Questions tagged [laplace-transform]
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159
questions
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Examining the Hilbert transform of functions over the positive real line
$\DeclareMathOperator\supp{supp}$Let $H:L^{2}(\mathbb{R})\to L^{2}(\mathbb{R})$ be the Hilbert transform. Let suppose we have a compaclty supported function $f \in L^{2}(\mathbb{R})$ such that $\supp(...
1
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0
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142
views
Laplace transform
\begin{equation}
\begin{cases}\mathbb{D}_t^\beta u(x, y, t)=-a(x)\left(u_x(x, y, t)+u_y(x, y, t)\right)+\ell(x, y, t, u(x, y, t)), & x>0, y>0, t>0 \\ u(x, y, 0)=0, & x>0, y>0 \\ ...
0
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0
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47
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Inverse Laplace transform of Dirac delta function [migrated]
I am trying to understand how to identify, or at least derive some properties of the inverse Laplace transform of the Dirac delta function, i.e. a function $\eta$ s.t.
$$
\int_0^\infty dt\, \eta(\...
33
votes
8
answers
3k
views
Motivation and physical interpretation of the Laplace transform
Concerning the one-sided Laplace transform,
$$\mathcal{L}\{f\}(s) = \int_0^\infty f(t)e^{-st} dt$$
what is a motivation to come up with that formula? I am particularly interested in "physical&...
2
votes
0
answers
134
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Beyond Watson's lemma
Suppose $f:[0,1]\rightarrow \mathbb{C}$ is a smooth function, which I wish to approximate near $0$. Watson's Lemma implies that I can find a smooth function $a:[0,1]\rightarrow \mathbb{C}$ such that:
$...
0
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0
answers
45
views
Computing the Laplace transform of an expression
I would like to find the Laplace transform of the following expression with respect to the Laplace parameter s
$ \int_{z=u}^{\infty} e^{-az/c} g^{'}(\dfrac{z-u}{c}) \int_{x=0}^{\infty} \varphi(z-x)dF(...
0
votes
1
answer
300
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Necessary conditions for convergence of convolution
In math.SE, I've asked a question about the convergence of convolution of two functions which have bilateral Laplace transform and also have disjoint Region Of Convergence (ROC) but the question didn'...
5
votes
1
answer
235
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How to evaluate inverse Laplace transform of $e^{- \sqrt{s}} $ using series?
I tried to find an inverse Laplace transform by series as follows
$$ f(t)=L^{-1}_s\left(e^{-\sqrt{s}}\right)(t)=L^{-1}_s\left(\sum_{k=0}^{\infty}\frac{(-1)^k}{k!} s^{\frac{k}{2}}\right)(t)$$
and by ...
3
votes
1
answer
179
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Laplace transform of Brownian motion functional
Let $(B_r,r\geq 0)$ be a standard Brownian motion on $\mathbb{R}$ started at $0$. I am interested in the quantity
$$g(s,t) = \mathbb{E}_0\left[ \exp \left(- \beta \int_s^t \left\vert \frac{B_r}{r}\...
0
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1
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98
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Explanation for Tauberian theorems for Laplace transform
I am struggling with the following theorem in Feller's book "Probability Theory and its Applications". The tauberian theorem is written as follow :
Let $F : [0,\infty) \to \mathbb{R}$ of ...
0
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0
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72
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Inverse Laplace of the Complex conjugate of the Laplace transform
Let the Laplace transform of f(t) be F(s) and let the inverse Laplace transform of F(s*) be g(t). is there a theorem relating f(t) and g(t)? Basically, looking for a way to calculate g(t) from f(t) ...
2
votes
1
answer
103
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Thinning of (mixed) binomial point process
Let $N= \sum_{i=1}^M \delta_{X_i}$ be a mixed Binomial process over $(\mathbb X, \mathcal X)$. I.e., $M$ is a $\mathbb Z_+$ valued random variable with probability mass function $q_M(m)$, $m=0, 1, \...
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55
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Infinite dimensional version of the Laplace transform and Gaussian integrals
This question is somehow related to my previous one Convergence of the Gaussian integral on $\mathcal{E}'$ for a mapping supported on $L^2$
Let $F : L^2(S^1) \to L^2(S^1)$ be a (nonlinear) Borel-...
1
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1
answer
50
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Possibility of bounding one functional by another functional (under certain constraints)
Suppose that we consider a class of $L^2(\mathbb{R}_+)$ functions $h$ such that $h$ can be expressed as a difference of two cumulative distribution functions $F$ and $G$ (whose corresponding densities ...
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1
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198
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Equivalence between the $L^2$ norm and the $L^2$ norm of Laplace transform
It is well-known that the Laplace transform, defined by $$\mathcal{L} \colon f(x) \in L^2(\mathbb{R}_+) \to \hat{f}(\xi) \in L^2(\mathbb{R}_+)$$ via $$\hat{f}(\xi) = \int_{\mathbb{R}_+} f(x)\,\mathrm{...
0
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28
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A condition for complete monotonicity
In D. V. Widder, The Laplace Transform, Chapter III, The Moment Problem, given a sequence $(\mu_n)_{n=0}^\infty$, it is defined that
$$\lambda_{k,m}:= {k\choose m}(-1)^{k-m}\Delta^{k-m}\mu_m, \, \...
0
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0
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101
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Laplace transform of a stochastic process
Let $R := (R_1, R_2)$ be a two-dimensional diffusion process defined by the following SDE:
$$\mathrm{d}R_{1,t} = -\lambda_1 R_{1,t} \, \mathrm{d}t + \lambda_1 \sigma(R_{1,t}, R_{2,t}) \, \mathrm{d}W_t$...
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Inverse Laplace transform of the hypergeometric function 2F1
In the book Integrals and Series: Inverse Laplace Transforms by A.P. Prudnikov, the inverse Laplace transform of the hypergeometric function 2F1 defined as
$$
_{2}F_{1}(p+a,b;c;-\omega), \qquad [b,p+a\...
1
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0
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106
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Relating $f(x)$ to its Laplace Transform for values other than $x=0$?
Suppose $X\in (0,1]$ is a random variable where $f(x)$ is its CDF and $g(t)$ is the Laplace Transform of $f(x)$. Tauberian theorems (Theorem 2.3 in Coqueret's "Approximation of probabilistic ...
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Contour integral with two essential singularity
I'm solving problems on the Gamma random variables and there is this question where it wants me to calculate the Mellin transform of sum of two independent Gamma variables from their moment generating ...
5
votes
1
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307
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Long tail property of Laplace transforms
A function $F: \mathbb R_+ \rightarrow \mathbb R_+$ is said to be long tailed if $F(\infty)=0$ and for all $y \geq 0$ $$\frac{F(x+y)}{F(x)} \rightarrow 1, \quad x\rightarrow \infty.$$
Let $\mu$ be a ...
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46
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Bounding the ratio of two functions given their Laplace Transforms
Suppose $h(i)$ is a probability density that's "nice" in some sense, and $g(i)=E[f(i,x)]=\int \mathrm{d}i\ h(i)f(i,x)$
How could I bound the following ratio $r(t)$ from above?
$$
r(t)=\frac{...
5
votes
1
answer
197
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Upper bound for an inverse Laplace transform
Can anyone see how to get a tight upper bound for the function defined in terms of the inverse Laplace transform below?
$$g_\text{sgd}(t)=\mathcal{L}^{-1}\left[\left(\frac{s}{2}+\frac{5 \sqrt{\frac{2}{...
2
votes
1
answer
170
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2D lattice sum with numerator
I've been struggling a bit with a double sum that arose as the trace of an operator:
$$\sum_{(j,k)\in Z^2 \setminus (0,0)} \frac{(j+k)^n}{(j^2+k^2)^n},$$
where $n$ is an even natural number. Is there ...
1
vote
1
answer
112
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Which kind of convergence can we get from Laplace transform convergence?
This question is a related question see this post Vague convergence VS Laplace transform convergence. But now we assume that
\begin{equation}
\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}...
1
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1
answer
215
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Vague convergence VS Laplace transform convergence?
If we assume that $\int_0^\infty e^{-sx}\mu_n(dx)\to \int_0^\infty e^{-sx}\mu(dx), \forall s\geq0$, it is possible to show that $\mu_n\to\mu$ vaguely. Where $\mu_n$ is a measure. Please check here for ...
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30
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Range of a Laplace-type transform
I'm interested in germs of functions $f(h)$ for $h\geq 0$ small. My question is, for what functions $f(h)$ can I write, for some $\delta>0$:
$$f(h) = \frac{1}{h} \int_0^{\delta} e^{-s/h} a(s)\: ds +...
0
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0
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90
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Modeling decay of a linear system with a mixing term
I'm trying to analyze convergence of $x\in \mathbb{R}^{+d}$ which follows the following recurrence
$$\mathbf{x}\leftarrow (\mathbf{1}-\mathbf{h})^2 \mathbf{x} + \mathbf{h}\langle \mathbf{x}, \mathbf{h}...
1
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0
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53
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Upper bounds on Inverse Laplace transform of a rational function
I need the inverse Laplace transform $\mathscr{L}^{-1}$ or a nice upper bound on $\mathscr{L}^{-1}$ for the following function:
$$f(y)=\frac{\left(\sum_i \frac{u_i}{y-a_i}\right)^2}{1-\sum_i u_i \frac{...
2
votes
1
answer
101
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Injectivity of two sided Laplace transform
Let $\mu,\nu$ be finite Borel measures on $\mathbb R$.
Assume that there is an open interval $(a,b)$ on which the Laplace transforms exist and coincide:
$$
\int_{-\infty}^\infty e^{-tx}\,d\mu(x) = \...
4
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1
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466
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$\frac {f (0)}{2}+ \sum_{k=1}^{\infty}f (k)=\sum_{n=-\infty}^{\infty} \mathcal{L} \{ f \} (2 \pi i n)$
I obtained the very strange formula above and at begining I was just wanted know how to interpretate it. But now when I know what is this with help of @Carlo Beenakker, I am leaving it as a proof. BTW ...
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1
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114
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What kind of functions can be represented as infinite linear combinations of exponential functions?
Let $f(x)$ be a real-valued function defined in $(0, \infty)$. I am curious what kind of $f(x)$ has the following representations:
$$
f(x) = \sum_{j=0}^\infty a_j e^{-jx}, \quad \forall x \in (0, \...
1
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1
answer
72
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Discrete uniqueness sets for the two-sided Laplace transform?
Let $f : \mathbb R_+ \to \mathbb C$ be a measurable and integrable function where $\mathbb R_+ = [0,\infty)$. The Laplace transform of $f$ is given by
$$
Lf(s) = \int_0^\infty f(x)e^{-sx} \, dx.
$$
A ...
3
votes
0
answers
140
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Monotonicity of a function defined by an integral
The question below is motivated by the related question Integral of a function changing sign and the associated answer:
Can we study the monotonicity of the following function on $(0,1)$?
$$\small f(x)...
0
votes
1
answer
116
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Approximate inverse Laplace transform in terms of the moments of a function
If $F(s)$ is the Laplace transform of $f(t)$ and
\begin{equation}
F(s)=\frac{1}{1-aG(s)}
\end{equation}
where $G(s)$ is the Laplace transform of a known probability density distribution $g(t)$ whose ...
3
votes
1
answer
311
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Integral of a function changing sign
By some numerical tests, we can see that the following function is negative on $(0,1)$:
$$\small f(x)=\int_0^\infty\frac{s^{x-1} e^{-2 s} (\pi \cos(\pi x) (s^{2 x}+(0.1)^2)-\sin(\pi x) \ln(s) (s^{2 x}-...
2
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102
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Evaluate action of $f(\frac{d}{dx})$ using the Fourier/Laplace transform
Consider a function $f(x)$ that is numerically defined in $-1 \leq x \leq 1$ interval (assume $N$ samples). I am trying to compute the action of $f(d/dx)$ on a function $g(x)$ using the Fourier ...
5
votes
1
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430
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Hardy–Littlewood Tauberian theorem for Laplace transform
The Hardy–Littlewood Tauberian theorem for Laplace transform in Chapter XIII in "An Introduction to Probability Theory and Its Applications" by Feller reads as follows
Let $F : [0,\infty) \...
6
votes
2
answers
324
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On frequency decay of an integral transform of a function
Suppose $f \in C^{\infty}_c((-1,1))$ and assume that there exists constants $a,b>0$ such that
$$
\bigg|\int_{\mathbb R} f(t) \,e^{\tau t^2+i\tau t}\,dt\bigg| \leq a\,e^{-b|\tau|},$$
for all $\tau \...
1
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0
answers
159
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Discretizing a differential operator which is a function of the derivative operator
Assume that $p(x)$ and $f(x)$ are sufficiently smooth functions and $D\equiv \frac{d}{dx}$. My question is concerned with the discretization of $p(x+D)f(x)$.
As an example, let $p(x)=x^{2}+2x$. Then ...
5
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2
answers
179
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Limit of the extremal process of i.i.d. Gaussians see from the tip
I'm trying to calculate the weak limit of $\mathcal{E}_N(x)=\sum_{k=1}^{2^N}\delta_{x-Z_k}$ , with $Z_k=X_k-\max_{k\leq 2^N}X_k$, $\{X_k\}$ being $2^N$ copies of i.i.d. Gaussians with mean zero and ...
1
vote
1
answer
75
views
Initial and final Theorem for upper and lower limits?
Let define $F(s)=\int_0^\infty f(u)e^{-su}du$. If $f$ is bounded and $\lim_{t\to 0}f(t)$ exists. Then we can get $\lim_{t\to 0}f(t)=\lim_{s\to\infty}sF(s)$.
Can we use upper limits or lower limits to ...
6
votes
1
answer
221
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Convergence speed of the tail of distribution using Tauberian remainder theorem
This question may be related to this one.
Now I try to make some statistical estimator using Laplace transform, but I face the following serious problem.
Let $f$ be some one-sided probability ...
-1
votes
1
answer
60
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Finding the nth value of a dual series [closed]
I have a problem which i remember solving using Z transform in my uni. time, but i don't recall the EXACT way.
I have 2 series of numbers:
X[n] and Y[n]
X[0] = Y[0] = 1
X[n+1] = aX[n] + bY[n]
Y[n+1] = ...
0
votes
1
answer
110
views
Unique zero solution to a difference equation via Laplace transform
We want to prove that
the unique solution to the following difference equation is the null one:
$$
au(x)+b\mathbf{1}_{(0,\frac{1}{2})}(x)u(x+\frac{1}{2})+c\mathbf{1}_{(\frac{1%
}{2},1)}(x)u(x-\frac{1}{...
0
votes
1
answer
200
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Laplace transform injectivity for different values of $p$
Let $y\in L^{2}(0,1)$ and let $\widetilde{y}$ be its extension on $(0,\infty
).$ Assume that there exist $p_{0},p_{1}\in
%TCIMACRO{\U{2102} }%
%BeginExpansion
\mathbb{C}
%EndExpansion
,$ $p_{0}\neq ...
1
vote
0
answers
36
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Solving an equation containing Laplace transform
Consider the equation
\begin{equation}
\frac{f(p)}{f(s_{1})}\mathcal{L}(y)(s_{1})+\frac{g(p)}{g(s_{2})}\mathcal{L}%
(y)(s_{2})=\mathcal{L(}y)\mathbf{(}p),
\end{equation}
where $\mathcal{L}$ is the ...
2
votes
1
answer
500
views
What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?
A function $q(x)$ is said to be completely monotonic on an interval $I$ if $q(x)$ has derivatives of all orders on $I$ and $(-1)^{n}q^{(n)}(x)\ge0$ for $x\in I$ and $n\ge0$. See Chapter 1 in the ...
2
votes
1
answer
180
views
Inverse Laplace transform of $\frac{1}{s^a + 1}$ with $0 < a \leq 1$
Problem
I am looking for the following inverse Laplace transform,
$$
f(t) = \mathcal{L}^{-1}\left[\frac{1}{s^a + 1}\right]
\;\;\;\;\;
\text{with}
\;\;\;\;\;
0 < a \leq 1.
$$
What I understand
...
4
votes
2
answers
486
views
Is it possible to use the Laplace Transform to calculate eigenvalues?
The relationship of Eigenvalues with Gradient Descent
Let $A$ be a symmetric (and thus diagonalizable) matrix, with diagonalization
$$A=VDV^T.$$
Let us define the quadratic function
$$f(x) = x^T A x.$$...