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The Laplace–Stieltjes transform, named for
Pierre-Simon Laplace Pierre-Simon, marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French scholar and polymath whose work was important to the development of engineering, mathematics, statistics, physics, astronomy, and philosophy. He summarized ...
and Thomas Joannes Stieltjes, is an
integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than i ...
similar to the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...
. For
real-valued function In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real ...
s, it is the Laplace transform of a Stieltjes measure, however it is often defined for functions with values in a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
. It is useful in a number of areas of mathematics, including
functional analysis Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined ...
, and certain areas of
theoretical A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be ...
and
applied probability Applied probability is the application of probability theory to statistical problems and other scientific and engineering domains. Scope Much research involving probability is done under the auspices of applied probability. However, while such res ...
.


Real-valued functions

The Laplace–Stieltjes transform of a real-valued function ''g'' is given by a Lebesgue–Stieltjes integral of the form :\int e^\,dg(x) for ''s'' a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the for ...
. As with the usual Laplace transform, one gets a slightly different transform depending on the domain of integration, and for the integral to be defined, one also needs to require that ''g'' be of
bounded variation In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
on the region of integration. The most common are: * The bilateral (or two-sided) Laplace–Stieltjes transform is given by \(s) = \int_^ e^\,dg(x). * The unilateral (one-sided) Laplace–Stieltjes transform is given by \(s) = \lim_ \int_^\infty e^\,dg(x). The limit is necessary to ensure the transform captures a possible jump in at , as is needed to make sense of the Laplace transform of the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
. * More general transforms can be considered by integrating over a contour in the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...
; see . The Laplace–Stieltjes transform in the case of a scalar-valued function is thus seen to be a special case of the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...
of a Stieltjes measure. To wit, :\mathcal^*g = \mathcal(dg). In particular, it shares many properties with the usual Laplace transform. For instance, the
convolution theorem In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e. ...
holds: :\(s) = \(s)\(s). Often only real values of the variable ''s'' are considered, although if the integral exists as a proper
Lebesgue integral In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Le ...
for a given real value , then it also exists for all complex with . The Laplace–Stieltjes transform appears naturally in the following context. If ''X'' is a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
with
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
''F'', then the Laplace–Stieltjes transform is given by the
expectation Expectation or Expectations may refer to: Science * Expectation (epistemic) * Expected value, in mathematical probability theory * Expectation value (quantum mechanics) * Expectation–maximization algorithm, in statistics Music * ''Expectation' ...
: :\(s) = \mathrm\left ^\right The Laplace-Stieltjes transform of a real random variable's cumulative distribution function is therefore equal to the random variable's
moment-generating function In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution. Thus, it provides the basis of an alternative route to analytical results compar ...
, but with the sign of the argument reversed.


Vector measures

Whereas the Laplace–Stieltjes transform of a real-valued function is a special case of the Laplace transform of a measure applied to the associated Stieltjes measure, the conventional Laplace transform cannot handle
vector measure In mathematics, a vector measure is a function defined on a family of sets and taking vector values satisfying certain properties. It is a generalization of the concept of finite measure, which takes nonnegative real values only. Definitions and ...
s: measures with values in a
Banach space In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between ve ...
. These are, however, important in connection with the study of
semigroup In mathematics, a semigroup is an algebraic structure consisting of a Set (mathematics), set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplication, multiplicatively ...
s that arise in
partial differential equations In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to ...
,
harmonic analysis Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an e ...
, and
probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
. The most important semigroups are, respectively, the heat semigroup, Riemann-Liouville semigroup, and
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
and other infinitely divisible processes. Let ''g'' be a function from ,∞) to a Banach space ''X'' of strongly bounded variation over every finite interval. This means that, for every fixed subinterval g(t_i)-g(t_) \right \">_X < \infty where the infimum and supremum, supremum is taken over all partitions of [0,''T'' :0=t_0 < t_1<\cdots< t_n=T. The Stieltjes integral with respect to the vector measure ''dg'' :\int_0^T e^dg(t) is defined as a Riemann–Stieltjes integral. Indeed, if π is the tagged partition of the interval [0,''T''] with subdivision , distinguished points \tau_i \in [t_i, t_] and mesh size , \pi, = \max \left , t_i - t_ \right , , the Riemann–Stieltjes integral is defined as the value of the limit :\lim_ \sum_^e^ \left (t_)-g(t_i) \right /math> taken in the topology on ''X''. The hypothesis of strong bounded variation guarantees convergence. If in the topology of ''X'' the limit :\lim_ \int_0^T e^dg(t) exists, then the value of this limit is the Laplace–Stieltjes transform of ''g''.


Related transforms

The Laplace–Stieltjes transform is closely related to other
integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than i ...
s, including the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
and the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the ...
. In particular, note the following: * If ''g'' has derivative ''g' '' then the Laplace–Stieltjes transform of ''g'' is the Laplace transform of ''g′''. \(s) = \(s), * We can obtain the Fourier–Stieltjes transform of ''g'' (and, by the above note, the Fourier transform of ''g′'') by \(s) = \(is), \qquad s \in \R.


Probability distributions

If ''X'' is a continuous
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the p ...
with
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ev ...
''F''(''t'') then
moment Moment or Moments may refer to: * Present time Music * The Moments, American R&B vocal group Albums * ''Moment'' (Dark Tranquillity album), 2020 * ''Moment'' (Speed album), 1998 * ''Moments'' (Darude album) * ''Moments'' (Christine Guldbrand ...
s of ''X'' can be computed using :\operatorname ^n= (-1)^n \left.\frac \_.


Exponential distribution

For an exponentially distributed random variable ''Y'' with rate parameter ''λ'' the LST is, :\widetilde Y(s) = \(s) = \int_0^\infty e^ \lambda e^ dt = \frac from which the first three moments can be computed as 1/''λ'', 2/''λ''2 and 6/''λ''3.


Erlang distribution

For ''Z'' with Erlang distribution (which is the sum of ''n'' exponential distributions) we use the fact that the probability distribution of the sum of independent random variables is equal to the convolution of their probability distributions. So if :Z = Y_1 + \cdots + Y_n with the ''Yi'' independent then :\widetilde Z(s) = \widetilde Y_1(s) \cdots \widetilde Y_n(s) therefore in the case where ''Z'' has an Erlang distribution, :\widetilde Z(s) = \left( \frac \right)^n.


Uniform distribution

For ''U'' with
uniform distribution Uniform distribution may refer to: * Continuous uniform distribution * Discrete uniform distribution * Uniform distribution (ecology) * Equidistributed sequence In mathematics, a sequence (''s''1, ''s''2, ''s''3, ...) of real numbers is said to be ...
on the interval (''a'',''b''), the transform is given by :\widetilde U(s) = \int_a^b e^ \frac dt = \frac.


References

*; 2nd ed (1974) . *. *. *. *. {{DEFAULTSORT:Laplace-Stieltjes Transform Integral transforms Stieltjes