Questions tagged [fourier-analysis]
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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Different ways to prove $\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$ (the Basel problem)
As I have heard people did not trust Euler when he first discovered the formula (solution of the Basel problem)
$$\zeta(2)=\sum_{k=1}^\infty \frac{1}{k^2}=\frac{\pi^2}{6}$$
However, Euler was Euler ...
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Fourier transform for dummies
What is the Fourier transform? What does it do? Why is it useful (in math, in engineering, physics, etc)?
This question is based on the question of Kevin Lin, which didn't quite fit in Mathoverflow. ...
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What is the difference between Fourier series and Fourier transformation?
What's the difference between Fourier transformations and Fourier Series?
Are they the same, where a transformation is just used when its applied (i.e. not used in pure mathematics)?
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Connection between Fourier transform and Taylor series
Both Fourier transform and Taylor series are means to represent functions in a different form.
What is the connection between these two? Is there a way to get from one to the other (and back again)? ...
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Difference between Fourier transform and Wavelets
While understanding difference between wavelets and Fourier transform I came across this point in Wikipedia.
The main difference is that wavelets are localized in both time and frequency whereas ...
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Fourier Transform of Derivative
Consider a function $f(t)$ with Fourier Transform $F(s)$. So $$F(s) = \int_{-\infty}^{\infty} e^{-2 \pi i s t} f(t) \ dt$$
What is the Fourier Transform of $f'(t)$? Call it $G(s)$.So $$G(s) = \int_{-\...
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Is this similarity to the Fourier transform of the von Mangoldt function real?
Mathematica knows that the logarithm of $n$ is:
$$\log(n) = \lim\limits_{s \rightarrow 1} \zeta(s)\left(1 - \frac{1}{n^{(s - 1)}}\right)$$
The von Mangoldt function should then be:
$$\Lambda(n)=\...
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Fourier transform of function composition
Given two functions $f$ and $g$, is there a formula for the Fourier transform of $f \circ g$ in terms of the Fourier transforms of $f$ and $g$ individually?
I know you can do this for the sum, the ...
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Why do we use trig functions in Fourier transforms, and not other periodic functions?
Why, when we perform Fourier transforms/decompositions, do we use sine/cosine waves (or more generally complex exponentials) and not other periodic functions? I understand that they form a complete ...
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How to interpret the adjoint?
Let $V \neq \{\mathbf{0}\}$ be a inner product space, and let $f:V \to V$ be a linear transformation on $V$.
I understand the definition1 of the adjoint of $f$ (denoted by $f^*$), but I can't say I ...
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Real world application of Fourier series
What are some real world applications of Fourier series? Particularly the complex Fourier integrals?
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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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Did Joseph Fourier ever make a pure mathematical mistake?
Cited by "Imre Lakatos and the Guises of Reason" John David Kadvany, 2001:
It is remarkable that the nineteenth century was a time of error for
mathematics: not trivial oversights or amateur ...
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Derivative of convolution
Assume that $f(x),g(x)$ are positive and are in $L^1$. Moreover, they are differentiable and their derivative is integrable. Let $h(x)=f(x)*g(x)$, the convolution of $f$ and $g$. Does the derivative ...
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Non-power-of-2 FFT's?
If I have a program that can compute FFT's for sizes that are powers of 2, how can I use it to compute FFT's for other sizes?
I have read that I can supposedly zero-pad the original points, but I'm ...
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Fourier transform of unit step?
I don't understand what's wrong with my derivation below...
$\delta(t) = u'(t)$
$\mathcal{F}(\delta)(\omega) = 1 = \mathcal{F}(u')(\omega) = i\omega \times \mathcal{F}(u)(\omega)$ (since the ...
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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 e^{...
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What is the Fourier transform of the product of two functions?
Given $x(t) = f(t) \cdot g(t)$, what is the Fourier transform of $x(t)$? If possible, please explain your answer.
The motivation behind the question is homework, but this is a basic principle in the ...
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Sobolev space $H^s(\mathbb{R}^n)$ is an algebra with $2s>n$
How do you prove that the Sobolev space $H^s(\mathbb{R}^n)$ is an algebra if $s>\frac{n}{2}$, i.e. if $u,v$ are in $H^s(\mathbb{R}^n)$, then so is $uv$? Actually I think we should also have $\lVert ...
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Closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$.
What is the closed form for $\sum_{n=-\infty}^{\infty}\frac{1}{(n-a)^2+b^2}$? We can use Fourier series of $e^{-bx}$ ($|x|<\pi$) to evaluate $\sum_{n=-\infty}^{\infty}\frac{1}{n^2+b^2}$. But this ...
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Intuitively, why is the Gaussian the Fourier transform of itself?
It's a standard exercise to find the Fourier transform of the Gaussian $e^{-x^2}$ and show that it is equal to itself. Although it is computationally straightforward, this has always somewhat ...
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What are all the generalizations needed to pass from finite dimensional linear algebra with matrices to fourier series and pdes?
I've studied Linear Algebra on finite dimensions and now I'm studying fourier series, sturm-liouville problems, pdes etc. However none of our lecturers made any connection between linear algebra an ...
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Functions that are their own Fourier transform
In Stein's Fourier Analysis, there's the following exercise:
The function $e^{-\pi x^2}$ is its own Fourier transform. Generate other functions [presumably in the Schwartz space $S(\mathbb{R})$] ...
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How to calculate the Fourier transform of a Gaussian function?
I would like to work out the Fourier transform of the Gaussian function
$$f(x) = \exp \left(-n^2(x-m)^2 \right)$$
It seems likely that I will need to use differentiation and the shift rule at some ...
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Instructive proofs in functional analysis
I am beginning to learn functional analysis (from Folland and Royden), but I am from a non-mathematical background, so I often encounter techniques in proofs that I am not familiar with (for example ...
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Fourier - Are sinusoidals strictly required?
We can define all signals as a sum of sinusoidals by taking fourier transform of the signal. Thats OK. My question is, why sinusoidals.? Can there be an another transform like Fourier somewhere in the ...
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Proof of the unbounded-ness of $\sum_{n\geq 1}\frac{1}{n}\sin\frac{x}{n}$
For any $x\in\mathbb{R}$, the series
$$ \sum_{n\geq 1}\tfrac{1}{n}\,\sin\left(\tfrac{x}{n}\right) $$
is trivially absolutely convergent. It defines a function $f(x)$ and I would like to show that $f(x)...
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What is the difference between the Discrete Fourier Transform and the Fast Fourier Transform?
Can anybody answer this question?
Thank you.
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Explicitly reconstructing a function from its moments
Let $f$ be an integrable real valued function defined on $[0,\infty)$. Let $$m_n=\int_0^\infty f(x)x^n \mathrm dx$$ be the $n^{th}$ moment, and suppose that all of these integrals converge absolutely....
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Is a Fourier transform a change of basis, or is it a linear transformation?
I've frequently heard that a Fourier transform is "just a change of basis".
However, I'm not sure whether that's correct, in terms of the terminology of "change of basis" versus "transformation" in ...
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Concrete FFT polynomial multiplication example
I have read a number of explanations of the steps involved in multiplying two polynomials using fast fourier transform and am not quite getting it in practice. I was wondering if I could get some help ...
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Is there a function whose autoconvolution is its square? $g^2(x) = g*g (x)$
I am looking for a function over the real line, $g$, with $g*g = g^2$ (or a proof that such a function doesn't exist on some space like $L_1 \cap L_2$ or $L_1 \cap L_\infty$). This relation can't hold ...
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Double integral separating real and imaginary parts of $\zeta (\sigma+i t)$
Recently, I stumbled upon what I believe to be a new representation of $\zeta(\sigma+i t)^b$ by chance, and thus have no proof of it, and I am wondering if it is possible to prove.
Let $\eta (s) = \...
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Why do Fourier Series work?
I would like to have an intuitive understanding of Fourier Series. I mean, I know the formulas:
$$
f(t) =\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos(n\pi tL)+\sum_{n=1}^\infty b_n \sin(n\pi tL)
$$
And ...
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Can a non-periodic function have a Fourier series?
Consider two periodic functions. Assume their sum is not periodic.
The periodic functions can be represented by a Fourier series. If you add up the Fourier series, you get a series that represents ...
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Applications of Pseudodifferential Operators
I am very interested in just about anything that has to do with PDE's, and inevitably pseudodifferential operators comes up. Its obvious that such a novel way of looking at PDE's would be important, ...
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Why is Fourier Analysis effective for studying uniform distributions
On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim:
When it comes to the primes, we find that we do not have a good ...
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Proving commutativity of convolution $(f \ast g)(x) = (g \ast f)(x)$
From any textbook on fourier analysis:
"It is easily shown that for $f$ and $g$, both $2 \pi$-periodic functions on $[-\pi,\pi]$, we have $$(f \ast g)(x) = \int_{-\pi}^{\pi}f(x-y)g(y)\;dy = \int_{-\...
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Why do Fourier transforms use complex numbers?
I know that the Fourier transform is as follows:$$\hat{f}(\xi)= \int_{-\infty}^{\infty}\exp(-\mathrm ix\xi)f(x)\mathrm{d}x$$ but I couldn't understand why we should use the complex number $i$ in the ...
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Fourier transform of the indicator of the unit ball
What is the Fourier transform of the indicator of the unit ball in $\mathbb R^n$?
I think it is known as one of special functions, so I would be happy to know which one.
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Tensor products of functions generate dense subspace?
Let $X$ and $Y$ be two spaces in certain category, $F(\cdot)$ a functor associating each space with a function space (with certain topology). Assume that for any $f\in F(X)$ and $g\in F(Y)$, $f\otimes ...
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What is Fourier Analysis on Groups and does it have "applications" to physics?
I am trying to be as specific as possible, but I am extremely unclear about this topic (Fourier Analysis on Groups).
In Reed-Simon Vol II (Fourier Analysis, Self-Adjointness) there is some ...
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What is the difference between a function and a distribution?
I remember there was a tongue-in-cheek rule in mathematical analysis saying that to obtain the Fourier transform of a function $f(t)$, it is enough to get its Laplace transform $F(s)$, and replace $s$ ...
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Calculate the Fourier transform of $\log |x| $
How can one prove that the Fourier transform of $\log |x|$ is
$$-\pi \mathrm{pf} \frac{1}{|\xi|} +C \delta,$$
where $\mathrm{pf}\frac{1}{|x|} = D(\mathrm{sign}(x)\log|x|)$ (in the sense of ...
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Learning algebra and harmonic analysis
I've revised my question a bit in response to the (very helpful) advice so far--
I have an engineering background but am interested in learning abstract harmonic analysis. My interest is rather ...
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Fourier transform of even/odd function
How can I show that the Fourier transform of an even integrable function $f\colon \mathbb{R}\to\mathbb{R}$ is even real-valued function? And the Fourier transform of an odd integrable function $f\...
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On the vector spaces of Taylor Series and Fourier Series
Taylor series expansion of function, $f$, is a vector in the vector space with basis: $\{(x-a)^0, (x-a)^1, (x-a)^3, \ldots, (x-a)^n, \ldots\}$. This vector space has a countably infinite dimension. ...
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How is the Fourier transform "linear"?
A "linear" function usually means one who's graph is a straight line, or that involves no powers higher than 1. And yet, many sources will tell you that the Fourier transform is a "linear transform".
...
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Accessible proof of Carleson's $L^2$ theorem
Lennart Carleson proved Luzin's conjecture that the Fourier series of each $f\in L^2(0,2\pi)$ converges almost everywhere. Also, Richard Hunt extended the result to $L^p$ ($p>1$).
Some time ago I ...
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What is the square root of a Fourier transform?
Given the Fourier transform defined like
$$
\mathcal F[f](\omega) = \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^\infty f(t) e^{-iwt} \mathrm{d}t
$$
how could one define a square root $\mathcal G$ of the ...
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