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 to evaluate $\int_{-1/2}^{1/2}e^{2\pi m i x}e^{\frac{2\pi n i}{x}}dx $?

I'm interested in the integral $$\int_{-1/2}^{1/2}e^{2\pi m i x}e^{\frac{2\pi n i}{x}}dx ,$$ where $m$ and $n$ are arbitrary nonnegative integers. Is there any possibility to evaluate this ...
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41 views

Inverse Fourier transform of cut off of Fourier transform

Suppose we have a function $f(x)$ such that $$|\frac{d^n}{dx^n}f(x)| \leq C(1 + |x|)^{-n}$$ Take the Fourier transform $\hat{f}(\xi)$ and consider the function $g(\xi) = \chi(\xi)\hat{f}(\xi)$, where ...
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How can I make the mean of samples be approximately equal to the mean of actual continuous signal?

Suppose there is signal f(t) that is continuous and periodic. It is known that this f is T-periodic. (but it's not necessarily a single cosine f(t).( I'd like to make the mean of samples be ...
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16 views

Frequency scaling property for Fourier series

For Fourier transform, there is an equation connecting time-scaling with frequency-scaling. (By scaling, I mean multiplying by constant for time or frequency) Is there such a relation for Fourier ...
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33 views

Fourier Tansform of derivative on Wolfram Alpha

If I'm not mistaken, the Fourier Transform of the nth order partial derivative of a function with respect to x, using the transform variable k is: (i*k)^n * [F(k)] so for the 1st order derivative ...
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83 views

Can anyone suggest a book on Fourier Analysis containing many good problems

I am taking a basic course in Fourier Analysis in my undergrad Analysis class and I know the theory and related theorems. However, this is a relatively new zone for me and I would like a book that ...
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44 views

Using Riemann-Lebesgue Lemma to show a function is continuous

I have been studying for a preliminary exam using old exams. Many of them ask the following question which we have unfortunately not talked about in the class at all. It would be nice to see an ...
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34 views

Entire function of exponential type $1$ bounded by $1/(1+|x|)$

Let $f$ be an entire function of exponential type $1$ such that $|f(x)| \leq \frac{1}{1+|x|}$ for all $x\in \mathbb{R}$. First, I have to show that $|f(z)| \leq \frac{Ce^{|Im(z)|}}{1+|z|} z\in ...
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8 views

Fourier Amplitude Sensitivity Test (FAST)

I am new in the domain of sensitivity analysis, I am trying to investigate the global sensitivity analysis method FAST (Fourier Amplitude SEnsitivity testing). I read alot about this subject, starting ...
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21 views

Convergence property of DTFT toward DFT when function is periodic

from Wikipedia: When the input data sequence $x[n]$ is $N$-periodic, DTFT can be computationally reduced to a discrete Fourier transform (DFT), because: $ X_{1/T}(f)$ converges to zero ...
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38 views

Find the Fourier series representation of $f(t)=\sin(3\pi t)$

Find the Fourier series representation of $$f(x)=\sin(3\pi t)\qquad \text{for }-1\leq t\leq1$$ When I calculate the coefficients, I always get $0$. Why is that? Is the series indeed zero?
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98 views

If $u\in H^2(R^n)$, how to prove $\|D^2u\|_{L^2}$ is equal to $\|\Delta u\|_{L^2}$ using Fourier transforms?

Problem: If $u\in H^2(R^n)$, how to prove $\|D^2u\|_{L^2}$ is equal to $\|\Delta u\|_{L^2}$ using Fourier transforms? My first question: Is it right to prove this using integration by parts as ...
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1answer
23 views

Why must the Fourier transform of a compactly support function not have compact support?

I've heard this stated several times, most recently as a motivation for using the Schwartz space as test functions. I think I can just about prove it using Heisenberg's uncertainty principle, but ...
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How to check periodicity of $f(t)$ using samples

Suppose that we know that signal $f(t)$ is $T_1$-periodic. Let $f_1 = 1/T_1$. But we want to know whether signal is $T_2$-periodic also. Let $f_2 = 1/T_2$, and $f_2$ is positive integer multiples of ...
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34 views

What is a window function with positive spectrum?

I need a real, symmetric window function $x(t) = x(-t)$ whose Fourier transform $\hat{x}(\omega)$ (also real and symmetric) is non-negative $\hat{x}(\omega) \ge 0$ for all $\omega$. The function does ...
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Proving $\sum_{n =1,3,5..}^{\infty }\frac{4k \ \sin^2\left(\frac{n}{k}\right)}{n^2}=\pi$

Proving $$\sum_{n =1,3,5..}^{\infty }\frac{4k \sin^2\left(\frac{n}{k}\right)}{n^2}=\pi$$ Where $k$ any number greater than $0$ I tried to prove it by using the Fourier series but I couldnt ...
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20 views

Is there anything similar to DTFT for Fourier series?

So if sampling condition is met well, with aperiodic signals we have discrete-time Fourier transform (DTFT) that allows us to get frequency-domain data that resemble continuous-time fourier transform. ...
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2answers
41 views

Fourier integral problem?

Show that $$ \int_0^{\infty} \frac{\sin \pi \omega \sin x\omega}{1-\omega^2}d\omega= \begin{cases} \frac{\pi}{2}\sin x,&\mbox{ if } 0\leq x\leq\pi\\ \quad\\ 0,&\mbox{ if } x\geq\pi ...
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1answer
22 views

Is it possible for continuous fourier transform of a function to have values only on finite number of frequencies?

Is it possible for continuous fourier transform of a function to have values only on finite number of frequencies? Or do these values necessarily impulse values, not complex numbers?
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1answer
76 views

Spectral convergence of coefficients of a Fourier series

I have seen claims that if a smooth function $f(x)$ is represented by its Fourier series, $f(x)=\sum_{n=-\infty}^\infty a_ne^{i(nt)}$, then as $|n|\rightarrow\infty$, then $|a_n|\rightarrow 0$ ...
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24 views

Smoothness-and-decay relationship of the Fourier transform

Recall that a function $f\colon \mathbb{R} \to \mathbb{C}$ is said to be rapidly decreasing if $$\sup_{x \in \mathbb{R}} \big|x^k f(x)\big| < \infty \quad \text{for all} \quad k = 0, 1, 2\dotsc$$ ...
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Is the DTFT of a sampled Gaussian a positive function?

I have an infinite sequence $x_{n}$ for $n \in \mathcal{Z}$ which is a sampled Gaussian function $x_{n} = \exp(-n^2/a)$ with a > 0. I need to check whether its DTFT $x(\theta) = \sum_{n \in ...
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25 views

What is the relationship between DTFT and continuous fourier transform?

As title says, what is the relationship between DTFT and continuous fourier transform? Let's say there is continious signal $f(t)$. Continuous Fourier transform convert this into $F(\omega)$. Now let ...
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23 views

Support of a convolution with the help of Titchmarsh theorem

I have to use Titchmarsh theorem in order to prove that : if $f\in L^1[-1,1]$, and $supp(f*f*f*f-f*f)\subset [-1,1]$ then $supp(f)\subset[-1/4,1/4]$. Does anyone have an idea ? Thank you.
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Prove a lower bound on $\left|\int_{-\infty}^{+\infty}k(t) f(t) e^{\lambda_n ti}dt\right|$.

Let $k(t)$ be any function absolutely integrable over $(-\infty,+\infty)$, let $$K(u)=\int_{-\infty}^{+\infty}k(t) e^{-uti}dt$$ and let $$f(t)=\sum_n a_n e^{-\lambda_n t i}, \ \ \ \lambda_n\in\mathbb ...
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1answer
25 views

How does equality in Bessel's Inequality prove an orthonormal complete sequence?

I've been searching around for an answer to this question on the web for some time, but I keep coming up short (it may very well be that I don't have the right terms to be searching with). In any ...
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1answer
30 views

Why is boundedness of the ball multiplier equivalent to the convergence of Fourier transform in Lp?

Let $\mathcal{F}$ be the fourier transform operator and let $T_R$ = $\mathcal{F}^* \chi_R\mathcal{F}$ where $\chi_R$ is the indicator function on the ball of radius $R$. Hence $T_R$ is the fourier ...
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60 views

Calculating value of integral of convolution using Fourier transform

Calcuate the integral $$I=\int_{-\infty}^\infty\frac{\sin a\omega\sin b\omega}{\omega\cdot \omega}d\omega.$$ First I noticed that $$\mathcal{F}(\mathbb{1}_{[-h,h]})(\omega)=\frac{\sin h ...
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1answer
31 views

Find the Fourier Coefficients that minimize the error [duplicate]

I know that the coefficients that minimize the expression are the ones that make it's derivative 0. I have also expanded the whole expression and taken it's derivative, but still I can't figure out ...
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1answer
24 views

fourier transform and principal values

Fourier transform and principal values Can anyone tell me from how can i get the fouries transformation of prinicipal value of (1/x) $$p.v\int \frac{1}{x}\Bigg(\int e^{-wix}\varphi(w)dw\Bigg)dx$$
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Making it possible to do a Fourier transform on it: $\frac{1}{(k+w)^2(a^2 +w^2)}$

Sorry for all the edits, I'm very stressed and not so used to Latex. Full question: consider a filter with impulse response $$h(t)=e^{-bt} u(t)$$ where $u$ is the unit step function. The input ...
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1answer
29 views

Show that f and fourier are in S(R)

I am dealing with fourier tranforms and have come to some problems when I need to show the following things: How can I show that if $f\in S(R)$, then $\hat{f}\in S(R)$? And how will properties ...
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36 views

Fourier Transform Properties - Proving

How do I go about proving the following properties of fourier transforms? I do not have a textbook (professor didn't issue one) so I is very hard for me to understand these concepts. ...
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20 views

Fourier Tranform Properties Established

I have a function, f, in $L_1(R)$. I need to establish the following fourier transforms and I don't know how to do so. Can anyone guide me through one of these that would then help me get the other ...
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derivation of an equation concerning Fourier transform of a wave packet

My question arises from Fourier transform of a wave packet in quantum mechanics. In the following context, $\{\phi_i, E_i | i=1,2,...,n\}$ are eigenstates and eigenvalues of a $n\times n$ Hermitian ...
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16 views

Proving orthonormality of system by sum of fourier coefficients

Let $f\in L^2(\mathbb R)$. Prove the system $\{f(t-n)\}_{n\in\mathbb{Z}}$ is orthonormal if and only if $$\sum_{k\in\mathbb{Z}}|\hat{f}(\omega+2\pi k)|^2\equiv 1$$ I have no clue how to prove ...
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Prove $\int_0^\infty f(t) \frac{1}{t+x} dt$ is its own Fourier cos transform if $f(t)$ is its own Fourier cos transform

The problem says to use the fact that $g(x) = \int_0^\infty f(t) e^{-xt}$ is its own Fourier sine transform if $f(x)$ is its own cos transform. My working so far: $F_c(\int_0^\infty f(t) ...
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1answer
23 views

Does Nyquist-Shannon sampling theorem require real-valued function $f(t)$? [closed]

For nyquist-shannon sampling theorem, is it required for function that is being sampled to be real-valued, that is $f:\mathbb{R}^n \to \mathbb{R}^n$? Or is it possible to be $f:\mathbb{R}^n \to ...
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24 views

Discretizing a set of functions while preserving orthogonality: general method?

Say I have a set of functions, $\left\{ \psi_j \right\}_{j \in \mathbb{N}}$ where $\psi_j : \mathbb{R} \mapsto \mathbb{C}$. Furthermore, these functions are orthogonal on some interval $A \subset ...
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Stuck on this integration $\int_0 ^\infty \frac{1}{1+x^2} cos(kx) dx =\frac{\pi}{2}e^{-k}$ [duplicate]

I'm not sure how to show this $$\int_0 ^\infty \frac{1}{1+x^2} \cos(kx) \ \mathrm dx =\frac{\pi}{2}e^{-k}$$ I tried by parts but I'm not getting anywhere, I'd really appreciate the help
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334 views

Fourier integral/ Fourier transformation of an oscillatory function with FFT

$f(x) = \cos(x^2)$ and $g(k) = \sqrt\pi \cos((\pi k)^2 - \pi/4)$ are a Fourier pair. I want to reproduce $g(k)$ by Fourier integrating $f(x)$ using FFT, i.e. approximating ...
3
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1answer
42 views

Solving Laplace $\nabla^2 \phi=0$ in $x,y \geqslant 0$

I'm trying to solve $\nabla^2 \phi=0$ in $x,y \geqslant 0$ $\phi(x,y)=0 $ as $x^2 +y^2 \rightarrow \infty$ $\phi_x(0,y)=0$ and $\phi(x,0)= \frac{1}{1+x^2}$ I know the solution is ...
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9 views

Result obtained on deletion of finite number of Fourier Coefficients

I want to know the answer to the following question. If a finite (but fixed) number of Fourier coefficients (of any choice) of a Fourier series are made $0$, then will the new series be a Fourier ...
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1answer
28 views

Find the coefficients of the Fourier series that minimise the error.

I am having a little trouble understanding what I have to actually do here. What does differentiate with respect to bn? I thinks after differentiation I must use some calculus theorem about extreme ...
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2answers
22 views

DTFT and its convergence

In the textbook "signals and systems", by prof. Simon Haykin, it says:   If $x[n]$ is not absolutely summable, but does satisfy square summable, then it can be shown that the following equation ...
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6 views

Fourier Analysis of a p2 continous Galerkin Scheme for the Laplace & Poisson Equation

Background: I am obtaining residual calculations for the 3D Laplace and Poisson Equation using finite element continuous galerkin scheme with lagrange polynomial basis functions for p1, p2, p3 and ...
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26 views

Find complex Fourier coefficients of $f(-x), f^*(x)$

For $f(-x)$ i have tried to replace the $k$ with $k'=-k$ but still i can't find any relationship between the coefficients. What could be a better way to approach this problem?
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4answers
212 views

Fourier transform of the wave equation

On the LHS side of the highlighted expression should it not read: $\displaystyle \frac {d^2 \hat{u}}{dt^2}$ as the Leibniz Integral rule requires you to transform the partial derivative to a ...
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1answer
45 views

What does coefficient before Forier integral and integration limits depends on?

I've read a couple of sources on Fourier transform. All of them give different coefficients and integration limits. Wikipedia: 1, -infinity, +infinity. Russian Wikipedia: 1/sqrt(2*pi), -infinity, ...
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21 views

$e^{t\Delta }f $ grows at the same rate as $[\int f]G_t$ in $L_2$

This is a question that I am asking to help myself with a homework question. Assume $f$ has is smooth with compact support ( so that $f$ and $\hat f$ are in $\mathscr{S}$), and $G_t$ is the heat ...