Questions tagged [fourier-transform]

The Fourier transform is important in mathematics, engineering, and the physical sciences. It is a tool that breaks a waveform (a function or signal) into an alternate representation, characterized by sine and cosines. The Fourier Transform shows that any waveform can be re-written as the sum of sinusoidal functions. It is named after French mathematician Joseph Fourier.

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Solving $\widehat{f}=f$ with $f\in L^2$.

The following question arose. What functions do they satisfy? $\widehat{f}=f$ with $f\in L^2(\mathbb{R})$? Only the function $f=0$?
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Fractional Fourier Transform of $\sqrt{c} x( c(t - \tau))$

I am trying to figure out what the Fractional Fourier Transform of the signal $\sqrt{c} x(c(t-\tau))$ would be with respect to that of $x(t)$. According to the paper "The Fractional Fourier ...
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Solving the discrete Fourier Transform of $\sin(x)+\sin(2x)$

I'm new to the discrete Fourier Transform and have managed to do a few examples by hand, (for example the Fourier Transform of $\sin(x)$). I now want to try the transform on signals that are made up ...
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The Fourier Transform of a Radial Function on $\mathbb R^n$

I am trying to understand the following computation for the Fourier Transform of a Radial Function on $\mathbb R^n$. I shall ask questions in-line. Suppose $n\ge 2$, $f\in L^1(\Bbb R^n)$, and $f(x) = ...
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Counstruct a sequence of Schwartz functions

Here is an exercise from Wolff's lectures on harmonic analysis, p26: Using translation and multiplication by characters, construct a sequence of Schwartz functions $\lbrace\phi_n\rbrace$ so that Each ...
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How can I do a Fourier expansion for a waveform with uneven square pattern

The pattern of the wave is as follows: 0.0 - 0.5 => low, 0.5 - 1.0 => high, 1.0 - 2.0 => low, 2.0 - 2.5 => high, 2.5 - 4.0 => low, 4.0 - 4.5 => high, 4.5 - 6.5 => low, 6.5 - 7.0 =...
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Is the convolution of $L^2$ functions continuous? [duplicate]

Is the answer to the following question positive? The convolution of two $L^2(\mathbb R)$ functions is continuous I briefly recall it here: Take $f$ and $g$ $\in L^2(\mathbb R)$, then I want to show ...
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Using Fourier transform to solve a heat equation on an infinite bar with two different boundary values

I have the given PDE problem \begin{equation} u_t=\alpha u_{xx} \ \ \ \ 0<x<L, t>0 \\ u_x(0,t)=0 \\ u(x,0)= \begin{cases} 0 \ \ \ \ 0<x<L \\ Q \ \ \ \ L<x<\infty \end{cases} \end{...
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Formula connecting Fourier transforms of function and its derivative

Suppose $f: \mathbb{R} \to \mathbb{R}$ is continuous, differentiable and suppose $f$ and $f'$ are both summable. Then $$\mathcal{F}(f')(x) = -ix \mathcal{F}(f)(x)$$ (Here $\mathcal{F}$ represents ...
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Cesaro summation of the inverse Fourier transform.

Let $f\in \mathcal{L}(\mathbb{R}^1).$ Prove that $$f(x)=\frac{1}{\sqrt{2\pi}}\lim_{T\to+\infty}\frac{1}{T}\int_{0}^{T}\int_{-t}^{t}e^{ixy}\hat fdy\,dt$$ for almost all x including points of continuity ...
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Let $f \in C^1(\mathbb{R})$ be such that both $f$ and $f'$ belong to $L^2(\mathbb{R})$. Show that $\hat{f} \in L^1(\mathbb{R})$

The exercise is the following: Let $f \in C^1(\mathbb{R})$ be such that both $f$ and $f'$ belong to $L^2(\mathbb{R})$. Show that $\hat{f} \in L^1(\mathbb{R})$. What I want to prove is that $\int_\...
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Inverting Fourier Transform for Heat Kernel

I am trying to show that the Fourier Transform of the heat kernel is given by a simple Gaussian but I have no idea how to handle the prefactor 1/sqrt(2pi). The first line is from my lecture notes. ...
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Fourier transform of $sinc(cos(t)) $

Given Linear system $Q $ with input $x(t)$ and output $ y(t) $ and impulse response to $ \delta(t-\tau) $: $$ h\left(t,\tau\right)=sign\left(\cos\left(\omega_{0}t\right)\cos\left(\omega_{0}\tau\right)+...
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When does Inverse Fourier transform look close to a positive definite function?

Let $G$ be a commutative locally compact group, and $\hat{G}$ be its dual group, consisting of all continuous characters (continuous homomorphisms from $G$ to the circle group $\mathbb{T}$) . I can ...
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integration by parts on a fourier transformation

Given is a function $$ \dfrac{1}{2 \pi} \int\limits_{- \infty}^{+ \infty} \mathrm{e}^{-itx}\dfrac{f(t)}{g(-t/h)} \, dt.$$ In the paper I'm working with it's written that by integration by parts, we ...
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Fourier transform of trigonometric/polynomial function

I'm completely stuck in trying to calculate the following Fourier transform: $$ \frac{x\cos(x)-\sin(x)}{x^2} $$ I have no idea how to proceed since it is impossible to integrate elementarily $\frac{e^{...
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Connection between "Cauchy" delta function and "Fourier" delta function?

A delta function has the sampling property, it picks out the value of a function $f(x)$ at a point $a$ $$ f(a) = \int_{-\infty}^{\infty} \delta(x - a) f(x) dx $$ The Fourier integral theorem says $$ f(...
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Relationship between a vector and its discrete fourier transform

Suppose we have a real vector $a=(a_0,\ldots,a_{n-1})$ and let $b=(b_0,\ldots,b_{n-1})$ be its discrete fourier trasnform. So $$\mathscr{F}(a_k)=b_k=\sum_{i=0}^{n-1}{a_i}e^{\frac{-j2\pi i k}{n}}$$Let $...
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Fourier transform of a beat

What is the Fourier transform of a beat? For example, I want to calculate the Fourier transform of the function $$f(t)=\cos((\omega_p+\omega_v) t)+\cos((\omega_p-\omega_v)t),$$ where $\omega_p+\...
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Infinite series Sum of zeroth order Bessel Functions of first kind

I am trying to find the upper bound of $$ \sum_{n \geq 1} J_0(an) J_0(bn) \sin(cn) \sin(dn) $$ where $J_0(x)$ is the zeroth order Bessel function of first kind, and $a,b \geq 0, \textit{ and } c,d \in ...
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Inverse Fourier transform of a function that changes at f=0 [closed]

I am given $$G(f)= \left\{\begin{array}{rl} 1,&f>0\\ 1/4,&f=0\\ 0,&f<0 \end{array}\right.$$ for which I have to compute the Fourier transform. How can I do that?
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How to compute the Fourier transform of a zero-centered Gaussian pulse over a specified range, say $[0,1]$?

Given an even Gaussian kernel $f : \Bbb R \to \Bbb R^+$ $$f(x) = \exp \left(- \frac{1}{2\sigma^2} x^2 \right)$$ and a probability density function (or, if you discretize $[0,1]$, then a probability ...
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Banach algebra $L^1(\mathbb{R})$ and its Fourier transform.

We know $L^1(\mathbb{R})$ is a Banach algebra whose product is defined to be convolution: $$f*g(x)=\int f(x-t)g(t)dt.$$ In fact, it is a $*$-Banach algebra (Don't confused with the convolution), with $...
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How to compute the discrete Fourier transform of f(x) = 1?

I am learning about the discrete Fourier transform in the context of digital image processing and not in the context of complex analysis so I might have some slightly different notation/formulas. Our ...
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Support of Fourier transform and Inverse Fourier Transform

Suppose $f \in L^2(\mathbb{R})$ and let $X \subseteq \mathbb{R}$ be a set of finite measure. Here let $\mathcal{F}$ and $\mathcal{F}^{-1}$ denote the Fourier transform. Q: If the support of $\mathcal{...
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Fourier transform on a multivariate function along a line

Given a function $f(\vec{x}) :\mathbb{R^3} \rightarrow \mathbb{R}$, it is always possible to compute the Fourier transform to compute of f, $F(\vec{\omega}): \mathbb{R^3} \rightarrow \mathbb{C}$. ...
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What is the Fourier transform of $xf(2x)$?

Given $f\in L^1(\mathbb{R})$ is a generic function, how do you find the Fourier transform using tables? Thus far I've gotten: $F(xf(2x))=F(xg(x))=(\frac{i}{2n})^1*\frac{d}{d\omega}\hat{g}(\omega)$ $\...
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Function in $H^1(\mathbb{R})$

Consider the function $$f=|x|^pe^{-x^2},$$ where $p$ is a real constant. The function $f$ is in $L^2(\mathbb{R})$ iff $p>-1/2$. The function $f$ is in $H^1(\mathbb{R})$ iff $p>1/2$ or $p=0$. I ...
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How to prove $|-\Delta_{x} ((-x)^{\alpha} \phi(x))|\leq A_{j,\alpha}(1+|x|)^{-n-1}$

How to prove $|-\Delta_{x} ((-x)^{\alpha} \phi(x))|\leq A_{j,\alpha}(1+|x|)^{-n-1}$ Hi all, i am reading Pseudo-differential Operators, singularities, applications. Y Egorov, on page 2. The inequality ...
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Dispersive Waves on a Semi-Infinite String

I am having an insane amount of trouble figuring out this problem that I solved probably ten years ago. Googling leads to solutions that make use of D'Alembert's formula, but that doesn't work for ...
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Solve $g_1(x) \mathcal{F}[e^{-c_1 f(x)}] = g_2(x)\mathcal{F}[e^{-c_2 f(x)}]$ for $f(x)$

Is there any way to approach such a problem? Let $f:\mathbb{R} \to [0,\infty]$ and $g_1, g_2:\mathbb{R} \to \mathbb{R}$ be smooth, compactly supported functions, and let $c \in [0,\infty]$. Denote the ...
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Frequency peak always appearing at half of Nyquist frequency in Fourier transform

When taking the FT of a signal I always get a sharp peak at exactly half the Nyquist frequency. My signal is shown here: and its FT here: The Nyquist frequency is 36.7 KHz. As can been seen in the ...
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Fourier transform with an additional term in the power

Consider this relation $$\sigma(\omega_{I})=\int_{-\infty}^{+\infty} <\psi(0)|\psi(t)>e^{i(E_i/\hbar+\omega_I)t}dt$$ I know, by inverse Fourier transform, I can compute for $$\int_{-\infty}^{+\...
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Wigner function derivation of a multicomponent signal

A multicomponent signal can be expressed in exponentials can be expressed as follow, $$x(t)=\sum_{k}c_k e^{2\pi f_0 t}$$ The Fourier transform of $x(t)$ is easy $$\mathcal{F}_x(f)=\sum_k c_k \delta(f-...
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Finding solutions to $ \sin\left(n \pi \frac{T_p}{T}\right) - \sin\left(n \pi \frac{T_0}{T}\right) = \sin\left( \frac{{n \pi}}{2}\right) $

I'm trying to figure out how I can find a solution(s) to the equation below. $$ \sin\left(n \pi \frac{T_p}{T}\right) - \sin\left(n \pi \frac{T_0}{T}\right) = \sin\left( \frac{{n \pi}}{2}\right) $$ ...
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Applying Fourier transforms in two variables separately

Let $G(x, t) := (4\pi t)^{-d / 2}e^{-|x|^2 / 4t}$ for all $x \in \mathbb{R}^d$ and $t \in (0, \infty)$. The function $G$ is not integrable on $\mathbb{R}^d \times (0, \infty)$ since $$ \int_0^\infty \...
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Laplace equation on a rectangle with inhomogeneous boundary conditions

I am trying to solve Laplace equation in cartesian coordinates, on a rectangle defined by $x_1<x<x_2$ and $-y_0<y<y_0$: $$\nabla^2 g=0$$ with $$g(x,y=\pm y_0)=f(x)\\ g(x_1,y)=f(x_1) \\ \...
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Poisson Summation at Half Integers

In the proof of Poisson Summation, for a Schwarz function $f$, you define $$ F(x)=\sum_{n\in\mathbb{Z}}f(x+n) $$ and you show that $$ F(x)=\sum_{n\in\mathbb{Z}}\hat{f}(n)e^{2\pi inx} $$ Then plugging ...
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How to solve integral equation of convolution using Fourier transform

I'm having trouble solving the following exercise: Use the Fourier transform to solve the integral equation $$f(x) = \int_{-\infty}^{\infty} e^{-|x-\xi|}u(\xi)\,d\xi$$ Then verify your solution when $...
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Fourier transform of exponentially modified gaussian

So i have to calculate the fourier transform of an exponentially modified gaussian $$f(x, A, \lambda, \mu, \sigma)=\frac{A\lambda}{2}e^{\frac{\lambda}{2}(2\mu+\lambda\sigma^2-2x)}erfc(\frac{\mu+\...
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Example of Fourier Transform that's not in $L^p$.

For $f \in L^1(\mathbb{R}^d)$, define $\hat{f}(u)=\int f(x)e^{iux}dx$. I know that $\hat{f} \in L^{\infty}$ and that if $f \in L^2$, then $\hat{f} \in L^2$. I also know examples where $\hat{f} \notin ...
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Periodicity of sub columns in Hadamard matrix

Let's consider the Hadamard transform $H_n$ where $H_{ij} = (-1)^{i.j}$. I want to count the number of repeated sub-columns of length $l$ in this matrix. Does it exist any formula or combinatorial ...
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Derivation of Wigner Function of trigonometric functions with phase

The Wigner Function of $x(t)$ is $$ W(t,f) = \frac{1}{2\pi}\int x\Big(t+\frac{\tau}{2}\Big)x^*\Big(t-\frac{\tau}{2}\Big) e^{-j2\pi f\tau}\;d\tau $$ I know how to get the $W(t,f)$ of $x(t)=\cos(2\pi ...
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Evaluate $\int_0^{\infty } k e^{-\left(x \left(k^2-\frac{\text{k1}^4}{k^2}\right)\right)} \sin (k r) \, dk$

I'm trying to do the 3D Fourier transform of the function given below, $$\phi(k)=\frac{1}{k^{2}+k_{1}^{2}-\frac{k_{1}^4}{k^{2}}}$$ To get back the real space function, $$\phi(r)=\int\frac{d^{3}k}{(2 \...
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Fourier tranform of signum in several dimensions

I want to calculate the Fourier transform of $$ f: \mathbb{R}^d \to \mathbb{R}, \; x \mapsto \mathrm{sgn}(x \cdot \theta), $$ where $\theta \in \mathbb{R}^d$ is fixed. As $f$ is not in $L^1$ I ...
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Solution of Integral equation via Fourier Transform

Solve for f(t) using Fourier transform: $$ \int_{-\infty}^\infty f(s)f(t-s)\,ds - 2\sqrt{2} \int_{-\infty}^\infty e^{-s^2/\pi}f(t-s)\,ds = -\sqrt{2}\pi e^{-\frac{t^{2}}{2\pi}} $$ Now, I get the ...
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2 votes
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Fourier series of $\sqrt[3]{\sin x}$

So I've done some experiments with how to add distortion to audio, and one of the methods I'm proposing is to take the cube root of an audio signal as a way to add overdrive. As the waveform that you'...
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3 votes
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Proving (or disproving) that a Hierarchical Distribution of the Stable Distribution Location Parameter is also Stable

The Normal and Cauchy distributions belong to a class of distributions known as the Stable distributions. For the Normal and the Cauchy, if you design a hierarchical model where their location ...
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Why we ignored the $e^\infty$ term in the result? [closed]

In Question (2.20) of Griffiths' Quantum Mechanics book, they have given this Solution. In the Solution of question 2.20(b), they omitted $e^{(ik-a) \infty}$ (or may have considered $e^{(ik-a) \infty}=...
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Fourier transform of $\dfrac{e^{-ax}}{x}$

I am required to find the Fourier transform of the following function: $$f(x) = \dfrac{e^{-ax}}{x}$$ Where $a$ is an arbitrary constant. This is what I've tried, but I'm not quite sure: The Fourier ...
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