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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How to change the fundamental frequency of a sample signal?

So I am dealing with a 60Hz signal that is sampled at 1kHz. This 60Hz signal has many other harmonics (eg, 120 Hz, 180Hz..... and more). For some reason, we would like it to be 50Hz. Could we ...
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36 views

Is $ \frac{2}{1+e^{t^2}} $ a characteristic function?

I'm trying to establish whether the following is a characteristic function of some random variable: $$ \phi(t) = \frac{2}{1+ e^{t^2}} .$$ It satisfies all basic characteristic function properties, ...
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52 views

Problem with (classical) Fourier transform

Problem Find the Fourier transformation of $u(x) = \frac{1}{1+x^2}$ I want $\int_\mathbb R e^{-itx} \frac{1}{1+x^2} dx$. Let $f(z) = e^{-itz} \frac{1}{1+z^2}$, $z \in \mathbb C$, let's integrate ...
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43 views

Fourier transform of Gaussian - sin/cos/Heaviside step function.

I would like to drive the Fourier transform of the following equations: $f_1(x)=e^{\frac{x^2}{2σ^2}}\cos(nx)$, $f_2(x)=e^{\frac{x^2}{2σ^2}}\sin(nx)$, where $n=2πf$ ...
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11 views

Discrete Fourier Transform actual frequency of pure tone function

Given that a signal s is amplified at $200Hz$ for $10$ seconds, yielding a sequence $s_t$ for $t = 0, 1, ..., 1999$. We have $s_f = \frac{1}{\sqrt{2000}} \sum_{t=0}^{\infty} ...
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16 views

Fourier Transform Inverse of 1 / (jw - a)

I want to find the inverse fourier transform of $$ \frac 1 {j \omega - 1} $$ The fourier transform of $$ e^{-at} u(t) $$ is $$ \frac {1}{j \omega + a} $$ This result if true ONLY if a > 0. If a ...
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7 views

Show the Hankel transform for Gaussian function

We have Gaussian-like function in $k$-domain $$G(k)=2\lambda^2 \sqrt{2K/\pi}e^{-\frac{\lambda^2 k^2}{2\pi}} $$ using the expressions for the Fourier transform ...
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33 views

Fourier Transform - Limit not existing

I was wondering what happens when calculating the fourier transform of a fonction, the limit does not exist. Let me explain this with an example : I have the function difined by : ...
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22 views

How to decompose tempered distribution by entire analytic functions?

Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with $ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1$ if $|\xi|\leq 1.$ Let $j\in \mathbb N$ and ...
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17 views

unilateral Fourier transform exists?

Is there one-sided Fourier transform (unilateral Fourier transform)? For example we have two-sided and one-sided Laplace transform.
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22 views

confusion about Morlet Wavelet: What is it exactly?

I was trying to follow and implment a method propose on the research paper. And currently, I have having some trouble to understand the wavelet transform. In particular, the paper I am looking at is ...
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42 views

Schwartz Space is closed under differentiation and multiplication by polynomials.

Schwartz space is closed under differentiation and multiplication by polynomials. In addition, if $f$ is a smooth function will all derivative bounded and $\psi$ is a Schwartz function, then ...
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22 views

What are $f(x)$ and $f(y)$ here in this heat equation problem?

Question: What is $f(x)$ and what is $f(y)$? You are given (no need to check) that the function $G(x-y,t)$ defined by $$G(x-y,t)=\frac{1}{\sqrt{4\pi c^2 t}}e^{-(x-y)^2/4c^2t}$$ satisfies the ...
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11 views

Wavelet transform closed form expression [closed]

What is the wavelet transform of a gaussian $exp(\frac{-x^2}{2\sigma^2})$?
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22 views

Characteristic functions and conditional distributions?

Say X and Y are random variables and we're interested in the conditional distribution of X given Y, can we make this calculation using only characteristic functions in a straightforward manner? If so ...
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21 views

Condition for uniform convergence of Fourier series

Let $f$ be a Lebesgue summable periodic function on $[-T/2,T/2]$. I read in Kolmogorov-Fomin's (p.414 here) that if $f$ is bounded on a set $E\subset[-T/2,T/2]$ and for any $\varepsilon>0$ there is ...
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$p$-stable Random Variables for $p>2$?

I will preface this by saying I am certainly no expert in Probability theory. My actual problem is an interpolation one, in which I am considering interpolation of bandlimited functions with shifts ...
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30 views

$f$ absolutely contininuous $\Rightarrow f\cdot\sin$ absolutely continuous?

I wonder whether, if $f:[a,b]\to\mathbb{C}$ is an absolutely continuous function, multiplying it by $\cos\frac{2\pi nx}{b-a}$ or $\sin\frac{2\pi nx}{b-a}$ results in another absolutely continuous ...
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9 views

Fourier transform frequency shift proof from duality and translation property

I read that you can prove the frequency shift property of the Fourier transform from the translation property using the duality property. I have tried substituting things into one another, ...
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17 views

what does support of convolution of functions says geometrically?

Let $f,g \in L^{1}(\mathbb R)$ we define $f\ast g(x)= \int_{\mathbb R} f(x-y)g(y) dy $ for all most all $x,$ and denote $\text{supp} (f)$ the support of $f.$ Fact: If $A$ is the closure of $\{x+y: ...
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65 views

Show that the Fourier Transform is differentiable

This might be a silly question. For $f$ an integrable, complex-valued function, its Fourier transform is $$ \hat{f}(s) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty} e^{-isx}f(x)\, \mathrm{d}x $$ I ...
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Proof the Fourier Transform is Unitary/Not Unitary

I couldn't think of a better title, but could someone provide a proof as to the unitarity of the Fourier Transform from time to angular frequency with the $\frac{1}{\sqrt{2\pi}}$ in front of the ...
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23 views

Hardy-Littlewood maximal function $Mf$ is greater than $f$

Suppose that $f\in L^{1}(\mathbb R)$ and $x\in \mathbb R.$ We denote $B_{x}$ by the ball in $\mathbb R$ with centred $x,$ and $|B_{x}|=$ length(Lebsgue measure) of $B_{x}.$ Put, $Mf(x)=\sup ...
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13 views

Fourier analysis on stock price history

What are some ways by which a person might use Fourier Analysis on stock quote price histories? In particular, I want to learn more about the rate at which stock prices oscillate, and come up with ...
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20 views

Determining if two expressions are equal, in order to ensure a Fourier series is correct

Motivation: I have a question that asked me to find the Fourier series of some function $f(x) = \left\{\begin{array}A,\quad -1\lt x \leq 0 \\ Ax, \quad 0 \lt x \leq 1 \end{array}\right.$ periodic on ...
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28 views

Divergence $\int_{-\pi}^{\pi} |D_n(x)|dx$ for Dirichlet kernel as $n\to\infty$

Let $D_n(x)$ be the Dirichlet kernel defined by $$D_n(x):=\frac{\sin\frac{(2n+1)x}{2}}{2\pi\sin\frac{x}{2}}$$where $D_n(0)$ can be set to $\frac{2n+1}{2\pi}$ if we desire it to be continuous. Another ...
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8 views

How to compute multidimensional inverse Fourier transform

everybody. I am currently working on deriving solutions for Stokes flows. I encounter a multidimensional inverse Fourier transform. I already known the Fourier transform of the pressure field: ...
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64 views

Technical question about Strichartz estimate's proof.

I was studying the proof of Strichartz estimates from the book "Semilinear Schrödinger equation" of T. Cazenave. The proof is divided in several steps. Here we can assume ...
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46 views

Showing two things are equal by Fourier series

Given the Fourier series for the function: $$f(x) = x+\frac14x^2 \quad -\pi\leq x \lt \pi$$ $$f(x)=f(x+2\pi) \quad -\infty \leq x \lt \infty$$ is $$\frac{\pi^2}{12}+\sum \limits_{n=1}^\infty (-1)^n ...
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22 views

Improvement of weak type inequality for Hardy-Littlewood Maximal inequality

Let $B(x,R)$ denotes the ball in centered at $x\in \mathbb{R}^n$ with radius $R$. The centered Hardy-Littlewood maximal operator $M$ is defined by \begin{equation} Mf(x)=\sup_{B(x,R)} ...
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36 views

Find the $n$-th Fourier transform of $e^{-|x|}$

The objective is to find the $n$-th Fourier transform of function $e^{-|x|}$. So i started of with finding the first Fourier transform and the result is $\frac{2}{y^2+1}$. Now I wanted to find its ...
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10 views

Order of distribution of the zeros of the interference function of periodic oscillations?

Given a finite (or an infinite) number of periodic oscillations of different shapes but even functions, along the abscissa $x∈R$, every such periodic oscialltion may cross on some zero points the ...
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31 views

Accessible textbook about basic Fourier analysis in terms of integrals wrt measures

I am looking for a basic and accessible textbook (or set of lecture notes) that discusses basic fourier analysis but in terms of measures and integrals with respect to measures. Not sure if it is done ...
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37 views

Approximating Riemann integrable functions on $[-\pi, \pi]$ by continuous (periodic) functions and trigonometric polynomials

Let $f$ be a Riemann integrable function on $[-\pi,\pi]$, and let $\epsilon>0$. Prove: 1) There is a function $g\in C[-\pi,\pi]$, satisfying $$\int_{-\pi}^{\pi}|f(x)-g(x)| ...
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24 views

Why isn't $f_{tt}(\vec{x},t)g(\vec{x},t) - f(\vec{x},t)g_{tt}(\vec{x},t)$ always equal to zero?

Consider the following expression $$H(\vec{x},t) = f_{tt}(\vec{x},t)g(\vec{x},t) - f(\vec{x},t)g_{tt}(\vec{x},t)$$ where f and g are functions of time and space and the subscript " $_{tt}$ " denotes a ...
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34 views

A question on convergence of derivative of power series

This is a question from Fourier Analysis with Applications by Folland. First we write Fourier series for $$e^{\theta}=\sum c_ne^{in\theta}$$ We differentiate this series term by term to obtain ...
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Fourier Series Transformation

I have a question regarding the following: Compute the Fourier transform of $f(x)=xe^{-2x^2}$, $x\in\mathbb{R}$. The Fourier Transform of $f(x)$ is given by ...
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15 views

Deconvolution of two delta functions (solving $y(t) = A x(t-a) + B x(t-b)$)

I would like to calculate $x(t)$, when only $y(t)$ with $y(t) = A x(t-a) + B x(t-b)$ is known. Since this is a linear shift invariant operation (convolution), the inverse relation must be of the ...
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1answer
78 views

Approximation of a $L^1$ function by a dominated sequence of continuous functions

Consider $\mathbb{T}=\{z\in\mathbb{C}\mid |z|=1\}$ and the Lebesgue measure on it. Denote by $L^1(\mathbb{T})$ the set of integrable functions on $\mathbb{T}$ and by $C(\mathbb{T})$ the set of ...
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1answer
37 views

Fourier Series of Real-valued Functions

Context: For a $2\pi$-periodic bounded function $f:\mathbb{R}\to\mathbb{C}$, we define the complex Fourier coefficients of $f$ by $$ \hat{f_k}:=\frac{1}{2\pi}\int_0^{2\pi}f(x)e^{-ikx}\,dx. $$ We call ...
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12 views

Is there any way to use a Fourier Transform or a variant to find periodic increases?

Suppose I have a staircase function, which has a periodic increase but no periodic decrease. I've been playing with Fourier transforms recently, and I know one main use is to pick out frequencies ...
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24 views

Band-limited function vanishing on a set of positive measure

I'm working my way through a Fourier analysis textbook and came across the classic result that a function cannot be both band- and time-limited, that is, if both $f$ and $\hat{f}$ are compactly ...
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Probability - Characterizing goodness of moment matching method.

I have a question about how to characterize the goodness of approximating a distribution using its moments. Suppose I have a probability density function $p(x)$ (e.g., normal distribution), and I am ...
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11 views

Fourier Cosine series expansion for two dimensional function

I have a two dimensional function with its values and range. I need to expand the function in Fourier cosine series. The function as follows: $$f(x,y) = \begin{cases} A &, -\frac{L}{2} + 2nL < ...
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28 views

Find the Fourier coefficients of $cos^2(x)sin^2(x)+2cos^3(x)$

So I know that the Fourier coefficients are expressed as: $$a_0 = \frac{1}{\sqrt{2}\pi} \int_{-\pi}^{\pi}f(t)dt$$ $$c_k = \frac{1}{\pi}\int_{-\pi}^{\pi}f(t)\cos(kt)dt$$ $$b_k = ...
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1answer
19 views

Hilber transform on [0,1)

Let $\mathbb{T}=[0,1)$ and $H$ be a Hilbert transform on $L^p(\mathbb{T})$ when $2\leq p< \infty$. If $f$ is $L^p$ and $f_n$ is trignometric polynomial such that $f_n\rightarrow f$ in $L^p$ sense. ...
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21 views

Formulation of Fourier transform

I would like to know about Fourier transform more. I attended a standard lecture of mathematics, but we did not talk about Fourier transform on $L^2$ much, nor the theory of $L^2$. We only defined it ...
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10 views

A question about time series analysis with fast fourier transform

I have a question about time series analysis with FFT. from here I have understood, to calculate a periodogram of a time series we should do these to steps: Fast Fourier trasnfomation of time ...
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1answer
11 views

Fourier transform for specific signal

Could someone show me how to calculate fourier transform for the following signal? s(t) = e^(-3*t^2) Thanks!
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46 views

Find the density of the random variable with characteristic function $\varphi(t)=(1-|t|)^+$.

Suppose a $X$ is a random variable, I am asked to find the density of the random variable with characteristic function $\varphi(t)=(1-|t|)^+$. I am trying to use the inversion formula for the ...