# Necessery and Sufficient Condition for Extention of Convolution Semigroup

Let and be real - valued continuous functions and ( ptf,(qtg,p and q be cons-tants, where denotes a set of positive rational numbers. Convolution of +Q+Q()fpt, and , denoted by ()q,tg()()qtgp,∗tf, is defined by ()()()()∫−=∗tdvqvtgpvfqtgptf0,,,,, where denotes convolution operation, provided that...

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Main Author: Thesis EnglishEnglish 2007 http://psasir.upm.edu.my/id/eprint/5080/1/FS_2007_54.pdf No Tags, Be the first to tag this record!
Summary: Let and be real - valued continuous functions and ( ptf,(qtg,p and q be cons-tants, where denotes a set of positive rational numbers. Convolution of +Q+Q()fpt, and , denoted by ()q,tg()()qtgp,∗tf, is defined by ()()()()∫−=∗tdvqvtgpvfqtgptf0,,,,, where denotes convolution operation, provided that the integral exists. From the definition of convolution, we introduce a new relation as follows ∗ ()()()()qt,pgqp,tfq,tgp,tf++=∗or , where denotes an ordinary addition. The new relation is called extension of con-volution semigroup. Objective of the study is to discover the necessary and sufficient condition for the new relation. The study is based on Laplace transformable functions. Convolution Theorem in Laplace transform is used to verify the new relation. It is im-possible to achieve the new relation directly since most of the transforms are rational polynomial functions. Furthermore, any transform in terms of exponential function is different from one another. However, we overcome the problem by (a) Identity property under convolution such that ()()(tftδtf=∗, where()tf is a real - valued continuous function, which has Laplace transform and ()tδ is the delta function and it is the identity function under convolution. The Laplace transform of delta function ()tδ is 1. (b) Under certain condition, the delta function ()tδ is a convolution semigroup such that ()()()qp,tδq,tδp,tδ+=∗. (c) Delta function can be replaced by other function under certain condition. ()tδ With (a), (b) and (c), we discover the following results: Proposition 1 Let ()()tpfptfε=, and ()()tqgqtg=, for such that 0≥t()()tgtf≠ε and ()()1lim0==∫∫⊂→dttgdttfRRIεεε. Then ()()() ,,,qptfqtgptf+=∗if and only if ()[]0≠tfLε and ()[]1=tgL, or ()()() ,,,qptgqtgptf+=∗ if and only if ()[]1=tfLε and ()[], where pand are constants with q+Q111=+qp and is an interval of the point with neighborhood. εIε Proposition 2 Let and ()tfε()tg be given real - valued functions with ()()0==tgtfε for . Let 0