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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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
33 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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7 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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1answer
17 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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23 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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21 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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1answer
15 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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1answer
12 views

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
10 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
15 views

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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1answer
23 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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1answer
20 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
23 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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2answers
35 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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16 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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2answers
47 views

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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12 views

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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1answer
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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4answers
51 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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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
15 views

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

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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21 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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1answer
11 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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3answers
90 views

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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0answers
8 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 ...
3
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1answer
33 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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1answer
18 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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2answers
16 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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2answers
25 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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0answers
8 views

Fourier transformation of the complex exponential [duplicate]

How do I prove that the Fourier transformation of the complex exponential $$\exp\left[ i \pi(( a^2 x^2) +(b^2 y^2))\right]$$ is $$\exp\left[i {\pi}\left( {fx^2 \over a^2} + {fy^2 \over ...
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1answer
27 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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0answers
18 views

Prove the periodic total variation of f = $\sum_{n=0}^{N-1} |(f*h)[n]|$

Let $\Bbb{C}^N$ be the N-dimensional Euclidean space with its inner product defined as $$ \langle f,g\rangle=\sum_{n=0}^{N-1} f[n]g^*[n],\ \forall f,g \in \Bbb{C}^N$$ where $g^*[n]$ is the complex ...
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1answer
18 views

Hankel transform with Bessel functions of the second kind

The Hankel transform is defined for Bessel functions of the first kind (see e.g. http://en.wikipedia.org/wiki/Hankel_transform) I would like to know if it is possible to define a Hankel transform ...
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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 ...
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1answer
17 views

What would be the Z transform from fourier transform?

I am trying to get the z tranform from the fourier transform, so I am trying to get its equivalent in time to then, get the z transform, this is what I have: $$Y(w) = \left\{ \begin{array}{c l} 1 ...
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1answer
29 views

Choosing which variable to take the Fourier Transform with respect to

Why here do we take the fourier transform with respect to $x$ as opposed to $t$?
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1answer
44 views

Fourier transformation of complex exponential proof

How do I prove that the Fourier transformation of the complex exponential $$\exp\left[ i \pi(( a^2 x^2) +(b^2 y^2))\right]$$ is $$\exp\left[i {\pi}\left( {fx^2 \over a^2} + {fy^2 \over ...
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0answers
22 views

Independence of variables in fourier transform

How do we know that $x,t$ are independent and why does this imply $\omega$ and $t$ are also?
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0answers
22 views

Fourier transformation of exp[jπ(a2x2 + b2y2) [on hold]

Can somebody derive the Fourier transformation of exp[jπ(a2x2 + b2y2)] function
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15 views

About fourier series and tranforms [on hold]

I would like to know whether Fourier transform can help in image detection. I want to use it for edge detection in my algorithm.
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2answers
39 views

Find a function whose Fourier transform is the following

Find a function whose Fourier transform is the following: $$\frac{1}{(4+k^{2})(9+k^{2})}$$ I know that $f(x) = F^{-1}\{\hat{f}(k)\}$ so I get: $$f(x) = ...
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1answer
40 views

Show that {$\phi (x-n), n\in \Bbb{Z}$} is an orthonormal sequence in $L^2(\Bbb{R})$

Let $\phi$ be a compactly supported continuous function such that $\phi (x)=0$ outside of some finite interval. $\phi$ satisfies the following refinable equation: $$\phi (x)=\sum_{k=0}^{M}c_k\phi ...
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1answer
20 views

Fourier transform proof

I am trying to prove that $$\langle\tilde{s}(\omega), \omega^{2}\tilde{s}(\omega)\rangle = -\langle s(t), \partial_t^2 s(t)\rangle$$ Where $\tilde{s}(\omega)$ is defined as the Fourier transform of a ...
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1answer
20 views

The fourier transform of $f$ in $L^2$

Let $f$ is continuous and piecewise smooth, $f\in L^2$ and $f'\in L^2$. Show that $\hat{f}$ (the fourier transform of $f$) is on $L^1$.
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1answer
32 views

Suppose $f$ fulfils a unity condition. Prove $\hat{f}(0)=1$

Suppose that $ f \in L^1(\Bbb{R})\cap L^2(\Bbb{R})$ satisfies the following unity condition $$\sum_{k\in \Bbb{Z}} f(x-k) = 1\ \ \ \ , \forall x\in\Bbb{R}$$ Prove that $\hat{f}(0)=1$ Here $\hat{f}$is ...
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0answers
18 views

Approximation by trigonometric polynomials in two dimensions

Let $f(x,y)$ be continuous function on $[0,1] \times [0,1]$. Prove that there exists polynomial in form $$P_{n}(x,y)=\sum_{|k_1|,|k_2|<N}e^{i(k_1x+k_2y)}$$ that we have: $$\sup_{x,y ...
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0answers
12 views

Fourier Transform of $n$ functions

I would like to evaluate the Fourier Transform of $n$ functions. I am aware from the derivation of the convolution how this is done for the case of $n=2$. How could this be generalised for $n=3$? ...
2
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0answers
23 views

A trajectory for shortened k-space data acquisition MRI

Given a real function $f:\mathbb{R}^n \to \mathbb{R}$, denote by $\hat{f}$ its Fourier Transform. I have shown that $\hat{f}(\vec \omega)=(\hat{f}(-\vec \omega))^*$ where $^*$ denotes complex ...
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1answer
14 views

How to derive the discrete fourier transform of $(n+2)a^nu[n]$ where $|a| < 1$?

This is a rather simple question, but I'm stuck on one step. Here's what I've done: 1) $x[n] = (n+2)(\frac{1}{2})^nu[n] = n(\frac{1}{2})^nu[n]+2(\frac{1}{2})^nu[n]$ The discrete fourier transform is ...