restriction on $[-\pi, \pi]$ is $f(x)_{[-\pi, \pi]} = |x|$ It is easy to see that its fourier series converges uniformily to $f$ and is $$f(x) = \frac \pi2 ...
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14 views

Characterstic Functions and Recovery

Assume that I have a pdf, call it $f$, that is supported on $[0,2]$. Let $\varphi(t)$ be the corresponding characteristic function, which is known to me. Is there some common method to recover the pdf ...
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23 views

Fourier Series: even extension and Parseval Identity

I'm trying to solve this exercise but I have some problems, because I haven't seen an exercise of this type before. $f(x)= \pi -x$ in $[0, \pi]$ Let's consider the even extension of f(x) in ...
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Example of continuous function whose Fourier series doesn't converge on an uncountable dense set.

According to a well-known theorem (Theorem 5.12 in Rudin's Real and Complex Analysis), there is a dense $G_\delta$ set of continuous periodic functions $f:\mathbb{R}\to\mathbb{C}$ such that the ...
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20 views

Fourier series: $\lim_{n\to\pm\infty} n^p \hat{f}(n) = 0$

Let $f:\mathbb{R}\to\mathbb{C}$, $f\in C^\infty$ (differentiable infinitely many times) and periodic,$T=2\pi$. Prove that for every $p>0$: $$ \lim_{n\to\pm\infty} n^p \hat{f}(n) = 0$$ So I ...
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Fourier series: Show that $f$ is a trigonometric polynomial

Let $N\in\mathbb{N}$ and $f_m:\mathbb{R}\to\mathbb{R}$, continuous functions and periodic, $T=2\pi$. Let's assume that $f_m \to f$ uniformly and for all $m\ge 1$: $$\left| \hat{f_m}(n)\right| \le ...
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20 views

Even or odd function. Fourrier coefficients

This is probably a very easy question, but I can't find the answer to it.. I'm working on Fourier coefficients and whether or not the integrals become zero. As far as i'm concerned this integral ...
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48 views

Fourier coefficients intuition?

I just learned about Fourier series, and this is how I interpreted them: The complex exponentials form a basis for all periodic functions, and the Fourier series essentially decompose the function ...
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55 views

Proof of the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series

I have tried and failed to prove the Dirichlet–Dini Criterion for Pointwise convergence of Fourier series which is as follows (and is described here: ...
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103 views

Double sum and zeta function

This is a personal research that came to an end , since the results were not those which were being anticipated. I was unable to come up with a solution therefore I post the topic here: Prove (it ...
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1answer
26 views

fft phase plot of pure sine function, why so messy?

I am plotting the phase plot of $sin(2*pi*60*x)$ in the frequency domain. Ideally, we should only see two peaks. How come this is not the case in matlab? ...
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9 views

Phase difference of two signal of different frequency

Currently, I have two signals, the main components of both signals are 60Hz, but both also have weaker response at 180Hz + small amount of noise. As shown in the photo below, I want to find the phase ...
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27 views

How are sinusoids and roots of unity related to each other?

The discrete Fourier transform (DFT) is often teached as being a transform that decomposes a given signal or sequence of numbers into sinusoids with frequencies $\large\frac{k}{N}$ where $k \in [0, ...
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16 views

Sum involving the “distance to the nearest integer function”

I want to prove that if $||x||$ is the distance between $x$ and the nearest integer to $x$, $\{\alpha_1,\ldots, \alpha_N\}$ are points in $\mathbb{R}$/$\mathbb{Z}$ and we define $$S(y) = ...
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Why do Fourier Series work?

I would like to have an intuitive understanding of Fourier Series. I mean, I know the formulas: $$ f(t) =\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(n\pi tL)+\sum_{n=1}^\infty b_n \sin(n\pi tL) $$ And ...
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General Fourier coefficients and smoothness

Suppose $f\in L^2([0,1],\lambda)$. Are there assumptions on the smoothness of $f$ which translate into the particular behavior of Frourier coefficients. Namely, I have arbitrary complete orthonormal ...
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40 views

Fourier sine series of $f = \cos x$

Let $f:(0,\pi) \to \mathbb{R}$ defined by $x \mapsto \cos x $ Show that the Fourier sine series of (odd extension) is given by $$\sum\limits_{n=2}^\infty \frac{2n(1+(-1)^n)}{\pi(n^2-1)}$$ So far, ...
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138 views

Derivation of fourier series equation

No matter where I search, every time if there's an article about Fourier series derivation, the first step made by author is to present the following formula: $$f(x) = \frac{a_0}{2}+\sum_{n=1}^\infty ...
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Fourier series - different equations

There are two very popular forms of Fourier series equation. $$f(x) = \frac{a_0}{2}+\sum_{n=1}^\infty \left(a_n \cos(\frac{2\pi}{T}nx) + b_n \sin(\frac{2\pi}{T}nx)\right)$$ and $$f(x) = ...
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34 views

Fourier transform and non-standard calculus

The Fourier transform is not as easily formalizable as the Fourier series. For example, one needs to introduce tempered distributions to define the Dirac delta-function. Also, it is impossible to view ...
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49 views

Help with proof of Poisson summation formula

I am trying to understand a proof of the Poisson summation formula and I cannot understand a vital part of it which the author seems to think is obvious, but is not obvious to me. If anyone can fill ...
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8 views

inserting absolute value in Hilbert transform and a discrete version of Hilbert transform

It is well known that the Hilbert transform $H(f)(x)=p.v. \int\frac{f(x-y)}{y}dy$ is bounded on $L^p(\mathbb{R})$ for $p\in(1,\infty)$. I want to consider some variants of $H$. 1) What happens if we ...
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25 views

Heat flow in 1D bar fourier series problem

I am stuck on this problem: The temperature $T$ in a one-dimensional bar whose sides are perfectly insulated obeys the heat flow equation $$ \frac{\partial T}{\partial t} = \kappa ...
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30 views

How to prove these Fourier-series identities?

The first series is : $$\sum_{n=1}^{\infty }(-1)^n\frac{4}{(n\pi )^2}\{(\cos(A(n\pi) )-\cos(B(n\pi ))\left. \right \}=(A^2-B^2)$$ Where $A$, $B$ are positive real numbers less than $1$. I need a ...
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42 views

Fourier Series and differential equation with epsilon

Happy New Year! I am stuck for days on expressing the solution of a differential equation using Fourier series. The question is: Consider the equation: $$\ddot{x}+x+\epsilon\left(\alpha ...
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16 views

Is the spherical harmonic representation of a 2D field independent of grid?

What I am currently unable to understand is whether the spherical harmonic representation of a 2D field is in any way tied to the nature of the grid on which decomposition/composition is performed. I ...
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1answer
17 views

Fourier Series Reduced Form: Phase Angle and Spectra

Im very confused regarding how to determine the angle on the reduced or harmonic form representation of the Fourier series. Some books state the following: $$f(t)=F_0+\sum_{n=1}^\infty |F_n ...
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30 views

Is there a trigonometric Fourier transform formula?

I wonder if one can express the Fourier transform in the trigonometric approach like, say, in the case of the Fourier series, where we can write it as: $Sf(x)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left ...
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56 views

What is the Fourier Series of a piecewise constant wave?

I am looking for the Fourier Series of this function: This is a winding function method for calculation of rotor inductances. The distance between each stator slot (each segment) is $10$ degrees or ...
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Norm Inequality (Vinogradov Notation)

I'm going through a proof of differentiability of fourier series on the d-dimensional torus and while proving the following inequality: $$ ...
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69 views

Does the phrase “orthogonal” mean the same thing when used in the terms “orthogonal function” and “orthogonal vector”?

I was reading about Fourier series when I came across the term "orthogonal" in relation to functions. http://tutorial.math.lamar.edu/Classes/DE/PeriodicOrthogonal.aspx#BVPFourier_Orthog_Ex2 I've ...
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Why is $\|f-s_n(f)\|_2=\inf_{T\in\mathcal{T}_n}\|f-T\|_2$

I am working through some examples in my book in the section on Fourier Series. Why is $\|f-s_n(f)\|_2=\inf_{T\in\mathcal{T}_n}\|f-T\|_2$? where $f$ is a continuous $2\pi$ periodic function, $T$ is ...
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65 views

Sum of $\sum_{n=1}^{\infty }\frac{1}{\pi n }\sin ^k\left(\frac{2\pi n}{k}\right)$

We have: $$S_k=\sum_{n=1}^{\infty }\frac{1}{\pi n }\sin ^k\left(\frac{2\pi n}{k}\right)$$ where $k$ is an odd number greater than $1$. I was able to find the sum of the series when $k=3,5$ as ...
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27 views

How does the Fourier transform get you the frequency amplitude

I understand that the Fourier transforn gets you the function which gives the amplitude of each frequency. But I don't understand how that is possible by multiplying it by an exponential. How is that ...
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56 views

wave equation on a square domain

I'm stuck on the following problem. Let $u(x, y, t)$ denote a solution to the linear wave equation $k^2(u_{xx}+u_{yy}) = u_{tt}$ with $k = 2$ on a square domain with corners at (0, 0), (0, 1), ...
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41 views

Fourier series for a logarithm

Is there an explicit Fourier sine series for the function $f$ defined below (valid for $x\in[0,\pi]$) ? $$f(x) := \ln\big(\sqrt{1 + \sin x} + \sqrt{\sin x}\big)$$ In case this is well known, a ...
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1answer
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Find a non-Lipschitz Riemann integrable function that her Fourier series converge uniformly to her

This is my first question here. So I'll try to be short and to the point. I'm asked to find a Riemann integrable function $f$, that is not Lipschitz continuous but her Fourier series converge to $f$ ...