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 ...
Carlo Beenakker's user avatar
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 $...
Alexandre Eremenko's user avatar
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)...
Michael Hardy's user avatar
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 ...
Michael Engelhardt's user avatar
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....
an_ordinary_mathematician's user avatar
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) ...
Liviu Nicolaescu's user avatar
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 ...
user479223's user avatar
  • 1,250
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\}...
Liding Yao's user avatar
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 ...
Tom Copeland's user avatar
  • 9,937
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 ...
Giuseppe Negro's user avatar
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\...
Christian Remling's user avatar
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 ...
Carlo Beenakker's user avatar
4 votes

The Fourier transform of the Liouville function?

Your first formula for $\lambda(x)$ is equivalent to $$\lambda(x)=\underset{N\to \infty}{\text{lim}}\left(\sum\limits_{z=-N}^N \cos(\pi(\Omega(z)+x-z))\, \text{sinc}(\pi(x-z))\right)\tag{1}.$$ ...
Steven Clark's user avatar
  • 1,061
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 $...
Christian Remling's user avatar
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 ...
Carlo Beenakker's user avatar
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)}...
Carlo Beenakker's user avatar
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 $-n$ ...
Bazin's user avatar
  • 15.1k
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 ...
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. ...
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, ...
Christian Remling's user avatar
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 (...
Joonas Ilmavirta's user avatar
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 ...
Bill Bradley's user avatar
  • 3,809
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 $k\omega'>0$.
Carlo Beenakker's user avatar
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 ...
an_ordinary_mathematician's user avatar
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 ...
Wahome's user avatar
  • 737
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/\...
Carlo Beenakker's user avatar
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 ...
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 ...
Michael Engelhardt's user avatar
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 $|t|\ge1$).
Iosif Pinelis's user avatar
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_{...
Carlo Beenakker's user avatar

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