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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Why the spatial/mathematician's Fourier Transform?

I was wondering why the sign-change in the exponential of the spatial/mathematician's Fourier Transform and why is it called mathematician's spatial in either case?
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derivative of a function using cosine transform

If we have a periodic function y(x) the derivative of function can be found using Fourier transform as $derivative = y'(x)$ taking Fourier transform $F(derivative) = -kiy(x)$ i is the complex ...
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Fourier transform with non sine functions?

Fourier says that any periodic function can be represented like a infinite sum of sine functions with their appropriate periods,amplitudes and phases. My question is: is it possible to represent the ...
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28 views

Question about Fourier transforms of gradient, curl and divergence

Consider a vector field $v:\mathbb{R}^3\rightarrow\mathbb{R}^3$. Denote by $F_u$ the Fourier transform of a scalar or vector field $u$. Can one finds an equality relation between ...
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Relation between discrete and continuous inner product of a function

Let $g_1^d$ be descrete (sampled) version of continuous function $g_1$ , same for $g_2$. So we have $$\left<g_1^d,g_2^d\right>=\sum_{n=-\infty}^\infty {g_1^d[n] ...
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Fourier transform of $f'(t)$

The Fourier transform of f′(t) is : $$\hat{f'}(\omega) = \int_{-\infty}^{+\infty}f'(t)e^{-i\omega t}dt = f(t)e^{-i\omega t}\bigg\vert_{-\infty}^{+\infty} ...
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Increasing the points in a time scale changes the shape of the fft-function

This question is derived by this question and the corresponding answer. The problem was that I had a $sech(x)$-function in a specific time interval, and I applied a ...
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Is there any pde whose solution evolves as a partial Fourier integral?

Is there any partial differential equation such that the its solution evolves as partial Fourier integral (continuous version of partial sum) of a function $f(x)$ which might be an condition or ...
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How can I show the approximate version of the fourier inversion formula?

Let f be $L^1(R) \cap C_0(R)$ and satisfies $|\hat{f}(\alpha)|\leq A\frac{1}{|\alpha|}$, for all non zero real $\alpha$, for some positive A. Then, show that for any $x \in R$, $f(x) = ...
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How can I show that there is M>0 for all positive a<A s.t $|{\int_{a}^A \frac{ \hat{f}(\alpha)}{\alpha} \ d\alpha}| <= M $? [closed]

Let f be $L^1(R)$ and odd function. Then, for any positive $a < A$, there is $M>0$ such that $$ \left|{\int_{a}^A \frac{ \hat{f}(\alpha)}{\alpha} \ d\alpha}\right| \leq M $$ ($\hat{f}$ is the ...
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51 views

Inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$

Does talking about the inverse Fourier transform of $\frac{1}{\sqrt{\xi}}$ even make sense? If it does, how can we conclude about the decay properties, support and smoothness of the inverse Fourier ...
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47 views

Compare between Short Time Fourier Transform and Wavelets

Fourier transform is localised in only frequency domain but Short time Fourier transform(STFT) is localised both in time and frequency domain same as in wavelets. I want to know How are STFT and ...
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334 views

Discrete Fourier Transform: Understand Negative Frequencies

I am trying to learn DFT on my own. I have been struggling for a while now around understanding the concept of negative frequencies and notably what happens when $k$ is greater than $N/2$ in the ...
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24 views

Fourier transform of the principal value distribution

I would like to compute the Fourier transform of the principal value distribution. Let $H$ denote the Heaviside function. Begin with the fact that $$2\widehat{H} =\delta(x) - \frac{i}{\pi} ...
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Fourier Transform of Constant Piecewise Function

Hello I am trying to find the Fourier Transform of the following piecewise function. $$ f(x)=\begin{cases} 1 \quad 0\le x\lt 1 \\ 2 \quad 1\le x\lt 2 \\ 3 \quad 2\le x\lt 3 \\ 0 \quad ...
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Weak limits and computing the Fourier transform of the Heaviside function

A common problem on this site is to compute the Fourier transform of the Heaviside function that is $0$ on the negative reals and $1$ on the positive reals. A standard technique is to consider the ...
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35 views

Extension of Fourier Transform and Plancherel Theorem

I'm very confused with the ideia of extension Fourier transform of $L^1(\mathbb{R}^n)$ to $L^2 (\mathbb{R}^n)$. I start with a $u\in L^1(\mathbb{R}^n)$ and I use the limit and the Banach property to ...
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Characterization of $H^k$ by Fourier transform

Let $u\in H^k(\mathbb{R}^n)$ a function with complex-valued. Question 1: If $u\in H^k(\mathbb{R}^n)$ and $D^\alpha u\in L_2(\mathbb{R}^n)$, by definition for each $|\alpha|\le k,$ then we have ...
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Prove that $\lim_{\substack{b\to\infty \\ a\to0+}}\int_a^b\frac{\hat{f}(\xi)}\xi d\xi=-\pi i\int_0^\infty f(x)dx$

Suppose $f\in L^1(\mathbb{R})$ and that $f$ is odd. Prove that $$\lim_{\substack{b\to\infty \\ a\to0+}}\int_a^b\frac{\hat{f}(\xi)}\xi d\xi=-\pi i\int_0^\infty f(x)dx$$ Here $\hat{f}$ denotes the ...
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Inverting a complex function

I am facing the following problem. I know that the following relationship holds $$A(\omega) = C + \int_{0}^{\infty} \frac{L(\tau)}{1 + i\omega \tau}\mathrm{d}\tau$$ where $C$ is a positive constant ...
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Fourier transform of the Fourier transform

Is the Fourier transform of the Fourier transform of $f(t)$: $$\hat{\hat{f(t)}} = f(-t)$$ or $$\hat{\hat{f(t)}} = 2\pi f(-t)$$ ? I have read the two versions here and here (respectively) for ...
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Where do the coefficient equations for Fourier series come from?

I don't see where the equations come from like: $$a_0= \frac{1}{2L} \int_{-L}^L f(x)~dx$$ And like wise for $a_n$ and $b_n$. Also where does the general formula for a Fourier series come from? If ...
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Problem with double integral using Fourier transform?

I have a doubt concearning the convergence of an integral. Let $\mathscr{S}(\mathbb R^n)$ be the Schwartz space on $\mathbb R^n$. Given $u\in \mathscr{S}(\mathbb R^n)$ we have an well defined ...
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limits and integrals

Show that $$ \lim_{n \to \infty} \int\limits_{0}^{h} \frac{\sin (n\varepsilon)}{\varepsilon} \;\mathrm{d}\varepsilon = \int\limits_{0}^{\infty} \frac{\sin (t)}{t} ...
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Integrability of fourier transform

Let $f\in L^1(\mathbb{R})$ such that there exist $R,\delta >0$ for which $f$ is bounded in $[-\delta, \delta]$ and $\hat{f}(\xi)\geq 0$ for $|\xi|\geq R$. Then $\hat{f}\in L^1(\mathbb{R})$. ...
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Uniform convergence for functions with jumps

We know that Fourier partial sums (integrals) do not converge uniformly for BV functions with jumps due to Gibb's phenomenon. Is there any other types of sums/procedures that use only Fourier ...
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Understandning Radial Fourier Analysis

I'm currently studying living cells. In order to characterize their form, we use "Radial Fourier Analysis" as described here. I can't, however, seem to find more information about this topic (Radial ...
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198 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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Fourier Transform Good Articles

I need to use FT in one of my programming projects, but I need to refresh myself on it. Any good books and articles? The last time I studied it was 12 years ago, when I was in college.
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Acceleration/Position signal correction

I have a set of data for a car position, velocity and acceleration. % my data time car_x car_velocity car_acc The problem is that these arrays have error and I ...
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Fourier transform of this complicated function

Given the discrete signal $$y(n)= x(n)+2 x(n)^2x^*(n)$$ I want to compute the Discrete Fourier Transform $y(n)$. My Solution Assuming that $$DFT[x(n)]= X(\Omega)$$ then the $$DFT[x^*(n)]= ...
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What is the equation for an anisotropic Hanning window (cosine wave) in two or three dimensions?

I do not exactly know how to ask this question, so I will explain myself thoroughly. I am really stuck on this one, and it is crucial for my research, so if anyone has any ideas on where I may find ...
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How to explain the topic of Fourier transform interactively? [closed]

This is a soft question . In the walk-in for the lectureship, I have decided to give demo lecture on the topic of Fourier transform. The principal of the institution ask me to take lecture ...
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79 views

What is the importance of phase spectrum in Fourier transform

For any given signal using Fourier transform, we can compute it's magnitude and phase spectrum. But I have found that while discussing Fourier transform ,only frequency spectrum or magnitude ...
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If$f \in L_1 (R)$ so that $xf(x), x \hat{f}(x) \in L_1 (\mathbb R)$, show that $\sum|\hat{f}(n)| \le \infty$ and finally Poisson summation formula

PROBLEM (1)$f \in L_1 (\mathbb R)$ so that $xf(x), x \hat{f}(x) \in L_1 (\mathbb R)$, show that $\sum|\hat{f}(n)| \le \infty$. (2)Also, show that $\sum_{n\in \mathbb Z} f(x-2\pi n)$ converges ...
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is the fourier transform of $\cosh (ax)=\sqrt \frac{\pi}{2} \delta(\omega-ia)+\sqrt \frac{\pi}{2} \delta(\omega+ia)$

Is the fourier transform F of $\cosh (ax)$: $ F(\cosh (ax))=\sqrt \frac{\pi}{2} \delta(\omega-ia)+\sqrt \frac{\pi}{2} \delta(\omega+ia)$ This is the answer wolfram alpha gives. I am looking up the ...
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Solving a wave equation using Fourier transform

Consider the wave equation $$ \partial^2_tu-\Delta u=F, \text{ on } (0,\infty)\times\mathbb{R}^m\\ u(0,\cdot)=f,\partial_t u(0,\cdot)=g $$ Show that for $f,g\in C^{\infty}_c(\mathbb{R}^m)$ and $F\in ...
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What is the precise mathematical definition of what a wavelet is and what is its relation to linear algebra?

I was reading on wavelets and it seems that its hard to find a precise mathematical definition of what this concept is. My confusion first arose due to Gilbert Stang's linear algebra book. In ...
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Need a closed form for Fourier coefficients (if it exists)

I have a set of $53$ Fourier coefficients. The dc term is $0$. The $26$ positive frequency amplitudes (coefficients) are given below. The $26$ negative frequency amplitudes are the same. $\{0.014451, ...
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A problem similar to $L^2$ Fourier transform, but in the setting of Borel measure.

Problem: Let $\mu$ be a finite Borel measure on the real axis, supported on a countable set $\mathbb{Q} \subset \mathbb{R}$ (I'm not sure whether here $\mathbb{Q}$ is all rational numbers ). And let ...
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Computing wavenumbers for discrete Fourier transform

I'm trying to implement a Fortran program to compute the derivative of a function using the FFT. To begin with, just to test my installation of fftpack, I computed the Fourier transform of ...
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Fourier transform of a radial function

Consider a function $f \in L^2(\mathbb{R}^n)$ such that $f$ is radial. My question is, is the Fourier transform $\hat{f}(\xi)$ automatically radial (I can see it is even in each variable $x_i$), or we ...
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Fourier series normalize

Let $ \mu \in \mathbb { R} $ and let $$ f ( x ) = e ^{ \mu x } ,\ x \in (- \pi , \pi ] . $$ i) Arguments for that the Fourier series $ \sum_ { k = - \infty } ^ \infty c_k e ^ { ikx } $ for $ f $ ...
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If $f \in L^1(-\infty, \infty)$ , and $G(u) = \frac{f(x+u) - f(x^+)}{\pi u}$, is $G(u) \in L^1(0^+, \infty)$ true?

To be more detailed, if function $f(x)$ satisfies $\int_{-\infty}^{\infty}|f(x)|dx < \infty$ and assume that $G(u) = \frac{f(x+u) - f(x^+)}{\pi u}$, is it true that $lim_{K \rightarrow \infty} ...
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Laplace transform,Fourier transform and Z transform mathematical equations

Fourier transform $x(w)$ of signal x(t) is given by $$ x(w) = \int\limits_{t=-\infty}^{+\infty} x(t) e^{-j w t} dt -(1)$$ Laplace transform $x(s)$ of signal x(t) is given by $$ x(s) = ...
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Littlewood-Paley theorem on an annulus

Suppose a smooth function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ satisfies $$\text{supp}~\hat{f}\subset \{\xi:1<|\xi|<2\}$$ and set functions $f_k$ by saying ...
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Identifying a function is even or odd or not even and odd. [closed]

Here I have a very confusing problem. I'm right now solving Fourier transform. In which different formulas has to be applied according to the nature of the function wether it is odd or even or not ...
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Recovering cosh(ax) from it's fourier transform

Let's say $f(x)=\cosh(ax)$, where $a$ is a complex number and $x$ is real. Then the fourier transform is $F(\omega)=\sqrt \frac{\pi}{2} \delta(\omega-ia)+\sqrt \frac{\pi}{2} \delta(\omega+ia)$. So ...
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Getting the frequency of a sawtooth wave that is contained within a non-trivial signal

If I had a signal that contained, say, a square wave and a sawtooth wave, how would I extract the frequency of the sawtooth wave without the higher harmonics that make the Fourier series converge on ...
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Is Fourier transform a generalisation of Fourier series?

Is the Fourier transform a generalisation of a Fourier series or an a different concept? I.e. Can Fourier transforms be used with periodic functions and will it reduce down to the Fourier series ...