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
29 views

uniform convergence of series and Fourier coeffient

let ${ \phi_{n} } $ be sequence of orthogonal functions on $[a,b]$ If the series $ \sum_{n=1}^{\infty } a_{n}\phi_{n}(x) $ converges uniformly to a function $f(x)$ on $[a,b]$ prove that for each $n ...
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1answer
30 views

Continuity/Differentiability of Fourier Series

Possibly stupid question: I'm wondering if there is some trick for evaluating the continuity/differentiability of a Fourier series. In particular, I'm looking at the function $f(x)=\sum_{n=0}^\infty ...
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1answer
39 views

Derive Fourier transforms from Fourier expansion. How are they related?

I am just trying to relate Fourier Series expansion to Fourier Transforms. If someone could show how one value on the middle of the table is derived (from expansion) as opposed to deriving their ...
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1answer
59 views

Diagonalization of circulant matrices

Why does the following hold?: $A$ circulant matrix iff it has a representation of the form $F^{-1}DF$ where $D$ is a diagonal matrix and $F$ is a discrete Fourier transformation. I get that $F^{-1}DF$ ...
3
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1answer
48 views

Show that for $0<t<1$, $\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$

Show that for $0<t<1$, $$\log\sin (t\pi)=\log(t\pi) + \sum_{n=1}^\infty\log\left(1-\frac{t^2}{n^2}\right)$$ So I derived the following Fourier series: ...
3
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1answer
95 views

What is the Fourier series of Dirichlet function?

Given the Dirichlet function defined as: $$f(x) = \begin{cases}0 & x \in \mathbb{Q} \\ 1 & x \in \mathbb{R} \setminus \mathbb{Q}\end{cases}$$ Find the corresponding Fourier series. Before ...
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0answers
32 views

DFT Zero-padding : what prepending with zeros does?

I am studying Fourier transform and want to understand better some point regarding zero-padding in DFT. All known sources say that padding is done by appending the data with zeros. However, if I add ...
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0answers
28 views

Fourier series/Limit

Assume $f$ satisfies the assumptions of Dirchlet's Theorem - i.e. $f$ is a piecewise continuous complex function that has one-sided derivatives at each point in $[-\pi,\pi]$. Determine the following ...
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0answers
42 views

Inverse Fourier transform on infinite series

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. I have derived that $$\hat{f}(y)=\sum_{n=-\infty}^\infty f(n)1_{[-\pi,\pi]}(y)e^{-iny}$$ in $L^2$ convergence. Let ...
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0answers
16 views

Hyprecomplex datapoint

I was working on 2D dataset 512$\times$512. To be more precise,its the datapoints are collected in dimensions t1 and t2. All the data points are complex in the form a+b$i$. To get the frequency ...
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2answers
94 views

Do the Fourier series of a function-valued Hermitian matrix converge?

Let $\mathbf{A}(t):\mathbb{R}\rightarrow\mathbb{C}^{n\times n}$ be a rank deficient periodic function-valued positive semi-definite Hermitian matrix. The entries $a_{ij}(t)$ of $\mathbf{A}(t)$ are ...
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1answer
29 views

How to extend a given function to an odd function with period 2 (Fourier Series)

This is the question. f(x) = 7x^2 + 4 I want to know how to extend the above fuction to become a odd function with period 2.
3
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1answer
246 views

Integrating Dirichlet's Kernel

Determine $\frac{1}{\pi}\int_{-\pi}^{\pi}\left[D_{m}(t)\right]^{2}dt$ for $m=100$ where $D_{m}(t)=\frac{1}{2}+\sum_{n=1}^{m}\cos{nt}$ (Dirichlet's kernel). Initially, I thought of using the ...
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1answer
32 views

If $f$ is identically zero then the coefficients are all zero

I am looking at the space: $$A:=\left\{f(x)=\sum_{k\in\mathbb{Z}}{a_ne^{inx}}:(a_n)_{n\in\mathbb{Z}}\in l^1(\mathbb{Z})\right\}$$ I want to say the following: if $f\equiv0$, then $a_n=0$ for all ...
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0answers
24 views

Fourier Transform of Fractional Laplacian

I'm trying to solve a PDE with a spectral method. The PDE has a fractional Laplacian... $\Delta^s$. In regards to a numerical implementation, will the "s" term simply become the exponent of the ...
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0answers
45 views

How do I find a function given its Fourier series?

I'm not completely sure this is how my problem is supposed to be done, but I think it will work. In the first part of a problem, I derived the transfer function to describe output voltage over input ...
2
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2answers
57 views

Fourier Series of what appears to be a sawtooth series

Find the Fourier series of \begin{equation} f(x)=\begin{cases} x-[x] \quad &\text{if $x$ is not an integer} \\ \frac{1}{2} \quad &\text{if $x$ is an integer} \end{cases} \end{equation} ...
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0answers
20 views

Fourier Series simple question

Let $f$ be a $\mathcal{C}^r$ function such that $f(0)=f(\pi)=0$, and define $a_n := \frac{2}{\pi}\int^\pi_0 sin(nx)f(x)dx$, its easy to show that exists $C>0$ such that $|a_n|\leq \frac{C}{n^r}$. ...
1
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1answer
46 views

What is the Fourier transform of an M like function

Given the function $$ f(x)= \begin{cases} \vert x \vert& \text{, for }\;\vert x\vert\le M \\ 0 & \text{, otherwise} \end{cases} $$ for some constant $M$. What would be the form for the ...
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0answers
24 views

Monotonically decreasing Fourier transform

What would be the conditions on $f(x)$ such that it's Fourier transform $F(k)$ would be monotonically decreasing from $k=0$ to half range ($F(0)$ would be the maximum, and it would "fall" on both ...
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1answer
27 views

Fourier series convergence without sines or cosines converging

Is it possible to have a Fourier series $$a_0 + \sum_{k=1}^{\infty} \left[a_k\cos(kx) + b_k\sin(kx)\right]$$ converge without either the cosines or the sines converging? Here is my work so far: Since ...
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1answer
30 views

Complex Conjugate (FourierSeries)

Let $f$ be a piecewise continuous complex function on the interval $[\pi,\pi]$ and \begin{equation} f(x) \sim \sum_{n=-\infty}^{\infty}c_{n}e^{inx} \tag{*} \end{equation} be its complex Fourier ...
2
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2answers
49 views

What is it called when you replace sine with cosine in a Fourier series?

Suppose you have a Fourier sine series: $$f(t) = \sum_{n=0}^{\infty} a_n \sin(n \omega t)$$ and you replace sine with cosine: $$g(t) = \sum_{n=0}^{\infty} a_n \cos(n \omega t)$$ or conversely, ...
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1answer
57 views

Inner product of function of period $2\pi$ with exponential

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be continuous with period $2\pi$. Prove that $$\lim_{N\rightarrow\infty}\dfrac{1}{N}\sum_{j=1}^Nf\left(\dfrac{2\pi j}{N}\right)e^{-2\pi ...
2
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0answers
42 views

Fourier Series, Parseval Identity

I need to prove $$\sum_{n=1}^{+\infty}\frac{1}{n^2-\alpha^2}=\frac{1}{2\alpha^2}-\frac{\pi}{2\alpha\tan(\alpha\pi)},$$ with $\alpha$ a non integer complex. I know that I have to use the Parseval's ...
2
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1answer
62 views

Fourier transform supported on compact set

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Show that $$\hat{f}(y)=1_{[-\pi,\pi]}(y)\sum_{n=-\infty}^\infty f(n)e^{-iny}$$ in the sense of $L^2(\mathbb{R})$-norm ...
2
votes
1answer
51 views

Fourier series formula with finite sums

Let $f\in C(\mathbb{R}/2\pi\mathbb{Z})$, meaning that $f$ is continuous with period $2\pi$. Let $x_N(j)=2\pi j/N$. Define $$c_N(n)=\dfrac1N\sum_{j=1}^Nf(x_N(j))e^{-ix_N(j)n}.$$ Show that for any ...
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1answer
563 views

Fourier series coefficients proof

Can somebody help me understanding the fouries series coefficients? I know that if we have: $$f(n) = \sum_{n=1}^N A_n \sin(2\pi nt + Ph_n) \tag{where $Ph_n$ = phase}$$ And because of the ...
2
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2answers
119 views

Fourier Series and periodicity

Let $f$ be a $2 \pi$-periodic piecewise continuous function and let \begin{equation} f(x) \sim \frac{a_{0}}{2}+\sum_{n=1}^{\infty}\left[a_{n}\cos{nx}+b_{n}\sin{nx} \right] \tag{*} \end{equation} ...
1
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2answers
90 views

Determine the trigonometric Fourier series

Consider the function $$ f(x):=\begin{cases}x(\pi-x), & x\in [0,\pi]\\-x(\pi +x), & x\in [-\pi,0]\end{cases} $$ and calculate its trigonometric Fourier series. Hello! So ...
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1answer
72 views

Fourier transformation: Determining the axis

I need some help with the Fourier transformation of my data. My original data is a Distance VS Time: upon doing a Fourier Transform, I get the following: I understand that normally after a ...
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0answers
20 views

Iteration of Fourier coefficient.

I would like to know if there's any work or if it's even interesting to look at an iteration of fourier coefficients. Here's what I mean, take $f\in L_2(-\pi,\pi)$ for simplicity, and compute its ...
0
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1answer
39 views

Multiple Characteristic Function and the Dirac Comb

Given the impulse train(Dirac comb): $$\Delta_T(t)=\sum_{k\in\mathbb{Z}}\delta(t-kT)$$ where $T$ is the signal period, $\delta(t)$ is the Dirac delta function and $\mathbb{Z}$ is the set of integers ...
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1answer
36 views

CT Fourier Transform

I need to find the Fourier Transform of the given signal below; $$ x(t) = \frac{\sin(\pi t)}{\pi t} \frac{\sin(2\pi t)}{\pi t}.$$ I know that if $ x(t) = \frac{\sin(Wt)}{\pi t} $ , then $ X(w) = ...
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5answers
3k views

What are some real world application of fourier series?

what are some real world application of Fourier series ? particularly the complex Fourier integrals
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3answers
179 views

Fourier Series coefficients/Trigonometric functions

I need some help about finding the Fourier Series coefficient of the given signal; $$ x(t) = \sin(10\pi t + \frac {\pi}{6} ) $$ I know that, $$ a_{k} = \frac{1}{T}\int_{0}^{T} x(t)e^{-jkw_{0}t}dt $$ ...
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0answers
38 views

Seeking better understanding of Fourier transform?

I'm quite confused on the one part of the Fourier transform. I don't understand what is the term $\left(u*x + v*y \right)$ mean. I mean $u$ and $v$ are the axis for frequency domain and $x$, $y$ are ...
2
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0answers
25 views

Convergence of the series $\sum_{\xi\in\mathbb Z^n} e^{2\pi ix\cdot \xi} a(x, \xi)\hat{f}(\xi)$?

I need some help with the following problem: let $a:\mathbb R^n\times \mathbb R^n\rightarrow \mathbb C$ be a smooth function and suppose there are constantes $C_{\alpha, \beta}$ and $M(\alpha, \beta)$ ...
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1answer
48 views

How can I know if my Fourier Series coefficients are correct?

I want to find Fourier Series coefficients ($a_n$ and $b_n$) for this signal: $$f(t) = \frac{A}{t_s}t[u(t) - u(t-t_s)] + A[u(t-t_s) - u(t-(t_s + t_{on}))] + ...
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0answers
94 views

How to properly prepare for a gradute level PDE course using Evans and Strauss ' book

For my undergrad background , I have cal 1-3, linear algebra , 1 semester of ODE, 1 semester of real analysis never have any PDE before. Thus I know this background is hardly enough to do well in a ...
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1answer
65 views

Calculation of coefficients of a Fourier series

Calculating the Fourier series of a periodic function I need to evaluate these integrals: $$1) \int_{-\pi}^{+\pi}dt\left(\cos^{-1}(\alpha t-1)+2(1-\alpha t)\sqrt{\frac{1}{2}\alpha ...
2
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1answer
49 views

Fourier Series and Summation

$\sum_{n=1}^\infty \frac{1}{n^2}$ can be computed in straight-forward way by computing the Fourier co-efficients of $f(x)=x$ and applying Parseval's identity. Likewise, $\sum_{n=1}^\infty ...
3
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2answers
54 views

Relationship between Fourier coefficients of $f\left(x\right)$ and $f^{-1}\left(x\right)$

Say I have a function $f\left(x\right)$, which can be expressed as a Fourier Series: $$f\left(x\right)=\sum_{k=-\infty}^{\infty} c_k e^{ikx}$$ Define the inverse of $f\left(x\right)$ as, ...
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2answers
75 views

Fourier Series Coefficient of a given signal

$$ {\rm x}\left(t\right) = \sum_{k = -\infty}^{\infty}\left[\delta\left(t-\dfrac{k}{3}\right) + \delta\left(t-\dfrac{2k}{3}\right)\right] $$ I need to find the Fourier series coefficient of x(t). I ...
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0answers
75 views

Fourier transform of a logarithm

How can one go about computing the 2d (or 1d, in either variable) Fourier transform of the function $$\ln(w^2-k^2)?$$
2
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1answer
104 views

Parseval equation for a Fourier series

Consider $f(x):=\lvert x\rvert, x\in [-\pi,\pi]$. Then the Fourier series is $$ f(x)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos((2n-1)x)}{(2n-1)^2}. $$ Now my task is to write down the ...
2
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0answers
50 views

Need help on computing odd, even extensions of a function

OK I am going over d'Alembert solutions. And I came across the following example. $$ f(x) = \begin{cases} \frac{3}{10}x &0 \le x \le \frac{1}{3} \\ \frac{3(x-1)}{20} & \frac{1}{3} \le x \le ...
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1answer
61 views

Show absolute and uniform convergence of a Fourier series

Hello and good evening! The Fourier series of $f(x):=\lvert x\rvert$ on $[-\pi,\pi]$ is $$ f(x)=\frac{\pi}{2}-\frac{4}{\pi}\sum_{n=1}^{\infty}\frac{\cos((2n-1)x)}{(2n-1)^2}. $$ I have to examine if ...
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1answer
21 views

transformation of DFT matrix

$\mathbf{F}$ is a unitary DFT matrix where the $(m,n)$-th entry of $\mathbf{F}$ is given by $\frac{1}{\sqrt{M}}e^{-\imath2\pi(m-1)(n-1)/M}$. Note that $\imath=\sqrt{-1}$. Let $\mathbf{A}$ be a matrix ...
3
votes
1answer
159 views

Basic Fourier Series Question

Let $f$ be a $2π$ periodic function where $$f(x) = \frac{π - x}2$$ over $[0, π]$. It is known that the Fourier series of $f$ is $$\sum_{n=1}^{\infty}\frac{\sin nx}n$$ At which points in $[-π, π]$ ...