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Parseval's theorem

In operational calculus there is Parseval's theorem, which states that if $ f(t) \doteqdot F(p), \varphi(t) \doteqdot \Phi(p) $ and both $ F(p) $ and $ \Phi(p) $ are analytical in $ Re p \geq 0 $, ...
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2answers
395 views

Sum of two independent random variables

Let $\xi, \eta, \eta'$ be non-negative random variables such that: $\eta \stackrel{\mathcal{L}}{=} \eta'$, $\xi + \eta \stackrel{\mathcal{L}}{=} \xi + \eta'$, $\xi$ and $\eta'$ are independent. ...
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2answers
70 views

a limit of the laplace transform and its derivative

If $\phi(s)$ is the Laplace tranfrom of $f(t)$, then $\lim_{s\rightarrow \infty} s\phi(s) = f(0^+)$. and also $\lim_{\rightarrow \infty} s\phi'(s) = \lim_{t\rightarrow 0^+}tf(t)$ since $\phi'(s)$ is ...
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1answer
58 views

Integral representation of the resolvent of a semigroup

Let $T(t)$ be a $C_{0}$-semigroup with the generator $A$. Now, does the so called integral representation of the resolvent $$ (\lambda - A)^{-1} = \int_{0}^{\infty} e^{-t\lambda}T(t) dt $$ hold for ...
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158 views

Laplace Transform in the context of Gelfand/Pontryagin

Question: Do quasi-characters properly generalize the Laplace transform or decompose functions in some setting in a way similar to how characters generalize the Fourier transform and decompose $L^1$ ...
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1answer
100 views

Extracting moments from a special Z-transform

Suppose I have a sequence of positive continuous random variables $\{X_k\}_{k=1}^\infty$ with (unknown) MGF's $M_{X_k}(s)$. Furthermore, it is known that ...
0
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1answer
79 views

Laplace transform - frequency differentiation property (generalization)

Let $\mathcal{L(f(t);s)}$ be the Laplace transform of a function $f$. It is known that the Laplace transform of $\mathcal{L}{(t^nf(t);s)}$ is given as (frequency differentiation property) ...
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0answers
76 views

Having problems with solving Lindley's equation in G/G/1 queuing?

Lindley's integral equation is as follows $$W(y)=\int_{u=-\infty}^{y}W(y-u)dC(u),$$for $y\ge0$; and $$W^{-}(y)=\int_{u=-\infty}^{y}W(y-u)dC(u)$$,for $y<0$. So we have ...
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0answers
44 views

Approximate Laplace (forward) transforms

Given f(t) let's write Lf(s) = INT_0^inf f(t) exp(-ts)dt for its Laplace transform. I lately find myself looking at expansions of the form SUM_i a_i exp(-t_i s) that converge to Lf of some given ...
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2answers
111 views

How do I estimate/bound the error in an inverse Laplace transform?

Suppose I have a Laplace transform, $$ F(s) = \int_0^{\infty}dx\ f(x)e^{-s x} \ . $$ I know that $$ F(s) \approx e^{A/(4s)} $$ (for $s$ real) where $A$ is very large, and I want to estimate ...
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1answer
138 views

An Integral Functional Equation

Let $f$ be a non-negative function supported and integrable on the positive real axis, such that $$\int_0^\infty f(x+y)p(y) dy = c[p] f(x), $$ where $c[p]$ a number (functional) dependent on function ...
4
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1answer
173 views

Characterization of the Laplace Transform

One of the main properties of the Laplace transform is given by the convolution theorem. $$\mathcal{L}(f*g)=\mathcal{L}(f)\cdot\mathcal{L}(g)$$ Question: Is there a full characterization of the ...
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1answer
417 views

Uncertainty principle for Mellin transform

Let $f:\mathbb{R}^+\to \mathbb{C}$. Let $Mf$ be its Mellin transform: $Mf(s) = \int_0^\infty f(x) x^{s-1} dx$. (a) Some time ago, I convinced myself that $f(t)$, $Mf(\sigma+it)$ and $Mf(\sigma-it)$ ...
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3answers
1k views

Does the inverse Laplace transform of the square root exist?

Does the inverse Laplace transform, defined by the integral, \begin{equation} F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds \end{equation} ...
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0answers
153 views

Inverse Laplace transform of the function: $F(s)=e^{-a\sqrt{s(s+b)}}$

I would like to find inverse Laplace transform of the function: $$F(s)=e^{-a\sqrt{s(s+b)}}$$ which $a$ and $b$ are positive real numbers and $s$ is a complex variable. It would be appreciated if ...
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1answer
140 views

Relations involving Stirling numbers of second kind

While inverting a Laplace transform using Post's inversion formula I found the following expression: $$ \sum_{k=1}^n S^n_k \ x^k(\alpha)_k $$ where $S^n_k$ is a Stirling number of second kind and ...
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0answers
147 views

Inferring asymptotic behaviour from the dominant pole of the Laplace transform

Hi, I am reposting the following question with the hope that a more detailed description will lead to a more descriptive response: dominant pole in the laplace transform I have a vector function ...
3
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1answer
277 views

Using a quadratic kernel instead of a linear kernel in the Laplace transform

Suppose $f$ is a bounded continuous function on $[0,\infty)$ such that $\int_0^\infty f(t) \exp(-xt) \: dt \rightarrow 0$ as $x \rightarrow 0^+$. Does it follow that $\int_0^\infty f(t) \exp(-xt^2) \: ...
3
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0answers
95 views

Lower bounds of laplace transform of characteristic functions

Cross-posted on maths.stackexchange I have the following integral: \begin{equation} f(\mu) = \int_0^\infty e^{-\mu t}\varphi_X(t)dt \end{equation} where $\varphi_X(t)$ is the characteristic function ...
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1answer
215 views

dominant pole in the laplace transform

hi, I have a function $X(t)$ whose Laplace transform $\hat{X}(s)$ has a unique pole of largest real part $x_0$, which is a real number. I am able to show that for each $t$, $X(t)$ is a convergent ...
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0answers
171 views

Wiener-Hopf Integral/Lindley's Equation

Lindley's equation is well known within queueing theory and is as follows $F(y) = - \int_0^\infty F(x)dH(y-x)$ However, many textbooks only consider the case where 0 $\le$ y $\le \infty$ (which ...
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2answers
189 views

On a certain generalization of the Laplace transform

Let $\alpha$ be a positive constant, $\mu$ be a Borel nonnegative measure in $\mathbb{R}^n_+$. We can define a transform $$ \tilde{L}\[\mu\](p) = \int\limits_{\mathbb{R}^n_+} e^{-(p_1 x_1 + \ldots ...
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0answers
696 views

An inverse Laplace transform involving Error function

Dear MOs, I need to calculate the inverse Laplace transform of the following function $$ g_a(z) = \frac{e^{a z}\: \text{erfc}(\sqrt{a z})}{\sqrt{z}-2},\quad a>0. $$ I have checked, among many ...
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2answers
517 views

Understanding the inverse Laplace transform of a function with essential singularities

I need to do an inverse Laplace transform of a function with essential singularities for a specific problem. I find it is very similar to an equation J. Noolandi worked out in one of his papers in ...
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2answers
2k views

How does the Laplace Transform work for circuit analysis? [closed]

I would like to understand how signals transformed from the time domain to the frequency domain for algebraic manipulation, can be transformed back to give solutions in the time domain. Knowing how to ...
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0answers
211 views

On the generalisation of the Laplace transform

I consider a measure transform $A$ given by $$ A\mu(x) = \int\limits_{\mathbb{R}^n_{+}} e^{-g(x,y)} \mu(dy) $$ where $g(x,y)$ is some positive smooth function, $\mu$ is a Borel measure. Is it a ...
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6answers
4k views

What does Mellin inversion “really mean”?

Given a function $f: \mathbb{R}^+ \rightarrow \mathbb{C}$ satisfying suitable conditions (exponential decay at infinity, continuous, and bounded variation) is good enough, its Mellin transform is ...
2
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1answer
150 views

Stieltjes Transform of $F^{*}PF$ as a function of the Stieltjes Transform of $P$ where $F$ is drawn from an $n \times n$ Gaussian-like random matrix distribution

I am trying to calculate the Stieltjes Transform of $F^{*}PF$ as a function of the Stieltjes Transform of $P$ where $F$ is drawn from an $n \times n$ Gaussian-like random matrix distribution. I am ...
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299 views

Laplace transform of a stopping time for stochastic volatility models

Let $V_t$ be a solution of the SDE $$dV_t=V_t(rdt+\sigma_t dW_t) $$ where $\sigma_t$ satisfies some other SDE $$d\sigma_t=\alpha(t,\sigma_t)dt+\beta(t,\sigma_t)dW^{\\ \prime}_t $$ and $W_t$ and ...
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3answers
989 views

When I can safely assume that a function is a Laplace transform of other function?

If I have a function and I want to represent it as being the Laplace transform of another, that is, I want to be sure that there is $\hat{f}(s)$ such that my function $f(x)$ can be written as: $f(x) ...
2
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0answers
270 views

inverse Laplace transform of $\delta_1(\cdot)$

Let's try to find a function $\psi(x)$ such that for Laplace transform $\tilde{f}(p)=\int_0^{\infty} f(y) e^{-py} dy$ one has $f(x)=\int_0^{\infty} \tilde{f}(p)\psi(px)dp$ (here we do not specify ...
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2answers
918 views

Getting a differential equation for a function from a functional equation of its Mellin transform

If $f$ is a locally integrable function then its Mellin transform $\mathcal{M}[f]$ is defined by $$ \mathcal{M}[f] (s) = \int_0^{\infty} x^{s - 1} f (x) dx . $$ This integral usually converges in a ...
2
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2answers
2k views

Laplacian operator and relation to the Laplace Transform

I'm trying to understand why the Laplacian operator is used in blob detection in image analysis. I must admit that in trying to figure out why the Laplacian is useful in this application, I've really ...
2
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1answer
498 views

method of moments and Laplace transform from Shepp and Lloyd

Again from the Shepp and Lloyd paper "ordered cycle lengths in a random permutation", I found this puzzling equality. This one might require access to the paper itself since it's quite a mouthful: In ...
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6answers
31k 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 ...
2
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2answers
542 views

Can we extract information about how fast a function decay from its Laplace transform?

My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform. More concrete case, let $f:\mathbb{R} ...
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5answers
4k views

Applied mathematics Books (graduate level)

What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems? There is a lot of books on ...
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2answers
503 views

Ansätze for solving PDEs with wavelets

It is common to solve PDEs with e.g. Fourier and Laplace Transforms. It is often said that Wavelets are a progression compared to them with many nice features. My question: Which Ans├Ątze do you know ...
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6answers
4k views

Intuition for Integral Transforms

It is well known that the operations of differentiation and integration are reduced to multiplication and division after being transformed by an integral transform (like e.g. Fourier or Laplace ...
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9answers
6k views

Motivating the Laplace transform definition

In undergraduate differential equations it's usual to deal with the Laplace transform to reduce the differential equation problem to an algebraic problem. The Laplace transform of a function $f(t)$, ...