Search results
Results from the WOW.Com Content Network
The Laplace transform is used frequently in engineering and physics; the output of a linear time-invariant system can be calculated by convolving its unit impulse response with the input signal. Performing this calculation in Laplace space turns the convolution into a multiplication; the latter being easier to solve because of its algebraic form.
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain ) equals point-wise multiplication in the other domain (e.g., frequency domain ).
This theorem is proved by applying the inverse Laplace transform on the convolution theorem in form of the cross-correlation. Let f ( t ) {\displaystyle f(t)} be a function with bilateral Laplace transform F ( s ) {\displaystyle F(s)} in the strip of convergence α < ℜ s < β {\displaystyle \alpha <\Re s<\beta } .
The Laplace–Stieltjes transform of a real-valued function g is given by a Lebesgue–Stieltjes integral of the form ()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.
A similar result holds for compact groups (not necessarily abelian): the matrix coefficients of finite-dimensional unitary representations form an orthonormal basis in L 2 by the Peter–Weyl theorem, and an analog of the convolution theorem continues to hold, along with many other aspects of harmonic analysis that depend on the Fourier transform.
The starred transform is a convenient mathematical abstraction that represents the Laplace transform of an impulse sampled ... Then per the convolution theorem, ...
In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L 1 function vanishes at infinity. It is of importance in harmonic analysis and asymptotic analysis .
Using the Green's function for the three-variable Laplace operator, one can integrate the Poisson equation in order to determine the potential function. Green's functions can be expanded in terms of the basis elements (harmonic functions) which are determined using the separable coordinate systems for the linear partial differential equation .