Necessery and Sufficient Condition for Extention of Convolution Semigroup - Mohd Jaffar, Mai Zurwatul Ahlam (2007)
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<t()()ptfptf−=ε, and ()()qtgqtg−=,. ()()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 ()[]0≠tgL, where p and q are constants and is an interval of the point with neighborhood. +QεIε Proposition 1 is called scale form of the functions f and g, while Proposition 2 is called shift form of the functions f and g. The extension of convolution semigroup is formed by a non - impulsive and an impulsive function such that the non - impulsive function is an approximation of the impulsive function under certain condition, where all functions in this study are both real - valued continuous and of exponential order. The study has shown that it is not necessary depend on the same function in order to get the new relation. This study is only true for the conditions described by Propositions 1and 2.