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 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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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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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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22 views

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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34 views

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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28 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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29 views

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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24 views

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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1answer
39 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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48 views

What does it mean that a sine wave is unchanged when added to another sine wave?

From the wikipedia article on sine waves: The sine wave is important in physics because it retains its wave shape when added to another sine wave of the same frequency and arbitrary phase and ...
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1answer
63 views

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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1answer
37 views

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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36 views

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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1answer
45 views

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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1answer
27 views

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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20 views

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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1answer
30 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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1answer
48 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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32 views

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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34 views

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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1answer
17 views

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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3answers
47 views

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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52 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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20 views

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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137 views

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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51 views

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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1answer
60 views

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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59 views

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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1answer
41 views

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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29 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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21 views

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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18 views

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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37 views

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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1answer
34 views

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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29 views

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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34 views

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} ...
3
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1answer
39 views

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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1answer
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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 ...
2
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1answer
48 views

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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1answer
26 views

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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22 views

$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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1answer
54 views

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 ...