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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In which case should a wavelet transform be applied instead of a Fourier transform?

I was wondering what are the advantages (and possibly the drawbacks) of using a wavelet transform instead of a Fourier transform for the signal processing, are there simple examples to illustrate that ...
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29 views

Fourier series of the function $1-e^{i\delta t}$

Expand the function $1-e^{i\delta t}$ ($t\in [\pi,\pi]$) in a Fourier series relative to the complete orthonormal system $\{1, \cos nt, \sin(n-1/2)t\}$. Fourier series: $$\sum_{n=-\infty}^\infty f_n ...
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91 views

Fourier Transform Help Needed

I need help with a Fourier Transform problem for a composite waveform for an assignment. I'm stumped with how to approach this one. The only way I could think of to solve this was by considering it ...
6
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1answer
136 views

Extend a function by convolution

Let $f \in \mathcal{C}^{\infty}(\mathbb{R})$ be a compactly supported function ($supp(f)\Subset\mathbb{R})$. I am wondering about the existence of a $g \in L^p(\mathbb{R})$, for some $p$, such that ...
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1answer
31 views

Discrete Time Fourier Tranform of $\frac{sin(n-2)\frac{\pi}{3}}{(n-2)\pi}$

I don't know how to go about solving this. I can see that this can easily be manipulated into the sinc function, but I don't know where that gets me.
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2answers
115 views

how to find fourier transform of $\exp(-x^2/2)$

How can we find the fourier transform of $e^\frac{-x^2}{2}$ where -$\infty $ < x < $\infty $. I tried applying the standard formulae but ended up in un defined form..
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37 views

Fourier Transform of inverse powers of the absolute value

I don't think this question has been asked previously, so here goes. I need to evaluate the following integrals - $$ ...
2
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1answer
38 views

Riemann-Lebesgue application

By the Riemann-Lebesgue lemma, I have shown that for any finite interval measurable set $I$ of finite measure, any $h \in \mathbb{R}$, $$\lim_{n \to\infty}\int_I \cos (n(x+h)) \mathop{dx} = 0.$$ I ...
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3answers
161 views

Complex Integral with exponential

I've been struggling with this: $$\int_{0}^{\infty }\frac{e^{-px}}{x^{2}+1}\mathrm{d}x, \; \; p\ge 0.$$
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1answer
132 views

Need help with a Fourier Transform Question

I need an way to solve this Fourier transform problem. $$ f(t)= \begin{cases} \cosh(t) & \text{ For } |t|<1\\ 0 & \text{ For }|t|>1 \end{cases} $$ The given answer for the ...
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1answer
51 views

How to apply Plancherel Theorem here?

Let f be a function on the real line R such that both f and xf are in L^2(R). Prove that f ∈ L^1(R) and the L^1 norm of f(x) is less than or equal to 8 times (the L^2 norm of f(x)) times the L^2 norm ...
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1answer
34 views

A Specific Example about Parabolic PDE

I am solving a PDE numerical problem. And I have already had a algorithm. However, it seems to be hard to find a specific example to test my solution. Could you please give me one? The equation ...
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1answer
37 views

use plancherel theorem to prove an integral inequality

Let f be a function on the real line R such that both f and xf are in L^2(R). Prove that f ∈ L^1(R). I'm sorry I don't know how to use Latex to post the problem. The origional problem is here: ...
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0answers
42 views

Absolute value of the Fourier Transform of Gaussian random variable

Assume you have a normally distributed random variable $x$ with zero mean $\mu$ and standard deviation $\sigma$. Now you take the Fourier transform of it. The resulting complex random variable ...
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0answers
8 views

Reason for restricted frequency range of a DFT

The frequency range of a continuous-time signal extends from $-\infty$ to $\infty$ while for a discrete time signal it's from $-\pi$ to $\pi$ (or 0 to $2\pi$). Why is the frequency range limited for a ...
2
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1answer
34 views

Equidistribution of $\{\xi_n\}$ where $\xi_n = <n\frac{p}{q}>$ $p,q$ rel. prime

I'm working from Stein's An Introduction to Fourier Analysis, and there's a question (chapter 4 number 6): Let $\theta = \frac{p}{q} \in \mathbb{Q}$ where $\operatorname{gcd}(p,q) = 1$. Assume ...
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1answer
42 views

Fourier analysis how do i calculate an equation

Struggle is an understatement! I'm trying to get my head around Fourier analysis and I have the equation : $$f(x)=2\pi^2+6x^2$$ unfortunately I have no idea where to start and my coursework depends ...
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1answer
42 views

Seasonal adjustment and Fourier analysis

I've been reading up on seasonal adjustment (removing "seasonal" periodic components from a time series) recently and although I see a lot of fancy work around ARIMA models and fancy ways to detect ...
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1answer
257 views

Is it possible to do a half-range sine expansion on the sine function?

Suppose $f(x)=sinx,\quad 0<x<π$. Can you do a half-range sine expansion on f(x)? I tried, but I got $a_0=a_n=b_n=0$. If you requrie me to show my steps (i.e. I should have not gotten ...
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0answers
25 views

Rate of convergence of Fourier series

I am having a bit of a confusion regarding convergence results. Suppose $f$ is Lipschitz, or $f \in C^\infty$ and let $S_{N}f$ be its truncated Fourier series. In the wikipedia page ...
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0answers
21 views

Fourier Transform of $sin(5t - \frac{\pi}{4})U(t+8)$

I have this function $$ sin(5t - \frac{\pi}{4})U(t+8) $$ I know the Fourier Transform of $sin(5t - \frac{\pi}{4})$, which is $$ \frac{e^{-\frac{\pi^2}{2}fj}}{2j}\left [\delta (f-\frac{5}{2 \pi}) ...
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23 views

Rearrangement of Fourier Series Sum to attain convergence

Let $f$ be a continuous function with diverging partial Fourier sums $S_N(f)(0)$ : $$ f(\theta) = \sum \limits_{k=1}^\infty \alpha_k P_{N_k}(\theta)$$ Let $f(x) \sim \sum \limits_{n=-\infty}^\infty ...
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17 views

Find $g\in C^\infty(\mathbb R^n)$ one-periodic satisfying the identity below..

Let $f\in L^1(\mathbb R^n)$ and $k\in C^\infty(\mathbb R^n\times \mathbb R^n)$ be a $1$-periodic function in both variables. Is there a $1$-periodic function $g\in C^\infty(\mathbb R^n)$ satisfying: ...
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2answers
71 views

Very narrow FFT window functions

What is the flat-top window function that provides the narrowest possible lobe width? I'm doing FFT analysis and I need the resulting main lobe of a sine wave to be as narrow as possible but avoiding ...
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0answers
27 views

Approximating the Fourier transform with DFT/FFT

Suppose I have a continuous function $f(x)$, $x\in[-L/2,L/2]$. Its $L-$periodic Fourier coefficients are given by $$ \hat{f}[k]=\frac{1}{L}\int_{-L/2}^{L/2}f(x)\exp(-2\pi ikx/L)dx $$ If I apply ...
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1answer
14 views

Is there a relation between $l_p$-norms of functions with same Fourier spectra but w.r.t different measures on the Hamming cube?

Informally, I want to ask if two functions $f$ and $g$ on the Hamming cube have the same Fourier spectra but w.r.t different measure and basis, then is $||f||_p$ related to $||g||_p$? (Where each ...
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43 views

Closure of Schwartz space in homogeneous Besov space

Let $\dot{B}^s_{\infty,\infty}(\mathbb{R}^d)$ denote the homogeneous Besov space of order $s$ with second and third index $\infty$, i. e. the homogeneous Zygmund space. Let $\mathcal{S}(\mathbb{R}^d)$ ...
2
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3answers
7k views

How to calculate the Fourier transform of a Gaussian function.

I would like to work out the Fourier transform for the Gaussian function: $f(x)=\exp(-n^2(x-m)^2)$. It seems likely that I will need to use differentiation and the shift rule at some point, but I ...
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1answer
73 views

How to use Parseval' identity( Plancherel)? [duplicate]

(May be this is very basic question for MO) Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put $$F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt \ (n=1,2,...)$$ Fix ...
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1answer
79 views

How to use Parseval' s( Plancherel' s) identity?

Let $f\in L^{2} (\mathbb R)$ with $\lim_{t\to \pm \infty} f(t)=0.$ Put, $F_{n} (x)= \frac{1}{2\pi} \int_{-n}^{n}e^{itx} f(t) dt, \ (n=1,2,...).$ Fix $\alpha \in (0, \infty)$ and we define $H_{n}(x)$ ...
2
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2answers
107 views

Morlet's wavelet reconstruction formula

The CWT (continuous wavelet transform) of a signal $x(t)$ is $$X_w(a,b)=\frac{1}{\sqrt{|a|}} \int_{-\infty}^{\infty} x(t)\psi^{\ast}\left(\frac{t-b}{a}\right)\, dt$$ In order to reconstruct the ...
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0answers
20 views

Fourier transform of a radial function [duplicate]

What is the Fourier transform of 1/|x| in $R^3$? It seems that it is $\sqrt{\pi/2}/|y|^2$ but I am not sure.
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2answers
32 views

Which function satisfies the conditions?

I'm solving a problem and, in order to run test case, I need a function $ b(x,y) $ that satisfies: $$ \int_0^L \int_0^H b(x,y) \, dx \, dy = 0 $$ and $$ \int_0^L \int_0^H b(x,y) \cos \left(\frac{n ...
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1answer
27 views

Fourier decomposition of Dirac-delta function under finite limit.

Dirac delta function is said to be Fourier transformation of 1, $$ \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty} dx\ e^{i bx} . 1\ =\ \delta(b) $$ This is usually shown by considering the definition ...
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10 views

decay of coefficients in the expansion into Bessel functions

Let us consider the generalized Fourier expansion into Bessel functions, as illustrated in the Wikipedia page: http://en.wikipedia.org/wiki/Generalized_Fourier_series. Let $J_0 (r)$ be the 0th ...
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1answer
46 views

Find coefficients of polynomial that has zeros at certain points

Given a list of values z0, z1, ..., zn-1 (possibly with repetitions), show how to find the coefficients of a polynomial P(x) of degree-bound n + 1 that has zeros only at z0, z1, ..., zn-1 (possibly ...
2
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0answers
28 views

a fourier multiplier inequality

I am reading "Estimates for translation-invariant operators" by Lars Hormander. Currently, I am stuck at the following argument which seems obvious to Lars ))). Let $\varphi(\xi)$ be such a function ...
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0answers
25 views

What does an image of Fourier Transformation of an image tell us?

First time studying image processing... I just don't understand what does fourier transformed image of an image describe? For example consider given following pictures, The first one is the image, and ...
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1answer
37 views

How does this phase shift in x-space affect the position of a spectrum in k-space?

I'm working on a new form of signal detection with which I hope to recover both the amplitude and phase of a very small signal. However, doing this requires the use of some Fourier maths that I don't ...
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0answers
14 views

Relating Fourier Transform to an Integral involving Sin(vt)

I have data for a function $S(Q)$ and I'm trying to find values for a different function $g(r)$ Now I know $g(r) = \int_0^{\infty} Q(S(Q)-1) \sin(Qr)\, dQ$ This is closely related to the sine ...
2
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1answer
57 views

For a given sequence $(a_k)$, there is no Riemann integrable function f such that $\hat{f}(k) = a_k \forall k$

I'm working out of Stein's Fourier Analysis: An Introduction, and am on chapter 3. There is an exercise that gives us a specific sequence $(a_k)$ and asks us to show that ...
2
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1answer
39 views

How to apply the method of stationary phase here?

Consider the following oscillating integral $$ I(\xi;t) = \int\limits_{\mathbb R}\int\limits_{\mathbb R^n} e^{it(\xi y - \theta f(y))}a(y) \, dy \, d\theta, \quad \xi \in \mathbb R^n \setminus 0, ...
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1answer
52 views

Inverse Fourier transform problem

Can anybody please guide me how to compute the following inverse Fourier Transform ? $$ p(x) = \frac{1}{2\pi} \int_{-\infty}^{\infty} \frac{1}{(1-j\omega\bar{x})^K}e^{-j\omega x}d\omega $$ I shall ...
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0answers
111 views

Singular measures - approximate characteristic function

One can decompose a $\sigma$-finite measure $\mu$ on $\mathbb{R}$ in three parts: $\mu_{ac}$: absolutely continuous $\mu_{sc}$: singular continuous $\mu_{pp}$: pure point A common example for a ...
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32 views

How to choose $\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha)$; so $\lambda ^{-1}\in L^{1}(\mathbb R)$?

We define, $$\lambda (x) =\int_{0}^{\infty} \frac{\sin^{2}x \alpha}{\alpha^{2}} d\mu (\alpha), (\mu(0)=0) $$ where $\mu (\alpha)$ is a non-decreasing function such that the integral converges for ...
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0answers
38 views

Solve ODE by Fourier transform, and versus by Laplace transform?

Regarding solving ODE by Fourier transform, I read a nice reply by O.L.. After applying Fourier transform to an ODE to obtain an algebraic equation, the reply showed that some terms involving the ...
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34 views

Fourier transform of partial derivative

I am currently doing some reading on using Fourier transforms to solve PDEs and I stumbled upon a property that I am not sure how to prove. Suppose we have a heat-equation $u_t(x,t)=\alpha^2 ...
0
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1answer
36 views

Upper bound for the norm of inverse Fourier tansform

Recall Hausdorff-Young inequality: For any $f\in L^p(\mathbb{R}^n)$, we have $||\hat{f}||_q\le ||f||_p$, where $p$ and $q$ are conjugate exponents and $p\in[1,2]$. It seems to me that it follows ...
0
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2answers
46 views

a differential equation equation related to fourier series

I am really struggling with this one. Any help is welcome! For equation $f''(z) + p(z) f'(z) + q(z) f(z) = 0$, where $p(z)$ and $q(z)$ are fixed polynomials. Given $f(0)=f_0$, $f'(0)=f_1$, prove that ...
2
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2answers
28 views

Let $F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha$. Is it true $F, F^{-1}\in L^{1}(\mathbb R)$?

Define $$F(x)= \int_{0}^{\infty} \frac{\sin^{2} (x\alpha)}{\alpha^{2}} d\alpha, \ (x\in \mathbb R).$$ It is clear to me that, the integral converges for every real $x$ (as near origin integrand is ...