Necessery and Sufficient Condition for Extention of Convolution Semigroup
Let and be real  valued continuous functions and ( ptf,(qtg,p and q be constants, 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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myupmir.508020130527T07:20:16Z Necessery and Sufficient Condition for Extention of Convolution Semigroup 2007 Mohd Jaffar, Mai Zurwatul Ahlam Let and be real  valued continuous functions and ( ptf,(qtg,p and q be constants, 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 convolution 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 impossible 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. Convolutions (Mathematics). 2007 Thesis http://psasir.upm.edu.my/id/eprint/5080/ http://psasir.upm.edu.my/id/eprint/5080/1/FS_2007_54.pdf application/pdf en public masters Universiti Putra Malaysia Convolutions (Mathematics). Science English 
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Universiti Putra Malaysia 
collection 
PSAS Institutional Repository 
language 
English English 
topic 
Convolutions (Mathematics). 
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Convolutions (Mathematics). Mohd Jaffar, Mai Zurwatul Ahlam Necessery and Sufficient Condition for Extention of Convolution Semigroup 
description 
Let and be real  valued continuous functions and (
ptf,(qtg,p and q be constants, 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 convolution 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 impossible 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. 
format 
Thesis 
qualification_level 
Master's degree 
author 
Mohd Jaffar, Mai Zurwatul Ahlam 
author_facet 
Mohd Jaffar, Mai Zurwatul Ahlam 
author_sort 
Mohd Jaffar, Mai Zurwatul Ahlam 
title 
Necessery and Sufficient Condition for Extention of Convolution Semigroup 
title_short 
Necessery and Sufficient Condition for Extention of Convolution Semigroup 
title_full 
Necessery and Sufficient Condition for Extention of Convolution Semigroup 
title_fullStr 
Necessery and Sufficient Condition for Extention of Convolution Semigroup 
title_full_unstemmed 
Necessery and Sufficient Condition for Extention of Convolution Semigroup 
title_sort 
necessery and sufficient condition for extention of convolution semigroup 
granting_institution 
Universiti Putra Malaysia 
granting_department 
Science 
publishDate 
2007 
url 
http://psasir.upm.edu.my/id/eprint/5080/1/FS_2007_54.pdf 
_version_ 
1747810346639294464 