Fourier analysis, also known as spectral analysis, encompasses all sorts of Fourier expansions, including Fourier series, Fourier transform and the discrete Fourier transform (and relatives). The non-commutative analog is (representation-theory).

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Y-Axis units on FFT graph

A 50Hz sinusoid wave with a voltage range of +/-20V is sampled at 512Hz for 1 second. No bias or phase shift are present. The signal is run through an FFT. The result is one spike at 50Hz on the ...
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2answers
182 views

Prove that the decomposition of a function $f(x)=f_{even}(x)+f_{odd}(x)$ on a sum of even and odd functions is unambiguous

How does one prove that the decomposition of a function $f(x)=f_{even}(x)+f_{odd}(x)$ on a sum of even and odd functions is unambiguous? I'm unsure of where to begin. Pertinent definitions: ...
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5answers
45 views

Proving a function $f(x + T)=k\;f(x)$ satisfies $f(x)=a^x g(x)$ for periodical $g$

I need to prove the following: If a function $\,f$ satisfies $$f(x+T)=k\;f(x), \forall x \in \mathbb R$$ for some $k \in \mathbb N$ and $T > 0$, prove that $\,f$ can be written as ...
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371 views

Fourier transform of squared exponential integral $\operatorname{Ei}^2(-|x|)$

Let $\operatorname{Ei}(x)$ denote the exponential integral: $$\operatorname{Ei}(x)=-\int_{-x}^\infty\frac{e^{-t}}tdt.$$ Now consider the function $\operatorname{Ei}(-|x|)$. ...
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1answer
95 views

Fourier Sine Series and Cosine Series

This is the Fourier Series representation for a periodic function with period 2p, given in my lecture note. $\dfrac{a_0}{2} + \sum_{n=1}^{\infty}(a_n cos(\dfrac{n\pi t}{p})+b_nsin(\dfrac{n\pi ...
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1answer
68 views

Prove the following relation between Fourier transform pairs

Lets say I have a function $f(x,y)$ and its corresponding Fourier transform $F(u,v)$. I'm trying to show that $f(x+by,dx + y) \leftrightarrow ...
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1answer
36 views

Transforming a function in the dirac delta function

I need help finding $k(\sigma)$ such that the family of functions $$ \delta_\sigma (x,y) = k(\sigma) e^{-\frac{1}{2}\frac{x^2 + y^2}{\sigma^2}} $$ defines the unit impulse $\delta(x,y)$ as $\sigma ...
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0answers
37 views

Is there a continuous and satisfying a certain condition but not integrable function?

Is there a continuous function $f$ on the interval $[0,1]$ which satisfies $$ |f(x+h)+f(x-h)-2f(x)|\leq \mathrm{const} \frac{|h|}{(\log\frac{1}{|h|})^{\beta}}\,\,\,\,\,\,\text{where}\,\,\,\, \beta ...
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1answer
22 views

Seperable functions have seperable Fourier transforms

How do I show that if $f(x,y)$ is separable into a product of a function of $x$ and a function of $y$, its Fourier transform $F(u,v)$ is also separable into a function of $u$ and a function of $v$?
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34 views

Fourier transform of following equation

Say I have the following equation $$ f(x,y) = \left\{ \begin{array}{lr} 1 & \text{if} \;|x|,|y| \leq 1 \\ 0 & \text{otherwise} \end{array} \right. $$ What is the ...
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39 views

Fourier transform of this oddly worded function

A function is equal to zero outside a unit area square centered at (0,0) and inside a central quarter-unit area square similarly oriented. Elsewhere the function is equal to unity. I am trying to find ...
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1answer
127 views

Expand $f(x) = x$ in a cosines serie

The task is: Expand $f(x) = x$ in a cosines serie. My approach was to use the trigonometry fourier formula $$ c_0 + \sum_{k=1}^{\infty } a_k \cos (k \Omega t) + b_k \sin (k \Omega t) $$ set $b_k = ...
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0answers
89 views

Fourier Cosine Series question

If I have even piecewise periodic function ($T=6$) $$x(t)=\begin{cases} 0 &-3\leq t \leq-2  \\ 2+t &-2\leq t \leq-1 \\ 1 &-1\leq t \leq 1 \\ -t+2 &1\leq t \leq 2 \\ 0 &2 \leq ...
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1answer
73 views

Is fourier transform or Wavelet transform better for this applicaiton?

I am currently designing an alogirthm that is either based on Fourier Transform approach, or the Wavelet Transform Approach, or the combination of the two. Since Wavelet is new to me, I am having ...
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1answer
23 views

Coefficient in defining Fourier Transform

I don't know why we define below coefficient in Fourier transform? thanks for your help
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196 views

Inverse Fast Fourier Transform to find the voltage across a capacitor of a RC circut

Fourier transform of a RC circuit The following example of a RC circuit describes the use of the fourier transform in order to receive the output voltage across the capacitor. My questions ...
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1answer
85 views

Centroid from Fourier Transform

I have a discrete fourier transform from an black and white image and would like to have a rough estimation of the centroid of an white shape in it. As far as I can tell from ...
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1answer
57 views

Can the sum of infinitely many zero amplitude sinusoids converge to any function?

I've read this in a post here (can't remember which - might even have been a comment) that I thought that was the most ridiculous thing I have ever heard. Can someone illustrate mathematically that ...
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0answers
22 views

Solve $\int\limits_{\mathbb{R}^n} e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$

How to calculate $$\int_{\mathbb{R}^n}e^{-2\pi i\langle\eta,x\rangle}e^{-a|\eta|}d\eta$$ where $\langle \cdot, \cdot \rangle$ denotes the canonical inner product in $\mathbb{R}^n$. I'm trying use ...
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1answer
121 views

integration concerning Fourier transform of homogeneous kernel(of degree 0)

Let $m\in C^\infty(\mathbb{R}^d-\{0\})$ be homogeneous of degree zero and has mean zero on the sphere$(S^{d-1})$. Then $m$ defines a tempered distribution and $\partial_j^dm\in S'(\mathbb{R}^d)$ is ...
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How to calculate Fourier Transform of logarithmic function?

Given a random variable (RV) $S$ equal to the sum of two mutually independent (RVs) $X_1,X_2$,i.e.$S=X_1+X_2$ and piece-wise probability density functions (PDFs) of $f_{X_1},f_{X_2}$ are as follow: ...
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1answer
39 views

Finding $\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$ using Plancherel’s theorem

Suppose $G(\mathbb R)\ni f(x),\mathcal{F}[f](\omega)=\frac{1}{1+|w|^3}$ find the value of $$\int_{-\infty}^\infty |f\ast f'|^2(x)\,dx$$ I thought using Plancherel’s theorem \begin{align} ...
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214 views

2D Fourier transform of $1/(x^2-y^2+q)$

How can I calculate the following 2D Fourier integral: $$ \iint \frac{{\rm e}^{{\rm i}(ax+by)}}{x^2-y^2+q} {\rm d}x\,{\rm d}y, $$ where $q$ is a complex number? If there was a "+" sign in the ...
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3answers
260 views

Finding the fourier series of floor function

Find the fourier series for $f(x)=\cases{x-[x]\quad x\in\mathbb{R\setminus Z} \\ \frac 1 2\quad x\in\mathbb{Z}}$ on $[-\pi,\pi]$ and its values for $x=1.5,3,5$. In order to find the series I need ...
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1answer
161 views

Why coefficients of Fourier series are countable, though the initial periodic function is described with an uncountable set of points

Coefficients in the Fourier series for any periodic square-integrable function $f(x)$ form a countable (though infinite) set, i.e., they have cardinality $\aleph_0$. As far as Fourier exponents form a ...
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1answer
61 views

Integrals using Parsevals Theorem

I've been assigned two integrals to calculate in Fourier Analysis: $$\int_{-\infty}^{\infty}\left(\frac{\sin x}{x}\right)^2dx$$ ...
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405 views

Inverting a Characteristic Function for half-cubic Student T entailing a Modified Bessel of 2nd kind

The Characteristic function of the Student T with $\alpha$ degrees of freedom, $C(t)=\frac{2^{1-\frac{\alpha }{2}} \alpha ^{\alpha /4} \left| t\right| ^{\alpha /2} K_{\frac{\alpha ...
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Finding the Fourier series of a piecewise function

I'm s little confused about Fourier series of functions that are piecewise. Here’s an example of such a function: $$f(x) = \begin{cases} x & -\frac\pi2 < x < \frac\pi2 \\[5pt] \pi - x & ...
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1answer
74 views

Differential equation with fourier transform and convolution

We have differential equation $3s(t)-2s''(t)=r(t)\,$ and $s(t)$ is convolution $s=g*r\,$ where $g(t)=ae^{-b\left | t \right |}\,$ $\\a,b\in\mathbb R+$ Solve constans a and b. I tried to solve ...
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1answer
763 views

Fourier transform of power function $t^\alpha$

While studying the 1/f noise, I found this webpage http://www.dsprelated.com/showarticle/40.php It gives the following Fourier tranform pairs However, there are no detailed explanation on how ...
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2answers
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How to proof $ \underset{n\to\infty}\lim \int_0^1 f(t)\, \mathrm{sgn}\big(\sin (2\pi n t)\big)\,dt = 0$?

I was wondering if you could provide hints which could lead me to a rigorous proof for the following: Given $\,f\in L^1([0,1])$, then $$ \underset{n\to\infty}\lim \int_0^1 f(t)\, ...
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1answer
69 views

Using the Fourier Series of $f(t)=(t-\frac{1}{2})^{2}$ to deduce the sum $\sum_{n=1}^{\infty }\frac{1}{n^{2}}$?

So this is a question in one of the previous tests: My approach (if you want just skip to step 3.):$$$$ 1. Formulation of the problem and calculating the constant term of the series $a_o$ I ...
2
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1answer
167 views

Combining Two Gaussian Filters

I am taking a class related to image processing and we were taught about Gaussian Filters that are related to the following Gaussian Function: $$G(u,v) = \frac{1}{2\pi\sigma^2}e^{-\frac{u^2 + ...
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What is a good book, or article, that explains the history of fourier analysis?

What is a good book on the history of Fourier Analysis? I'm looking for a book which explains how it came to be and what the mathematicians (or physicists) were thinking when they came up with it. If ...
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0answers
65 views

Convergence of Fourier series in $L^\infty$

So if $f\in L^1(\mathbb{T})$ and $S_Nf\rightarrow f$ in $L^\infty(\mathbb{T})$ ($S_Nf$ is the partial sum of the fourier series of $f$), then $f$ is continuous. How do we show that this is true? In ...
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1answer
40 views

Estimating the rate of convergence of $|S_Nf-f|$ given that $\|f\|_{H^s}\leq 1$

Given that the Soloblev space norm $$\|f\|_{H^s}^2=|\widehat{f}(0)|^2+\sum_{n\in\mathbb{Z}}|n|^{2s}|\widehat{f}(n)|^2.$$ and the inequality $$\|f(\cdot +\theta)-f\|_{L^2}\leq 2\pi ...
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1answer
69 views

Surface measure and the wave equation

I am having an annoying conceptual problem trying to solve problem 46 in Chapter 8 of Folland's "Real Analysis". I'll try to explain my problem as briefly as possible. Consider the wave equation ...
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2answers
67 views

Fourier Transform of mix partial derivative

I know FT{$\frac{\partial u}{\partial x}$} = (ik)FT{u}. Give a function $U(x,y)$. Is the following true? FT{ $\frac{\partial^2 U}{\partial y \partial x}$} = FT{$\frac{\partial U}{\partial y}$} ...
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2answers
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Convergence of Fourier Series in $L^1(\mathbb{T})$

Suppose $f \in L^1(\mathbb{T})$ and the sequence of partial sums of its Fourier series converges (in $L^1(\mathbb{T})$) to $g$. How can I prove $f=g$?
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When orthogonal polynomials form an Hilbert basis?

Let $\mu$ be a probability measure on $\mathbb R$, and consider the sequence of orthonormal polynomials in $L^2(\mu)$. These polynomials are constructed by applying Gram-Schmidt to the sequence ...
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Frequency response and low pass filtering

I have one small doubt. I am studying filter banks, and hit upon by the frequency response of Haar wavelets. Which has the following form: $$\frac{1}{2} + \frac{1}{2}e^{j\omega}$$ This drops to to ...
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1answer
177 views

Simplifying big expression

What to do with this? $$f(x) = \frac{\sinh(\pi)}{\pi} + \frac{2\sinh(\pi)}{\pi}\sum_{n=1}^\infty (-1)^n \left[\frac{\cos(nx)-n \sin(nx)}{1 + n^2}\right]$$ Can it be simplified?
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Calculate the fourier transformation of a function to determine the answer of another…?

I have an assignment which is: Show that $\int_{-\infty}^\infty (\frac{\sin x}{x})^2dx=\pi$ by calculate the Fourier Transformation of $ f(t) = \left\{ \begin{array} /1, |t| \leq 1 \\ 0, |t| > 1 ...
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0answers
28 views

Fourier inversion of $f(k) \Leftrightarrow \lim_{k \to -\infty } f(k)=0$

Let $k \to f(k)$ be a function and define its Fourier transform as $$ \hat{f} (u) = \int_{-\infty}^{\infty} e^{iux} f(x) dx $$ if $\hat{f} (u)$ is integrable we can get back $f$ by doing the inversion ...
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170 views

Fourier series with half range

Question What are $a_0,a_n,b_n$ equal to with range $-L\leq x \lt0$, rather than the standard $-L\leq x \leq L$? For example: $$f(x)=2x^2,\quad-1\leq x\leq0$$ Instead of $f(x)=2x^2,\quad-1\leq ...
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1answer
332 views

Fourier transform of Bessel Function Second Kind

How do I prove the following equation, $$\frac{1}{\sqrt{(x^2 + y^2)}}=\int_0^{\infty}\frac{2}{\pi}K_0(yt)(\cos(xt))\,dt $$ This is a Fourier transform of $K$, I proceeded as follows: \begin{align} ...
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1answer
33 views

Fourier transformation of $f(t) = \frac{1}{1+9t^2} $

The assignment is to determine the Fourier transform of the following function: $$f(t) = \frac{1}{1+9t^2} $$ I have some rules that I think I can use: $\frac{1}{1+t^2}$ has the transformation $\pi ...
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2answers
40 views

Evaluate Fouries transform using properties

$$x(t)=t\ \left[\frac{\sin(t)}{\pi t}\right]^2$$ How can I find the Fourier transform of the above signal without direct integral evaluation(using Fourier Transform properties) The answer will ...
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0answers
50 views

What is the Fourier Transform of $f'(x)/x$? [duplicate]

What is the Fourier Transform of $f'(x)/x$? Is it even possible to find? It's deceptively simple looking. What about $f(x)/x$?
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1answer
97 views

What is the Fourier Transform of $f'(x)/x$ [closed]

What is the Fourier Transform of $f'(x)/x$? Is this even possible to find? It's deceptively simple looking.