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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Fourier transform of $ (t^2-1)^n(\operatorname{sign}(t-1)-\operatorname{sign}(t+1)), $

I have trouble with finding the Fourier Transform of the following function: $$ (t^2-1)^n(\operatorname{sign}(t-1)-\operatorname{sign}(t+1)), $$ where $n\in N$. I know that the answer involve so ...
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96 views

Fourier transform of function decaying at $ae^{-bx^2+cy^2}$

I'm a bit stumped on a problem. The problem is as follows: Suppose $f(z)$ is entire and $|f(x+iy)|\leq ae^{-bx^2+cy^2}$ for $a,b,c>0$. If $\hat{f}(z)=\int_{-\infty}^\infty f(x)e^{-2\pi ixz}dx$, ...
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Clear explanation of heaviside function fourier transform

I know that fourier transform of Heaviside function is : $\hat{H}(x) = \pi \delta(\omega) + i (v.p. \frac{1}{\omega})$ How can i proof this result?
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1answer
164 views

Boundedness of supremum of an Integral operator

I am trying to find an $L_2$ - bound on a certain class of operators, and on my way I produced an estimate for which I need to show that \begin{equation} \sup_{x \in \mathbb{R}^n} \, ...
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Homogeneous function in bounded mean oscillation BMO($\mathbb R^n$) space

Let me recall some notations: The mean oscillation of a locally integrable function $u$ (i.e. a function belonging to $ L^1_{\textrm{loc}}(\mathbb{R}^n))$ over a cube Q in $\mathbb R^n$ (which has ...
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Conceptual/Graphical understanding of the Fourier Series.

I've been reading about how the Fourier Series works, so like how the orthogonality cancels out all but the one that we're looking for. I've read derivations of the Fourier Series. What I would like ...
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1answer
258 views

Showing $\sum_{n=-\infty}^\infty \frac{1}{(z+n)^2}=\frac{\pi^2}{\sin^2(\pi z)}$

I'm doing a homework problem, and so far I've proved $$\sum_{n=-\infty}^\infty \frac{1}{(z+n)^k}=\frac{(-2\pi i)^k}{(k-1)!}\sum_{m=1}^\infty m^{k-1}e^{2\pi imz}$$ for $k$ an integer $\geq 2$ and ...
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2answers
1k views

Computing a convolution using FFT

I have two sequences of the same length, $(x_i), i=1, 2, \ldots, N$ and $(y_i), i=1, 2, \ldots, N$ and a function $K(t) = -t \times \exp(-t^2 / 2)/ \sqrt{2 \pi}$. I need to compute the following ...
2
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1answer
182 views

the integral of the inverse of a Fourier series

Let $\{a_h\}$ be a double-sided complex sequence such that $\sum_{h=-\infty}^{\infty} |a_i| <\infty$ with $a_{0}\neq0$. Set $f(x) := \sum_{h=-\infty}^{\infty} a_h \exp(ixh)$ and assume that ...
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1answer
3k views

Fourier transform on $1/(x^2+a^2)$

I'm reading a book to review Fourier transforms, and I came across the following example, which is here: ...
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100 views

positively homogeneous asymptotic expansion associated to the symbol of a pseudodifferential operator

I am currently reading about pseudodifferential operators and their symbols, and I came across the notion of classical pseudodifferential operators. For these it is possible to find an asymptotic ...
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1answer
238 views

An elegant non-technical account on the work of Joseph Fourier.

It would seem difficult for a naive person to understand the beauty of work done by Fourier. So as far as I know, one can use the Fourier transforms, analysis and series to apply them for heat ...
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2answers
2k views

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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2answers
957 views

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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431 views

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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126 views

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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2answers
545 views

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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1answer
1k views

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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1answer
2k views

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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84 views

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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244 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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3answers
1k 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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12k views

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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1answer
780 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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100 views

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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336 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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245 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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184 views

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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1answer
781 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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259 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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1answer
137 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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956 views

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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162 views

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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3answers
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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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380 views

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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1answer
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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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3answers
4k views

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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1answer
849 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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83 views

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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1answer
156 views

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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1answer
265 views

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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3answers
5k views

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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1answer
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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873 views

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 ...