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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Basic Fourier analysis questions [closed]

Could someone point me in the right direction to prove the following: Let $f \in L^{1}(\mathbb{T})$ and suppose $S_{N}f \to g$ in $L^{p}(\mathbb{T})$ then $\|f-g\|_{L^{1}(\mathbb{T})}=0$, where $1 ...
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0answers
32 views

Why are complex exponentials orthogonal over sums?

Consider the following sum: $$ \sum_{j=0}^{J-1} \omega_j^k \bar{\omega_j^l} $$ Where $\omega_j^k=\exp \left( \frac{2\pi i j}{J} \right)$. It is easy to show that if $k \equiv l$ modulo $J$, then ...
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1answer
26 views

An example of a function which is not piecewise continuous, but has Fourier series

Would you Please give an example of a function which is not piecewise continuous, but has Fourier series? It means that the coefficient in the Euler-Fourier formulas can be computed. In fact, the ...
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2answers
36 views

Proving orthogonality of complex form of Fourier Series

I am lost when working on this complex Fourier Series question, I am sure it is a basic simple problem but I am not well versed in applied math: Show that $\{e^{\mathscr i n\pi x/\mathscr l}\}, n ...
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1answer
67 views

How to prove the convergence of this Fourier Series?

Suppose $f:\mathbb{R}^{n}\rightarrow \mathbb{C}$ is smooth and 1-periodic. Define $$c_{k} = \int_{0}^{1} \dots \int_{0}^{1} f(\vec{y})\operatorname{exp}^{-2\pi i \langle ...
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1answer
36 views

Help figuring out output signal of LTI system.

Would greatly appreciate any help in figuring out the output signal of my discrete time LTI system. My input signal is cos(ωn) and my frequency response is H(e^jω)=(1+e^−jω)/2.
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2answers
30 views

A property of the fourier series

Show that any periodic function $f(x)$ with period $2\pi$ which is both odd and satisfies $f(\pi-x)=f(x)$ has $b_{n}=0$ for $n$ even and so has a fourier series of the form $$f(x) = ...
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0answers
28 views

Issues proving a basis via wedge product

On a quiz I was given the problem" a series that is a basis for $[-1,1]$ is $ \sum_0^{\infty} c_n P_n $, where $ P_n $ is a polynomial and each polynomial $P_n$ is orthonormal to the others. Using the ...
2
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0answers
51 views

Fourier series using Bessel function

so Im stuck on the following problem; Use the identity $\exp(ix\sin\theta) = \sum\limits_{k=-\infty}^\infty J_k(x)\exp(ik\theta)$ to find the Fourier series of $\cos(\theta + 4\sin\theta)$, where ...
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0answers
29 views

Expansion of function in polar coordinates

I'd like to expand a function in polar coordinates to something that splits radius and angle $f(r,\theta)=\sum_i A_i(r)B_i(\theta)$ I've found some hints on the internet by the name of polar Fourier ...
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1answer
38 views

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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0answers
21 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
33 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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1answer
23 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 ...
0
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1answer
19 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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0answers
23 views

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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1answer
51 views

Finding limit under integral [closed]

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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3answers
116 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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0answers
40 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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1answer
33 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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0answers
36 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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30 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
34 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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1answer
41 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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0answers
16 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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1answer
99 views

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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1answer
28 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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1answer
21 views

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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1answer
26 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 ...
4
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1answer
55 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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3answers
160 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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1answer
415 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
42 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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30 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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1answer
32 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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1answer
42 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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5answers
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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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1answer
29 views

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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1answer
50 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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1answer
165 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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27 views

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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0answers
43 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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1answer
68 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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1answer
69 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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1answer
57 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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1answer
32 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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1answer
52 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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44 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
44 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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1answer
31 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 ...