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 ...
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 $...
20
votes
Motivation and physical interpretation of the Laplace transform
Interest is continuously compounded at rate $r,$ so that if you deposit $\\\$1$ now it will be worth $\\\$e^{rx}$ at time $x.$ How much do you need to deposit now in order to withdraw at rate $\\\$f(x)...
16
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 ...
7
votes
Is $\frac{|t|}{e^{a|t|}-e^{-b|t|}}$ the Fourier transform of a positive function
No for $0<a<b$ your function has a global maximum at $\pm x_0 \neq 0$. Then, if it was $\hat{\varphi} = f $ for some positive function $\varphi$, $f(x-y)$ would be a positive semidefinite kernel....
6
votes
Motivation and physical interpretation of the Laplace transform
I will not discuss the uses of the Laplace transform.
Myself I think of the analogy with sequences. A sequence $(a_n)_{n\geq 0}$ is determined by its generating function (convergence issues aside)
...
4
votes
Accepted
Characteristic exponent after Girsanov transformation
By Doob-Dynkin we have that $A_t=f(t,\{B_s\}_{0\leq s\leq t})$.
If $\mu_0$ is the law of Brownian motion then under the measure $\mu=\exp\left(\int_0^T A_sdB_s-1/2\int_0^T A_s^2 ds\right)\mu_0$ we ...
4
votes
Accepted
Any references for generalised square functions?
For $0<p<\infty$, $0<q\le\infty$ and $s\in\mathbb R$, the Triebel-Lizorkin space $F_{pq}^s(\mathbb R^n)$ is the set of all (tempered distributions) $f$ such that $\big\||\{2^{js}\cdot P_jf\}...
4
votes
Motivation and physical interpretation of the Laplace transform
Re "I have not yet encountered any good explanation of how the Laplace transform formula arises." I think it's useful to look at early origins of the LPT, relations to other transforms, and ...
4
votes
Motivation and physical interpretation of the Laplace transform
From R. N. Bracewell, "The Fourier Transform and Applications", McGraw Hill 3rd ed., pp.381:
"Advantages of the Laplace transform over the Fourier transform for handling electrical ...
4
votes
Accepted
Is there a compactly supported differentiable function whose Fourier transform is not in L1?
Yes, such differentiable compactly supported functions $G$ with $\widehat{G}\notin L^1$ exist. This follows from the fact the Fourier transform is not bounded as a map from $\{f\in C[0,1]: f(0)=f(1)=0\...
4
votes
Accepted
Predicting the peak "amplitude" of a damped sine wave in the frequency spectrum with FFT
You can just Fourier transform your signal
$$\hat{x}(\omega)=A\int_{0}^\infty \sin(\omega_0 t)e^{-\alpha t}\,e^{i\omega t}\,dt=\frac{A\omega_0}{\alpha^2-2 i \alpha \omega+\omega_0^2-\omega^2}.$$
The ...
4
votes
The Fourier transform of the Liouville function?
Your first formula for $\lambda(x)$ is equivalent to
$$\lambda(x)=\underset{N\to \infty}{\text{lim}}\left(\sum\limits_{z=-N}^N \cos(\pi(\Omega(z)+x-z))\, \text{sinc}(\pi(x-z))\right)\tag{1}.$$
...
3
votes
Accepted
Function with non Riemann-integrable Fourier transform
Yes. We can adapt the functional analytic argument for the existence of continuous functions with divergent Fourier series.
Fix a $\varphi\in C[-2,2]$ with $0\le\varphi\le 1$, $\varphi(\pm 2)=0$, and $...
3
votes
Theoretical/Practical Implications of DFT Eigenvectors
A method to construct a real and orthogonal eigenbasis of the DFT matrix has been developed by Dickinson and Steiglitz. There is no explicit closed-form expression, the problem is reduced to the ...
3
votes
Accepted
Is this integral solvable analytically?
there is a closed form solution for
$$I_n = \int_{0}^{\infty} e^{-\Lambda x} x^n e^{i \tau ( c_1 - c_2 e^{-c_3 x} ) } dx$$
for integer $n$, for example, for $n=0$:
$$I_0=\frac{1}{\lambda (c_3+\lambda)}...
2
votes
Accepted
Fourier transforms of homogeneous functions
Your function is, with $P_d$ homogeneous harmonic polynomial of degree $d$ in $n$ variables,
$$
u(x)=\frac{P_d(x)}{\vert x\vert^{n+d}}.
\tag{1}
$$
This is an homogeneous distribution of degree $-n$ ...
2
votes
Intersection of Fourier analysis (especially on the transform) and group theory, number theory, dynamical systems, etc
Firstly, please, do not regard me as any kind of authority in the topics of fourier analysis or number theory (I'm 17 y.o. without any formal degree in mathematics), but let me show you an example of ...
Community wiki
2
votes
Intersection of Fourier analysis (especially on the transform) and group theory, number theory, dynamical systems, etc
Fourier analysis indeed has deep connections with all listed topics. I recommend the classical article on this
Mackey, George W. Harmonic analysis as the exploitation of symmetry—a historical survey. ...
Community wiki
2
votes
Singular Integrals and $L^1$
We do have $v=|D|u\in L^1$ always, basically because $\widehat{|\xi|}$ decays like $1/x^2$, so the lack of smoothness of $\widehat{v}(\xi)$ near $\xi=0$ is not so serious after all.
More precisely, ...
2
votes
Accepted
Prove if the fractional Laplacian of a function is bounded
Yes, it is bounded.
This is perhaps easiest to see via the Fourier transform as you seem to have suspected.
The Fourier transform of your $f=\log(1+x^2)$, a tempered distribution, seems to be (...
2
votes
Theoretical/Practical Implications of DFT Eigenvectors
If you will forgive the self-serving pointer, I wrote a little paper examining this question from the perspective of "what's the sparsest basis of eigenvectors inside the DFT?". The paper's ...
2
votes
An integral similar to the Delta function
$$\int_{-\infty}^{\infty} e^{-k \omega' |\tau|} e^{i \tau(\omega'-\omega)} d\tau=\frac{2 k \omega'}{\left(k^2+1\right) \omega'^2+\omega^2-2 \omega\omega'},$$
for $k\omega'>0$.
1
vote
Examining the Hilbert transform of functions over the positive real line
The answer to the second question is negative as well. Take for example $g$ supported in $(-\infty,-1)$ and discontinuous in some point. If $f$ is supported in $\mathbb{R}_+$ and $y,z<-1$ it holds ...
1
vote
Motivation and physical interpretation of the Laplace transform
The following simple motivation for the formula was given in my undergrad class, which I hope I'm not misremembering:
Typically the way the Laplace transform arises in applications is when solving ...
1
vote
What is the asymptotic behavior of the Levy distribution $P (x)$ when the independent variable $x$ approaches $0$
$$P_{\alpha,\gamma}(x)=\frac{1}{\pi}\int_{0}^\infty \exp(-\gamma q^\alpha)\cos qx\,dq$$
$$=\frac{1}{\pi}\gamma^{-1/\alpha} \Gamma \left(1+\frac{1}{\alpha}\right)-\frac{1}{2\pi\alpha}x^2 \gamma^{-3/\...
1
vote
Intersection of Fourier analysis (especially on the transform) and group theory, number theory, dynamical systems, etc
You should seriously explore the literature for connections among the integral transforms, the Fourier, Mellin, and Laplace transforms, and the variety of fields you mention. Here are just a few of ...
Community wiki
1
vote
A question regarding Hermite polynomials and exponential operators $\exp[e^{x^2/2}p(\frac{d}{dx})e^{-x^2/2}]f(x)$
You can of course write
$$
k(x,t) = \exp \left[ e^{-t^2 /2} p\left( -\frac{d}{dt} \right) e^{t^2 /2} \right] \delta (x-t)
$$
Not clear whether that affords you any sort of simplification you may be ...
1
vote
Accepted
A probability distribution, with Fourier transform smaller than $C \exp(-ct^2)$
Yes. E.g., if $f(x)=\dfrac{1-\cos x}{\pi x^2}$ for real $x\ne0$, then $\psi_\mu(t)=\max(0,1-|t|)$ for real $t$ (which latter $=0$ if $|t|\ge1$).
1
vote
Closed form of a Fourier transform
For $d<4$ the Fourier transform with respect to the radial coordinate $y$ has a closed form expression in terms of a Bessel function:
$$\hat{G}(\xi,t)=(2\pi)^{d/2}\xi^{1-d/2}\int_0^\infty H(y,t)J_{...
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