All Questions
7 questions
3
votes
0
answers
144
views
Minimizing vertical integral of a Mellin transform
Let $\eta:[0,\infty)\to [0,\infty)$ satisfy $\eta(0)=1$ and $\int_0^\infty \eta(x) dx = 1$ (say).
Write $M\eta$ for the Mellin transform of $\eta$. Let $\epsilon>0$ be small.
What is the choice of $...
- 20.2k
6
votes
2
answers
336
views
On frequency decay of an integral transform of a function
Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that
$$
\bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$
for all $\tau \...
- 4,143
7
votes
1
answer
1k
views
Where does the Laplace transform come from?
The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform.
Q. What can we say concerning the Laplace transform?
- 4,058
1
vote
0
answers
100
views
Expressing 1-e^{-z} as a Fourier integral
According to the theory of screw functions and screw lines by John Von Neumann and Issai Schoenberg (see here), any function $F:\mathbb{R} \rightarrow \mathbb{R}$ such that $F(|x_i - x_j|) = \|f(x_i)-...
- 93
3
votes
0
answers
140
views
Decay of Laplace (or Mellin) transform beyond region of convergence?
Let $f:[0,\infty)\to \mathbb{R}$ be a piecewise differentiable function with $f(0)=0$ and $f'(t)$ of bounded variation. Its Laplace transform $\mathcal{L}f$ converges for $\Re s > 0$. Assume it can ...
- 20.2k
23
votes
0
answers
1k
views
Laplace Transform in the context of Gelfand/Pontryagin
Questions:
Is there a class of objects (presumably related to locally compact abelian groups) for which the quasi-characters canonically generalize the Laplace transform?
If not, is there a ...
- 1,124
41
votes
6
answers
87k
views
Fourier vs Laplace transforms
In solving a linear system, when would I use a Fourier transform versus a Laplace transform? I am not a mathematician, so the little intuition I have tells me that it could be related to the boundary ...
- 411