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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Question on Proof of Fourier's Uncertainty Principle

The Fourier Uncertainty Principle: If $f\in \mathscr L^2(\mathbb R)$ and $xf(x),\xi\hat f(\xi)\in \mathscr L^2(\mathbb R)$ then \begin{equation}\left\|xf(x)\right\|_2\big\|\xi\hat f(\xi)\big\|_2\ge ...
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366 views

3D fourier series

I wonder how I can write a function $f(\textbf{r})$ as a fourier series, when $f$ is periodic, in the sense that there exists a $ \textbf{T}_i \neq \textbf{0} $ so that $f(\textbf{r} + \textbf{T}_i) = ...
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Fourier transform of $\left|\frac{\sin x}{x}\right|$

Is there a closed form (possibly, using known special functions) for the Fourier transform of the function $f(x)=\left|\frac{\sin x}{x}\right|$? $\hspace{.7in}$ I tried to find one using ...
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Is it possible to calculate a single frequency bin in $O(\log N)$ time (considering the $N\log N$ performance of FFT algorithms)?

Fast Fourier transform (FFT) algorithms are able to calculate the discrete Fourier transform (DFT) in only $O(N\log N)$ asymptotical time. Since there is roughly $N\log N$ operations for computing $N$ ...
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55 views

Boundedness of continuous summable function

Let $f\colon\mathbb{R}\to\mathbb{C}$ be a continuous function. If we suppose that $f$ is a $L^1(\mathbb{R;C})$ function too, then can we conclude that $f$ is bounded? ADD: I asked the preceding ...
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Using a function in Matlab

Very new to MATLAB and Im trying to use the FFT function. I got a video which showed me that a function and a normal m file is needed. Created that but now dont know how to call the function from the ...
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113 views

Explicit example of a function with Fourier transform in $C_0^\infty(\mathbb{R})$

Is anyone aware of an explicit example of a (Schwartz, real-analytic, extending to an entire function with suitable decay properties along imaginary directions as for the Paley–Wiener-Schwartz ...
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343 views

Whose basis is {1,sin(x),cos(x),sin(2x),cos(2x),…}?

Whenever $f(x)$ is a (Riemann) integrable function on $[-\pi,\pi]$ we can define its Fourier series $f=a_0/2+\sum a_nsin(nx)+b_ncos(nx)$.But we give arbitrary sequence {$a_n$} and {$b_n$},I think ...
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105 views

Discrete Fourier Series: What Happens After N/2?

I am really confused! I started to study Fourier series. I think I understand the theory approximately (I am still new to it). I was curious so started to read about DFT which I thought would be ...
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Question on Fourier Transform

Fourier transform on $f$: $$\hat{f}(\xi)=\int_{\mathbb{R}^d}f(x)e^{-2\pi ix\cdot\xi}dx$$ $\xi\in\mathbb{R}^d$. How to show that $$\hat{f}(\xi)=\frac{1}{2}\int_{\mathbb{R}^d}[f(x)-f(x-\xi')]e^{-2\pi ...
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342 views

If $f$ is a function of moderate decrease then $\delta \int f(\delta x) dx = \int f(x) dx$

A function of moderate decrease is a map from $\mathbb{R}$ into $\mathbb{C}$ such that there exists $A \in \mathbb{R}$ such that $\forall x\in \mathbb{R}, \ |f(x)| \lt \frac{A}{1 + |x|^{1+\epsilon}}$. ...
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51 views

Is there a countable Fourier transform for infinite sequences?

There's the discrete Fourier transform and the continuous one, but where's the one for infinite sequences. Let $(a_i) \subset \mathbb{C}$ be a sequence of complex numbers. The naive ways of defining ...
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Looking for guidance on a Fourier integral

Working with a Fourier transform problem, I've encountered the following integral: $$ \int_{-\infty}^{\infty}\frac{\exp\left(-a^2x^2+ibx\right)}{x^2+c^2}dx $$ where $a$, $b$, and $c$ are real ...
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78 views

Harmonic Oscillator and Fourier Series

I am currently studying Fourier Series (on my own). I am using a few different references/sources. Some are more trying to give an intuition about Fourier Series and others are more rigorous. ...
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119 views

Interpreting the sign of fourier coefficients

I am studying the Fourier series right now. Hopefully it's going okay. Now I have been playing a little bit with taking the product of a wave function (a sine or cosine with some phase) with a sine ...
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90 views

Fourier Series: going from $a_n$ and $b_n$ to $c_n$

I sort of understand the principle of the Fourier series, but when I watch the wiki page I don't understand how to get from: ${a_0 \over 2} + \sum_{n=1}^N[a_n cos({2\pi n x \over P}) + b_n sin({2\pi ...
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260 views

Exercises about Distributions

I'm looking for references (books or pdf) about the following themes (especially the first two) : Fourier Series of Distributions. Distributional solutions of ordinary differential equations. ...
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What is $\int_{0}^{\infty}e^{ixw}dw$?

We know that $$\int_{-\infty}^{\infty}e^{ixw}dw=\delta(x)$$ More details, see http://en.wikipedia.org/wiki/Dirac_delta_function Now my question is $$\int_{0}^{\infty}e^{ixw}dw=?$$ Be grateful with ...
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How do they do this w.l.o.g. so freely (Fourier series).

Theorem 2.1. Suppose that $f$ is an integrable function on the circle with $\hat{f}(n) = 0$ for all $n \in \mathbb{Z}$. Then $f(\theta_0) = 0$ whenever $f$ is continuous at the point ...
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146 views

How do I interpret continuous time fourier transform plot?

Suppose I have a fourier transform $X(f)$ of an energy signal $x(t)$. Now how do I interpret that continuous fourier transform plot. For example if the input signal is a rectangular pulse the fourier ...
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399 views

Wave equation 1D inhomogeneous Laplace/Fourier Transforms vs Green's Function

I am trying to solve the following 1D inhomogeneous wave equation. Forgive me if I a miss any rigorous mathematical concept. $$ \frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 ...
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506 views

1D Fourier Transform of Piece-wise function

I have the following piecewise function: $$ x(t) = \begin{cases} 1 & |t| \le T_0, \\[6pt] 0 & |t| > T_0. \end{cases} $$ I apologize for the formatting. I need to compute the Fourier ...
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198 views

Asymptotic behavior of Fourier transform

Consider the function $f:\mathbb{R}^3\rightarrow \mathbb{R}$, $f(x) = |x|^{-1}$. It is locally integrable, and its distributional Fourier transform is $F(f)(k) = g(k) = 4\pi/|k|^2$. Intuitively, the ...
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Integral approximation of Fourier series.

Consider a function $f$ defined on the real line. Consider the restriction of the function to the interval $[-L,L]$ and periodically extend the function using a Fourier series $$f_L(x) = ...
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percentage of numbers starting with $2$ in $\{2^n\}$

I have once heard a professor telling (during a course on Fourier theory) that there is a way to determine the numbers starting with a $2$ in the sequence $\{2^n\colon n\in\mathbb{N}\}$. I asked him ...
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55 views

laplace transform and infinitely differentiation

This fact appears in my statistics textbook (Pg 543, statistical decision theory and bayesian analysis). it says : for normal distribution the generalized bayes estimator becomes \begin{align*} ...
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132 views

Fourier Series and Solving Differential Equations

I am getting stuck on how to use Fourier Series to solve ODE's. Take the problem where \begin{equation} E(t)=200t(\pi^2-t^2), \end{equation} for $t$ between $-\pi$ and $\pi$ (period of $2\pi$), ...
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41 views

Is it possible to show that the Gaussian is a fixed point of the Fourier transform using a fixed point theorem?

We know that $\text{exp}(-\alpha |x|^2)$ is a fixed point for the unitary Fourier transform if $\text{Re } \alpha > 0$. Is it possible to show this using a fixed point theorem?
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122 views

Solve integral equation using convolution

I'm trying to solve an integral equation by identify the convolution and then transforming, but I'm getting to a really confusing expression, where I'm not sure how to continue: $$ ...
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about Fourier transformation on zero-padded vector

I have a vector $x$ of n elements. I did a fft on it and return another vector of n elements also (i.e.$X = \text{fft}(x)$). Now I am trying to pad the $x$ vector by n zeros so to get $y$ $$ y = [x ...
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177 views

l2 norms, rapidly decreasing functions and fourier transforms

Let $f\colon \mathbb{R} \to \mathbb{C}$ be a rapidly decreasing (rd) function. Let $\mathcal{F}(f)$ be the Fourier transform of $f$. It is known that 1) $\| \mathcal{F}(f) \|_2 = \| f \|_2 $ ...
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221 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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175 views

The inverse Fourier transform of $1$ is Dirac's Delta

From the definition of the Dirac delta $\delta_0$ one can infer that its Fourier transform is identically equal to $1$. But going in the other direction is not as straightforward. How can one show ...
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135 views

Fourier transforms of Marcum Q function

I cannot find in the literature the following: $$\int_0^{\infty}\,Q_1(a,bx)\,\cos(\omega x)\,\,dx$$ and $$\int_0^{\infty}\,Q_1(a,bx)\,\sin(\omega x)\,\,dx$$ with $a,\omega>0$. $Q_1(a,x)$ is ...
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Fourier Series of $f(x) = x^n$ - Fast Method?

Is there a fast way to compute the real Fourier series of $$f(x) = x^n \ ?$$ How about the complex fourier series? If there isn't a fast way for arbitrary $n$, how about $n = 5$ or something at ...
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180 views

A Cauchy principal value integral, using contour integration and Plemjel.

I came across the following integral $$lim_{\epsilon->0+}\int_\mathbb{R}\frac{e^{-ax^2+ibx}}{x+i\epsilon}dx$$ with a,b>0. Using Plemjel's formula led me to evaluating ...
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63 views

Amplitude versus time producing unexpected patterns.

I am writing a program to generate audio frequencies in multi-channel PCM format. This question may be more suited on an audio forum but I would like to know what is going on mathematically. My ...
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257 views

FFT for Fourier Integrals of analytic Functions

So, I'm trying to implement a fourier-transform of an analytic function through DFT or FFT as I'm only interested in a certain frequency range. So far I've tested Numerical Recipes' FFT algorithm ...
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106 views

Fourier Transform of $ g(x) = (1 + x^{1/a} )^a $

Given a function $g(x) = (1 + x^{1/a} )^a \times 1_{\{x > 0\}}$ that is bounded that is bounded by zero, can I compute the fourier transform? Thanks!
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An estimate For the Laplacian semi-group

Let $S(t)$ be the semi-group generated by the Dirichlet Laplacian in $L^2(0,1)$, which is given, for $y\in L^2(0,1)$, by $$S(t)y=\displaystyle\sum_{n=1}^\infty e^{-n^2\pi^2 t} \langle y,\sin(n\pi x) ...
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Sum of Fourier Series

I need to find the Fourier Series for $f\in \mathcal{C}_{st}$ that is given by $$f(x)=\begin{cases}0,\quad-\pi<x\le 0\\ \cos(x),\quad0\le x<\pi\end{cases}.$$ in the interval $]-\pi,\pi[$ ...
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63 views

Periodic Fuctions - Signals -

If, in the periods, the two half's signal periodic have the same form and opposite phases, the periodic signal has symmetry of half wave. If the periodic signal $g(t)$, of period $T_0$, satisfy the ...
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For a real valued function $g(t)$, how to prove that $G^{*}(f) = G(-f)$ , where $G(f)$ is the fourier transform of $g(t)$?

Suppose real function g(t) has corresponding fourier transform G(f). In one text book I saw that the complex conjugate of G(f) equals G(-f). How to prove this? ie for a real valued function $g(t)$, ...
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86 views

These questions are all about Fourier analysis.

Please prove these equalities,these questions appear in the chapter of Fourier series. If you can use other methods,please tell me more about it, and I am glad to know how to solve the questions: ...
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I have an equality which I need to prove

This question apppear in the chapter of fourier series. Suppose $f$ is Riemann integrable function in $[a,b]$, and $g$ is a periodic function in $\Bbb R$. The period of $g$ is $T$, and $g$ is ...
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question about 1D & 2D Fourier transformation

I have 100 1D signals (each corresponds to a 1D array of 38684 elements). Those signals are all independent and I need to find the spectrum of each of them. In this case, I have to perform 1D Fourier ...
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165 views

Scale invariance and the Mellin transform?

The Mellin transform of a function is given by: $$\mathcal{M}[f](s) = \int_0^{\infty}x^{s-1}f(x)dx$$ Supposedly, the magnitude of the Mellin transform is invariant to scaling, analogous to how the ...
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Is there a link between divergent series and discontinuities in Fourier integrals?

Take the derivative of the function $$\frac{d}{dx}\frac{e^x-1}{e^x+1}= \frac{d}{dx} \left(1- \frac{2}{e^x+1} \right) = \frac{2e^x}{(e^x+1)^2}$$ At $x=0$, this is equal to $\frac{1}{2}$. However, ...
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319 views

Trigonometric interpolation

From http://en.wikipedia.org/wiki/Trigonometric_interpolation trigonometric interpolation can be calculated as follows: Now assume we have 6 data points (0, 0.1), (1, 0.3), (2, 0.4), (5, 0.3), (6, ...
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Fourier Coefficients : Frequency Shifting

The FS coefficients of a signal $x(t)$ is given by $$x(t) \longrightarrow C_k$$ The frequency shift property says that: if we multiply a signal $x(t)$ by $e^{j m\omega_0 t}$ the fourier series ...