the laplaceâstieltjes transform, named for pierre-simon laplace and thomas joannes stieltjes, is an integral transform similar to the laplace transform. for real-valued functions, it is the laplace transform of a stieltjes measure, however it is often defined for functions with values in a banach space. it is useful in a number of areas of mathematics, including functional analysis, and certain areas of theoretical and applied probability. ==real-valued functions== the laplaceâstieltjes transform of a real-valued function g is given by a lebesgueâstieltjes integral of the form :\int\mathrm{e}^{-sx}\,dg(x) for s a complex number. 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 on the region of integration. the most common are: * the bilateral (or two-sided) laplaceâstieltjes transform is given by ::\{\mathcal{l}^*g\}(s) = \int_{-\infty}^{\infty} \mathrm{e}^{-sx}\,dg(x). * the unilateral (one-sided) laplaceâstieltjes transform is given by ::\{\mathcal{l}^*g\}(s) = \int_{0^-}^{\infty} \mathrm{e}^{-sx}\,dg(x). :where the lower limit 0&minus; means ::\lim_{\varepsilon\to 0^+}\int_{-\varepsilon}^\infty. :this is necessary to ensure that the transform captures a possible jump in g(x) at x&nbsp;=&nbsp;0, as is needed to make sense of the laplace transform of the dirac delta function. * more general transforms can be considered by integrating over a contour in the complex plane; see . the laplaceâstieltjes transform in the case of a scalar-valued function is thus seen to be a special case of the laplace transform of a stieltjes measure. to wit, :\mathcal{l}^*g = \mathcal{l}(dg). in particular, it shares many properties with the usual laplace transform. for instance, the convolution theorem holds: :\{\mathcal{l}^*(g * h)\}(s) = \{\mathcal{l}^*g\}(s)\{\mathcal{l}^*h\}(s). often only real values of the variable s are considered, although if the integral exists as a proper lebesgue integral for a given real value s&nbsp;=&nbsp;Ï, then it also exists for all complex s with re(s)&nbsp;â¥&nbsp;Ï. the laplaceâstieltjes transform appears naturally in the following context. if x is a random variable with cumulative distribution function f, then the laplaceâstieltjes transform is given by the expectation: :\{\mathcal{l}^*f\}(s) = \mathrm{e}\left[\mathrm{e}^{-sx}\right]. ==vector measures== whereas the laplaceâstieltjes 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 measures: measures wi
Integral_transforms