#### Title

#### Identifier

etd-06092009-162436

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

In this dissertation we investigate, compute, and approximate convolution powers of functions (often probability densities) with compact support in the positive real numbers. Extending results of Ursula Westphal from 1974 concerning the characteristic function on the interval $[0,1]$, it is shown that positive, decreasing step functions with compact support can be embedded in a convolution semigroup in $L^1(0,infty)$ and that any decreasing, positive function $pin L^1(0,infty)$ can be embedded in a convolution semigroup of distributions. As an application to the study of evolution equations, we consider an evolutionary system that is described by a bounded, strongly continuous semigroup ${T(t)}_{tgeq0}$ in combination with a probability density function $pin L^1(0,infty)$ describing when an observation of the system is being made. Then the $n^{th}$ convolution power $p^{star n}$ of $p$ is the probability distribution describing when the $n^{th}$ observation of the system is being made and $E_n(x_0):=int_0^{infty}T(s)x_0,p^{star n}(s),ds$ is the expected state of the system at the $n^{th}$ observation. We discuss approximation procedures of $E_n(x_0)$ based on approximations of the semigroup $T$ (in terms of its generator $A$) and of $p^{star n}$ (in terms of its Laplace transform $widehat{p}).$

#### Date

2009

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

Zito, Kevin W., "Convolution semigroups" (2009). *LSU Doctoral Dissertations*. 2990.

https://digitalcommons.lsu.edu/gradschool_dissertations/2990

#### Committee Chair

Frank Neubrander