RESULTS ON THE COMPOSITION AND NEUTRIX COMPOSITION OF THE DELTA FUNCTION

Authors : Brian Fisher, Biljana JOLEVSKA–TUNESKA

Pages : 43-50

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Publication Date : 2014-01-01

Article Type : Other Papers

Abstract :The neutrix composition F insert ignore into journalissuearticles values(f insert ignore into journalissuearticles values(x););); of a distribution F insert ignore into journalissuearticles values(x); and a locallysummable function f insert ignore into journalissuearticles values(x); is said to exist and be equal to the distributionhinsert ignore into journalissuearticles values(x); if the neutrix limit of the sequence {Fn insert ignore into journalissuearticles values(f insert ignore into journalissuearticles values(x););} is equal to hinsert ignore into journalissuearticles values(x);,where Fn insert ignore into journalissuearticles values(x); = F insert ignore into journalissuearticles values(x); ∗ δn insert ignore into journalissuearticles values(x); and {δn insert ignore into journalissuearticles values(x);} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-functionδinsert ignore into journalissuearticles values(x);. It is proved that the neutrix composition δinsert ignore into journalissuearticles values(s); {[exp + insert ignore into journalissuearticles values(x); − 1]r } exists andδ insert ignore into journalissuearticles values(s); {[exp + insert ignore into journalissuearticles values(x); − 1]r } = rs+r−1k=0 insert ignore into journalissuearticles values(−1);s+k s!c rs+r−1,k2rk! δ insert ignore into journalissuearticles values(k); insert ignore into journalissuearticles values(x);, for r = 1, 2, . . . and s = 0, 1, 2, . . .. Further results are also proved.
Keywords : distribution, dirac delta function, composition of distributions, neutrix, neutrix limit