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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Fourier transformation of $\cos^2(\pi t)$

I need to fouriertransform the function $$f(t) = \cos^2(\pi t)$$ First question is if there is some kind of theorem to solve it in a very easy way but I don't know what will happen to the ...
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Fourier transform in reconstruction problem

I'm trying to solve Exercise 20 of Chapter 5 of Fourier Analysis by Stein. The problem is as follows: Suppose $f$ is of moderate decrease and that its Fourier transform $\hat{f}$ is supported in $I=[-...
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To bound a heat equation on a real line?

Let $\displaystyle\mathcal{H}_{t}(x)=\frac{1}{(4\pi t)^{1/2}}e^{-x^{2}/4t}$ be the Heat Kernel. The imposed initial condition for the heat equation on a real line is $u(x,0)=f(x)$ a function belongs ...
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$\{x^nf(x)\}_{n\in\mathbb{N}}\subset L_2(a,b)$ as a complete system

I read in Kolmogorov-Fomin's (p. 430 here) the statement, sadly left without a proof, that if function $f:(a,b)\to\mathbb{C}$, measurable almost everywhere on $(a,b)$, where $-\infty\leq a<b\leq\...
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96 views

Finding Fourier cosine series of sine function

I am trying to find Fourier cosine series of following function, but think that I am messing up somewhere. $$ f(x)=\sin \bigg ( \frac{\pi x}{l} \bigg ) $$ Fourier cosine series can be written as $$ f(...
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$\int_{\mathbb{R}}f(x)e^{-ixz}d\mu_x$ analytic for $f\in L_1$

Let $f\in L_1(-\infty,\infty)$ be a Lebesgue-summable function on $\mathbb{R}$ and let $x\mapsto e^{\delta|x|}f(x)$ also be Lebesgue-summable on all the real line. I have added the condition that $f\...
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Fourier transforms in combinatorics

I've got something like a "meta-question": In some parts of combinatorics, looking at Fourier transforms can be a very helpful tool. For example, an early proof of Roth's theorem (any sufficiently ...
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316 views

Multidimensional Fourier Transform

I'm having difficulty with multidimensional Fourier Transforms. I have the following problem for $u=u(t,x) \in \mathbb{R}$ $$ \frac{\partial u}{\partial t} = \sum_{m,n=1}^d a_{mn}\frac{\partial^{2}u}{...
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230 views

Derivative of Fourier transform: $F[f]'=F[-ixf(x)]$

Let us define the Fourier transform of the Lebesgue-summable function $f\in L_1(\mathbb{R},\mu_x)$ as $F[f](\lambda)=\int_{\mathbb{R}}f(x) e^{-i\lambda x} d\mu_x$, where $\mu_x$ is the Lebesgue linear ...
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Using Fourier analysis to show a function is positive

Let $$f(x)= \sum_{n=-\infty}^{\infty} \frac{\mathrm{e}^{i nx}}{n^2+1}$$ on $[-\pi, \pi]$. Prove that $f(x)>0$ for any $x \in [-\pi, \pi]$. How to use Fourier analysis to show that function is ...
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54 views

the Fourier transform of a constant

How to calculate the Fourier transform of a constant without the aid of duality property? In other words, how do I calculate $$ \int_{-\infty}^{\infty}e^{-j\omega t}dt? $$
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If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ has the distribution of $aX$, find all characteristic functions of $X$.

If $X_1,X_2,X$ are iid random variables with $X_1+X_2$ have the same distribution as $aX$ for some real $a$, what are the possible characteristic functions of $X$? Let $\varphi_X(t)$ be ...
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416 views

Laplace equation on infinite strip

I'm trying to solve the following problem using the Fourier transform: $$u_{xx}+u_{yy}=0$$ on the domain $\;0\lt y\lt b$ , $-\infty\lt x \lt \infty \;$ with the following conditions: $$ u(x,0)= \...
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90 views

Difference between the Rectangular “Window” Function and the Rectangle Function

I'm getting ahead in my differential equations textbook (Fundamentals of Differential Equations by Nagle et. al) and in the chapter of Laplace Transforms it states that the rectangular window function ...
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116 views

Show that Fourier transformation is differentiable if $\int|xf(x)|\,d\lambda<\infty$

Let $f\in\mathcal{L}^1(\mathbb{R},\mathcal{M},\lambda)$. Then we define the Fourier transform of $f$, denoted $\hat{f}$, by $$\hat{f}(t)=\int_\mathbb{R}e^{-itx}f(x)\,d\lambda(x),\;\;\;\;\;\;\;t\in\...
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42 views

Suppose $f$ is integrable on $\mathbb{R}$, and $g$ is locally integrable and bounded. Then $f*g$ is uniformly continuous and bounded?

Suppose $f$ is integrable on $\mathbb{R}$, and $g$ is locally integrable and bounded. Then $f*g$ is uniformly continuous and bounded? I don't even know where to start proving or disproving, but ...
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$\frac{\varphi(x+h)-\varphi(x)}{h}\to\varphi'(x)$ uniformly as $h\to 0$?

Let $\varphi$ be a bounded, differentiable function on $\mathbb{R}$ such that $\varphi'$ is bounded and uniformly continuous on $\mathbb{R}$. We want to prove that $\displaystyle\frac{\varphi(x+h)-\...
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47 views

Asymptotic behaviour of Fourier transform: $|F[f]|=|\lambda^{-k}F[f^{(k)}]|$ for absolutely continuous $f$

I read in Kolmogorov-Fomin's (p. 429 here) that if function $f:\mathbb{R}\to\mathbb{C}$ is such that $f^{(k-1)}$ [the $(k-1)$-th order derivative] on any finite interval and if $f,...,f^{(k)}\in L_1(-\...
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33 views

For $f(\theta)= e^{\theta}$. Is it true that $\hat{f}(n)(1-in)=0$ for all $n\in \mathbb Z.$?

(This is motivated from the following question) Fact: If $f \in C^1(\mathbb{T})$, then the Fourier coefficients $\widehat{f'}(n)$ of the derivative $f′$ can be expressed in terms of the Fourier ...
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28 views

convergences in $\mathcal {S'}$

strong textLet $\phi$ 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 $|\xi|\leq 1.$ My Question is: ...
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Sampling theorem.

Let us consider \begin{equation} \hat{f}(x)=\sum_{n\in \mathbb Z}\left\langle\hat{f},e^{i n x}\right\rangle_{L^2[-\pi,\pi]} e^{i n x} \ \ \ \ \ \ \ \ (1) \end{equation} where $\langle g, h\rangle_{L^2[...
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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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93 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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85 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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305 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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377 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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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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362 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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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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172 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 $f\psi$ ...
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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 $1$...
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193 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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64 views

$p$-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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$f$ 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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207 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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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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329 views

Show that the Fourier Transform is differentiable [duplicate]

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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92 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 \{|B_{x}|^...
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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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362 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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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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112 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)} \frac{1}{|B(x,R)|...
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58 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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252 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)| \mathop{dx}<\epsilon$...
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31 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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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 $$e^{\...
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44 views

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 $\mathcal{F}\{f(x)\}=\mathbb{...
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44 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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141 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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213 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 ...