43
votes
Why is the Fourier transform so ubiquitous?
From the point of view of physics, Fourier transforms are ubiquitous because they are expansions in eigenfunctions of the derivative operator - and the derivative operator is fundamental in many ...
- 6,696
30
votes
Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?
This is basically a lightly transformed version of Euler's cotangent partial fraction expansion
$$ \pi \cot(\pi z) = \frac{1}{z} + \sum_{n=1}^\infty \frac{1}{z-n} + \frac{1}{z+n}$$
(the log derivative ...
- 114k
28
votes
Anti-delta function?
Let $\beta\mathbb{R}$ be the Stone-Čech compactification of $\mathbb{R}$: any bounded continuous function $f$ on $\mathbb{R}$ extends uniquely to a (necessarily bounded) continuous function $f^\beta$ ...
- 32.4k
27
votes
Motivation and physical interpretation of the Laplace transform
The physical motivation for the Laplace transform is causality.
Consider the linear input-output relation
$$f_{\text{output}}(t)=\int_{0}^\infty R(t-t')f_{\text{input}}(t')\,dt'.$$
Causality dictates ...
- 188k
26
votes
Accepted
The $2\pi$ in the definition of the Fourier transform
The version number 2 is the only one that makes the Fourier transform both a unitary operator on $L^2$ and an algebra homomorphism from the convolution algebra in $L^1$ to the product algebra in $L^\...
- 488
25
votes
The $2\pi$ in the definition of the Fourier transform
I would like to add the point of view from a more general aspect.
In the general situation, the Fourier transform can be defined for any locally compact abelian group $G$.
Let $G$ be a locally ...
- 3,432
25
votes
Positivity of certain Fourier transform
A simple argument that the Fourier transform cannot be nonnegative for any $m>1$, integer or not, is given in my 1991 paper with Odlyzko and Rush:
Noam D. Elkies, Andrew M. Odlyzko, and Jason A. ...
- 79.9k
25
votes
Accepted
When are Fourier coefficients monotonic?
It suffices that $f$ be (the restriction to $[0,2\pi]$ of) a completely monotone real-valued function defined on $[0,\infty)$. Indeed, then for some finite measure $\mu$ on $[0,\infty)$ and all real $...
- 128k
25
votes
Motivation and physical interpretation of the Laplace transform
Besides the important physical motivation pointed out by Carlo Beenakker, there is another one, purely mathematical. Laplace transform is a generalization of a power series (and Dirichlet series).
In $...
- 91.8k
22
votes
Why is the Fourier transform so ubiquitous?
To add a representation theory perspective: if $G$ is a Lie group, and $f$ is a function (or more precisely a distribution) on $G$ then (under certain mild conditions on $f$ and $G$), the function $f$ ...
- 7,952
21
votes
Accepted
Where does the Laplace transform come from?
Set $A = L^1([0, \infty))$, equipped with the structure of a Banach $*$-algebra via convolution. The spectrum of this algebra is the half plane $\text{Re}(z) \geq 0$, and the Gelfand transform is the ...
- 29.2k
17
votes
Motivation and physical interpretation of the Laplace transform
The Laplace transform is the fundamental operation encoding the canonical ensemble in statistical mechanics. It converts the density of states $d(\varepsilon )$ (a non-statistical concept) into the ...
- 6,696
16
votes
Accepted
Positivity of certain Fourier transform
it is positive for $m=1$, but not for $m=2$, see this Mathematica output:
- 188k
16
votes
Positivity of certain Fourier transform
I think the result goes back to Polya, see "Some theorems on stable processes" by Blumenthal and Getoor. Another reference is Paul Lévy "Sur une application de la dérivée d’ordre ...
16
votes
Looking for sufficient conditions for positive Fourier transforms
please show me how to prove for $|\omega|>20$
With great pleasure. We shall just show that $F(y)=\int_0^\infty e^{-x^a}x^a\log x\cos(yx)\,dx>0$ for large enough $y>0$. Note that $\cos(yx)=\Re ...
- 61.9k
14
votes
Accepted
What is the difference (if any) between "fourier transform" and "SO(3) fourier transform"?
The usual (discrete) Fourier transform expands a function in the basis set $e^{i n\phi}$, $\phi\in(0,2\pi)$, $n\in\mathbb{Z}$. The $\text{SO}(3)$ Fourier transform uses as basis the Wigner D-functions ...
- 188k
11
votes
Pairs of elementary Fourier transforms in $L^2$
The following example is interesting
$$\widehat{\sin(x)\over x \sqrt{|x|}} =
\sqrt{2\pi} \ \left\{
{
\matrix{
\scriptstyle
\sqrt{|\xi+1|} + \sqrt{|\xi-1|} &
...
- 18.7k
11
votes
Function and Fourier transform vanish on an interval
A distribution for which both the distribution and its Fourier transform vanish on some interval is the Shah function aka Dirac comb
$$
Ш = \sum_{k=-\infty}^\infty \delta_k
$$
which is its own Fourier ...
- 12.7k
11
votes
Accepted
Fourier transform on Minkowski space
Well you seem to have worked it out but I wrote most of this before your comment happened: I claim there isn't any material difference between your "Minkowski space Fourier transform" and the usual ...
- 784
11
votes
Intuitive explanation why "shadow operator" $\frac D{e^D-1}$ connects logarithms with trigonometric functions?
The op $$T_x = \frac{D_x}{e^{D_x}-1} = e^{b.D_x},$$
where $(b.)^n = b_n$ are the Bernoulli numbers, is (mod signs) often referred to as the Todd operator (maybe originally given that name by ...
- 10.5k
10
votes
Accepted
Fourier transform either changes sign infinitely often far out or is continuous at $x=0$
The exercise is stated in Dym and McKean with a mistake. The correct statement is
If $f$ is real, even, the finite limits $f(x\pm 0)$ exist for all $x$,
and $\hat{f}$ does not change sign for $|x|>...
- 91.8k
10
votes
About the Fourier transform of the logarithm function
I thought that it might be instructive to present an approach to deriving the Fourier transform of $\log(|x|)$. The result includes a distributional interpretation of $\frac1{|x|}$. Finally, we show ...
- 201
10
votes
Why is the Fourier transform so ubiquitous?
From the point of view of engineering, sin and cos are eigenfunctions of LTI (linear time-invariant) systems, which makes the Fourier transform immanently important for system theory - and thus for ...
- 231
10
votes
Anti-delta function?
To elaborate on the comment, I would suggest to take $F(x)=x^{-2}\delta(1/x)$. Let me check for the representation
$\delta_{\epsilon}(x)=(2\pi\epsilon)^{-1/2}e^{-x^2/2\epsilon}$,
and $F_\epsilon(x)=x^{...
- 188k
10
votes
Who introduced the discrete Fourier transform?
You can go earlier than Gauss if you allow for a DFT involving only sines or only cosines: I quote from Gauss and the history of the fast Fourier transform
Alexis-Claude Clairaut (1713-1765) ...
- 188k
9
votes
Accepted
Function and Fourier transform vanish on an interval
Over at MSE (see linked question), user1952009 (= reuns on MO) has given a very elegant construction of a Schwartz function with this property. I am reproducing his/her answer here.
Let $g,h\in C_0^{\...
9
votes
Limit of an integral vs limit of the integrand
Care should be taken because of the pole at $k=0$, let me first take the principal value of the integral. I note that $I(\alpha,-r)=\bar{I}(\alpha,r)$ (complex conjugate), for convenience I will ...
- 188k
9
votes
Accepted
On the Fourier inversion formula
Note: I am not sure if I understand the word "converges" correctly.
This is completely analogous to the similar question regarding convergence of Fourier series, which is classical.
Let $$g(...
- 17.2k
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