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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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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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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29 views

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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46 views

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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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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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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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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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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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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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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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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How to use Fourier Transform with non-trivial boundary conditions such as in potential flow around a plate?

I'd specifically like to be able to solve this PDE with boundary conditions corresponding to flow around a line (plate cross-section), otherwise known as flow-tangency, with integral transforms. ...
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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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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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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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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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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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28 views

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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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 ...
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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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Dominated convergence to characteristic function

Let $\phi_m(x):=\chi_{[0,1]} * \chi_{[0,1]} *...* \chi_{[0,1]}$ be the m -times convolution (so $m+1$ characteristic functions are involved). Then the Fourier transform of this function is given by ...
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Fourier transform of a polynomial function with both real and complex roots

I am given the following function: \begin{equation} f(x)=\frac{x}{x^3-7x^2+16x-10} \end{equation} which has the following roots: \begin{equation} x_1=1 \in \mathbb{R}, \quad x_{2,3}=(3 \pm i) \in ...
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A more elegant way to find the Fourier transform

Let $f$ be defined analytically as : $$f(x)=\arccos \left ( \sin \left ( 2x \right ) \right ), x \in\left (0,10 \right ], f(x)=0, x\notin\left ( 0,10 \right ]$$ Here is a graph of the above ...
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A Simple Fourier Transform [duplicate]

I am studying about the randomprocess thesedays. I am stuck on solving the discrete signal to show the fourier transform the formula is that $$ w_b(k) = {N-|k|\over N} \quad \quad when\ \ |k| ...
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Parseval Identity - Fourier series for the function of $f(t) = e^{at}$ is equal

I need help for this proof: by the Fourier series the function $f:[-\pi,\pi]\rightarrow \mathbb{R}$ is defined by $f(t)=e^{at}\,,\,a\neq 0$ show that $$ \sum_{n=0}^{\infty} \dfrac{1}{a^{2}+n^{2}} ...
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Fourier transform of $f(x)=\frac{x}{1+x^4}$ and $g(x)=\frac{x^2}{1+x^4}=xf(x)$

Let $f(x)=x/(1+x^4)$, the improper integral of which exists. I computed the Fourier transform of $f$, to be: \begin{equation} ...
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Gaussian is the only radial function which is separable

One way to characterize the Gaussian $ae^{b x^2}$ is that its a $C^1$ function $h$ that is radial $h(x,y) = h(\sqrt{x^2+y^2})$ and also separable, that is expressible as a product of one-dimensional ...
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Inverse Fourier transform of $\frac{1-e^{-2\pi ift}}{2\pi if}$

I would like to calculate the inverse Fourier transform of the following $$H(f) = \frac{1-e^{-2\pi i f t}}{2\pi i f}$$ Can anyone tell me and explain to me how to do that? I don't want just an ...
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Solution to Schrodinger equation with non-Schwartz initial data

For $\xi_0\in\mathbf{R}^n$ compute the solution of the Schrodinger equation with initial data $$ i\partial_tu-\Delta u=0 \text{ in } (0,t)\times\mathbf{R}^n\\ u(0,x)=e^{i\xi_0\cdot x} \text{ for all } ...
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Fourier series convergence

Let $f_n \rightarrow f$ be a sequence of $2\pi$-periodic functions, where the convergence is in $L^1({\mathbb R}/2\pi{\mathbb Z})$. Then the Fourier-coefficients satisfy $|F(f_n) -F(f)| \rightarrow 0 ...
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$h_k(x):=\sum_{n \in \mathbb{Z}} F(f_k)(n)e^{-inx} \rightarrow h(x):=\sum_{n \in \mathbb{Z}} F(f)(n)e^{-inx} $?

Let $f_n \rightarrow f$ be a sequence of functions in the $L^1$ sense. Then the Fourier transform implies $||F(f_n) -F(f)|| \rightarrow 0 $ uniformly. Now, I was wondering. Does this imply that the ...
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Show equality of a given function with a series in $ℝ$

Show that: $$2x\cos x-\sin x=4\sum_{n=2}^\infty \frac{(-1)^n}{n^2-1}\sin(nx)$$ Supposedly, this can be proved by using Fourier series, by choosing the right function but I have been thus far unable ...
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Is the step function periodic?

Consider Example 3.5 in the following lecture notes on Fourier Analysis, on page 10 at the bottom. http://www.math.ku.dk/~schlicht/DL/2013/fourier-summary.pdf I cannot understand why it says that a ...
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What is the range on a fourier transform?

In particular, I want to know the range of the coefficients on the type-IV discrete cosine transform. Assuming no normalization factor or window is applied, what interval can I expect the coefficients ...
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Is the Fourier transform a conformal map on $L^{2}$?

I read that a conformal map is one that preserves the angles. I know nothing more about conformal maps. I don't know where to find a generalized definition of a conformal map, but I guess that if ...
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Principal value methods for fourier laplace etc.

I recently saw this here: $$\int_{-\infty}^{\infty} \frac{1}{\omega^2} e^{j\omega t} d\omega$$ and I was unable to understand how such an integral could be computed. I want to learn about this ...
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Fourier Transform: Understanding change of basis property with ideas from linear algebra

The notion of Fourier transform was always a little bit mysterious to me and recently I was introduced to functional analysis. I am a beginner in this field but still I am almost seeing that the ...
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Convergence of a subsequence in $(C(\mathbb{T}), \|\cdot\|_2)$

Problem: Define, $ \mathbb{T} := \mathbb{R}/{2\pi\mathbb{Z}} $. Consider a sequence of functions $(g_n)_{n\in \mathbb{N}} \in C^4(\mathbb{T})$ such that, $ \sup_{n \in \mathbb{N}}(\| g_n \|_2 + \| ...
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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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Fourier Transform of $x^p \cdot {{df^q} \over {dx^q}}$

What is the Fourier Transform of $x^p \cdot {{df^q} \over {dx^q}}$? This seems like an elementary question, but my CRC book of standard formulae doesn't have it. My attempt is rather trivial, but for ...
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Fourier transform of this?

I need to prove: $$\frac 1 {\Delta x} \int_0^\infty dk \, e^{-ik(\Delta t-i\epsilon)} \sin k\,\Delta x=\frac 1 {\left(\Delta t-i\epsilon\right)^2 -\Delta x^2} $$ A possible hint could be: ...
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Discrete time Fourier transform on decimated signals

If I have a signal $x[n]$ and its decimated version, $y[n]=x[2n]$, is there a known expression for the DTFT of $y[n]$, $Y(\theta)$, as a function of $X(\theta)$? Thanks,
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Proving Inverse DFT

I have trouble understanding the proof I was provided of the IDFT, here is what I have: $$ \nu_n = \frac{n}{\Delta N} \\ x(t) = \int_{-\infty}^{\infty}X(\nu)e^{i2\pi\nu_n t}d\nu \\ $$ the next step I ...
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An exercise regarding fourier inversion formula

I have to solve an exercise which looks like a more general form of Fourier inversion formula. But, I'm having hard time attacking it... A given function f is integrable on the real line and ...
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If $f \in L^2 \cap C_c$ then $\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0+…a_n \cos(2 \pi n \xi)$

Let $f \in L^2 \cap C_c$ , then I want to show that $$g(\xi):=\sum_{n \in \mathbb{Z}}|\hat{f}(\xi+2\pi n)|^2 = a_0 + 2 \sum_{n=1}^{N}c_n \cos(2\pi n \xi) $$ for some $N \in \mathbb{N}.$ Does ...