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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DTFT of a triangle function in closed form

I am sampling a continuous signal $x_c(t)$ that follows a triangle function in the time domain, meaning: $$x_c(t)=\left\{\begin{array}{rl}1-|t/a|,&|t|<|a|\\ ...
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Which function's Fourier transform is the function itself?

We know that the Fourier transform of a Gaussian function is Gaussian function itself. Can anyone give one or more functions which have themselves as Fourier transform?
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Fourier matrix - multiplicity of eigenvalues?

This question is Miscellaneous Exercise M.10 in Chapter 8 (Bilinear Forms) of Artin's Algebra. (The sentences in italics are due to me.) The row and column indices in the $n \times n$ Fourier ...
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How can one prove that integrating $\cos{(ix)}\cos{(jx)}$ cancels in Fourier Analysis?

This portion of the Wikipedia entry on Fourier Analysis details a formula, and later says that the terms for $j \ne k$ vanish. Could someone please provide a proof of this? I actually would like to ...
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Dirac delta forcing of a harmonic oscillator

Is it possible to solve this differential equation: $$\ddot{x}(t)+\omega^2x(t)=k\delta(t)$$ where $k$ is a constant and $\delta(t)$ the Dirac delta function? Is it possible alternatively, to know ...
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Computing Coefficients of Complex Form Fourier Series

I am having some trouble knowing how to correctly start a problem of finding the Fourier Coefficients using complex exponential form. The problem is given below: $$g_1(t)=\begin{cases} 1,~~\qquad ...
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Fourier transform of the characteristic function

My qustion is about the Fourier transform of the characteristic function $\chi_{[0,1]}$. How can I find what it is? The problem is I got something really messy, so I think I didn't get it right.
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Scale invariance and $1/f^2$ power spectrum

In the paper Occlusion Models for Natural Images : A Statistical Study of a Scale-Invariant Dead Leaves Model; Lee, A. B. Mumford, D. B. Huang, J.; International Journal of Computer Vision I read ...
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How to do a statistical analysis?

I am sorry for my profound knowledge of statistics and for this candid question. Your help is valued. I have the following data. Data1 (stress-I):: 24 35 53 15 40 37 58 11 ...
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227 views

Do Fourier transforms of $\min$ and $\max$ exist (in closed form)?

I am wondering if there are Fourier transforms of $\min(x,a)$ and $\max(x,a)$ functions. Please forgive me if this is a dumb question, I don't normally use Fourier transforms. I attempted to simply ...
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954 views

time-frequency domain

im confused on how these folks seems to like convert a frequency into a time function, and a time function into a frequency function. i know that time function uses amplitude that varies over time, ...
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How to find inverse Fourier transform

I have the function $$ \delta(f-2) $$ How can we inverse Fourier transform it? It's easy if $f$ is replaced with $w$. But based on my knowledge, $w = 2\pi f$. The correct answer is $$ e^{4\pi i ...
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756 views

Fourier transforms of cos and sin

I have the function of time $f(t)=\cos(t10\pi) + \sin(t10\pi)$ and i wish to transform it. By using the tables, i have $\pi [\delta(w-10\pi) + \delta(w+10\pi)] + (\frac{\pi}{j})[ \delta (w-10\pi) = ...
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Introductory book on Distribution theory [duplicate]

Possible Duplicate: Distribution theory book Is there a good alternative to Friedlander Introduction to the Theory of Distributions ? Many thanks !
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308 views

How to solve a linear PDEs with trigonometric function as coefficients

Is a general method for solving a system of linear partial differential equation with trigonometric function as coefficients exist ? For example something like that: $q$ is the unknown function, $2 ...
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232 views

Solution of the Dirichlet problem

I'm reading Jones' book Lebesgue integration on Euclidean space. Let $u(x, y)$ be a harmonic function on the half space $\mathbb{R}^n \times (0, \infty)$, with boundary condition $f(x) = u(x, 0)$. On ...
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What does it mean to carry out calculations modulo $S^{- \infty}$

I have trouble making sense of the following remark that is given in a set of lecture notes I am currently going through on my own: " Oscillatory integrals are used for the study of singularities of ...
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711 views

Meaning of polynomially bounded

I am currently reading through some lecture notes on Fourier Transform and Distributions on my own, and came upon the notion of a polynomially bounded function. I am not sure I understand this ...
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240 views

Functions whose Fourier transform vanishes outside of a small interval

Suppose $f(t)$ is a function whose Fourier transform $\hat{f}(\omega) = \int_{-\infty}^{+\infty} f(t) e^{- \omega t} dt$ is supported on the interval $[-\epsilon,+\epsilon]$. Is there a theorem to the ...
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133 views

Help understanding a proof on Taylor's formula in Schwartz space $S(\mathbb{R}^n)$

I am having trouble understanding a proof to establish a specific version of Taylor's formula. I'll first give the statement and then below cite the part where I am stuck, so here is what I'd like to ...
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Fourier transform of $x^\alpha$

Define $\hat{f}(\xi)\equiv F(f)(\xi):=\int_{\mathbb{R}}e^{ix.\xi}f(x)dx $ My question is: if we consider $x^{\alpha}$ as a distribution then what is $ F(x^{\alpha})(\xi)$ where ...
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Continuity of the Characteristic Function of a RV

Defining the Characteristic Function $ \quad \phi(t) := \mathbb{E} \left[ e^{itx} \right] $ for a random variable with distribution function $F(x)$ in order to show it is uniformly continuous I say ...
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Showing that a function is the sine function

How can I prove the following: If $ f: \mathbb{R} \rightarrow \mathbb{C} $ is a $2\pi$-periodic function of class $C^{\infty}$ such that $f'(0)=1$ and that for any $n\in \mathbb{N}, ...
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What are some good Fourier analysis books?

I have taken real analysis, but never learned Fourier analysis. What is a good book to get started? I'm not sure the Stein book would be good.
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Are these zeros equal to the imaginary parts of the Riemann zeta zeros?

Edit 8.8.2013: See this question also. The Fourier cosine transform of an exponential sawtooth wave times $e^{-x/2}$: $$\operatorname{FourierCosineTransform}(\operatorname{SawtoothWave}(e^x)\cdot ...
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Power Spectral Density excercise

Compute the power spectral density of $x(t) = \operatorname{sgn}(t)$ Hint: $$\lim_{t\to\infty} t(\operatorname{sinc}(ft))^2 = \delta(t).$$ Please help me I have to solve this exercise urgently. ...
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Discrete Cosine and Sine Transforms

Can anyone explain to me what is the point of using complex numbers to get the Discrete Fourier Transform when the Discrete Cosine Transform and Discrete Sine Transform exist and both use only real ...
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Fourier transform of sine and cosine function

For the sine function we can do the following formal computation: $$\mathcal{F} (\sin(2\pi kt))(x) = \int_{-\infty}^{\infty} e^{-2\pi i xt} \frac{e^{2\pi i kt}-e^{-2\pi i kt}}{2i}dt= \frac{i}{2} ...
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Why does $\sum_{n \in \mathbb{Z}}|\widehat{f(n)}|<\infty$ gives that the matching Fourier series uniformly converges?

I'd really love to understand why does the fact that the series of the absolute Fourier coefficient converges ($\sum_{n \in \mathbb{Z}}|\widehat{f(n)}|<\infty$) for a function $f$, leads to the ...
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Scaling property of Fourier series and Fourier Transform

This question about the intuition behind the scaling property of the Fourier transform made me wonder about the corresponding notion for a Fourier series. The Fourier transform of $f(ax)$ is ...
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743 views

Schwartz Space is a subspace of Sobolev Space, but how can I show that?

How can I see that $S(\mathbb{R}) \subset H^s(\mathbb{R})$, where the former is Schwartz and the latter is Sobolev space ? This should be obvious according to my notes but unfortunately I can't make ...
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The integral of a function over $S^1$

Let $S^1=\mathbb R/\mathbb Z,$ I was wondering how to calculate the integral of a function over $S^1$ and why. Like, $\int_{S^1}1 dx=?$ Given an "appropriate" function $f$, what is $\int_{S^1}f(x)dx?$ ...
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What are the local minima in this spectrum?

Edit 6.2.2012: The sequence to be transformed should be f = 0,1,2,3,4,5... which makes the mentioning of the von Mangoldt function less necessary. Edit 5.2.2012: I had the wrong plot of the insignal. ...
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Inverse Fourier Transform - left and right? [duplicate]

Possible Duplicate: Surjectivity of the Fourier Transform on Schwartz Space Consider the Fourier transform on Schwartz space, given by \begin{equation} \mathcal{F}(f)(\xi)= \hat{f}(\xi) = ...
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Surjectivity of the Fourier Transform on Schwartz Space

I understand that, for $f \in S(\mathbb{R})$ (the Schwartz space) the transform \begin{equation} \tag1 Tf(\xi) = (2\pi)^{-\frac{1}{2}} \int_\mathbb{R} e^{i\xi x}f(x) \,dx \end{equation} defines a ...
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Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
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62 views

delaying signal

What does delaying a signal mean? Graphically? Mathematically? Is it, advancing to the next numbers, or using the previous numbers? Suppose i have $x[n] = \{0,1,2,3,4,5\}$ and i use $x[n-m]$ (example ...
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How smooth is a smooth function?

Let's say a smooth function is a $\mathcal{C}^\infty$ function on $\mathbb{R}$. Some smooth functions are not analytic, the most notorious example being the bump functions. A non-analytic ...
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Fundamental role of the Fourier Transform

I am currently learning about the Fourier Transform and the associated Fourier Analysis. So far I realize that it has a number of applications, but more than that it seems to be central to Functional ...
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Condition for differentiablility

Suppose I have two functions that are Schwartz class, say $f,g \in S(\mathbb{R})$, and suppose I have another function $\psi(x)$ such that \begin{equation} g(x) = \psi(x)f(x) \end{equation} I would ...
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Help with understanding a proof in Fourier Analysis

I have a stack of lecture notes that I am currently going through to teach myself a little bit about Fourier Analysis. Now I struggle with the following Lemma, which is needed to talk about the ...
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Showing a function is Schwartz

This question is related to this post, for which I received a really good answer that gave a beautifull solution, yet I am still trying to understand one thing on the side that is not covered by the ...
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485 views

Chirp Transform and Convolution

I was reading about the discrete fourier transform from the CLR algorithms book and I came upon an exercise whose hint confuses me. The exercise reads as follows: The chirp transform of a vector ...
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295 views

Question on Schwartz function

I am seeing the Schwartz Space for the first time today and I have trouble understanding the following argument: Given that $f \in S({\mathbb{R}})$ with $f(x_0) = 0$ Taylor's theorem tells us that ...
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Quick question on Fourier Transform of $\exp(-\frac{x^2}{2})$

I am currently looking at an example of how to calculate the Fourier Transform for the function \begin{equation} f(x) = \exp\left({-\frac{x^2}{2}}\right) \end{equation} Now $f$ solves the differential ...
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Showing $e^{-x^2} \in \mathcal{S}(\mathbb{R})$

I am currently studying Fourier Analysis on my own and have just been started to look at the Schwartz Space of rapidly decaying functions. One example of such functions is given in the notes that I ...
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What is the Fourier transform of a phase modulated signal?

I'm studying some Fourier analysis and have for a while been trying to figure out how to apply the Fourier transform to a phase modulated signal. More rigorously stated, what is $$ ...
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What does the Fourier Transform mean in the context of images?

This is clearly a very important equation with tonnes of properties that I see come up a lot in image processing literature, but I don't understand why this equation is important, and what it is ...
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Fourier transform independent of kernel?

I've tried computing a windowed Fourier transform using various kernels that were all made from periodic signals of the form a + bi with b being a shifted version of a. I used square waves, sin waves, ...
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Uniform convergence of Fourier Series

I am currently studying Fourier Analysis on my own. In the Notes I use the following comment is made, which I unfortunately don't understand: Given that we know the series $f(x) = \sum c_k e^{ikx}$ ...