Multiplication of Distributions and Applications to Partial by M. Oberguggenberger

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By M. Oberguggenberger

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Additional info for Multiplication of Distributions and Applications to Partial Differential Equations

Example text

More precisely, we mean that if (u2 /V)GM(fi) and (ui ,v) , (ciui + C 2 U 2 /V) G M (fi) , and In particular, M (ft ) A if property appears in a pair belonging to 0«v = 0 . shared commutativity; If u*v = v u then (C1 U 1 + C 2 U 2 )•v = ciUi*v + C 2U 2 *v . vGP*(Q) at all, then ci,C2 ^ i/ by u-v all products exists, to be defined here then so does vu is , and . 1). 2) is understood in the sense products have the following locality properties: (a) If u-v (b) If subset exists, then ui'V 0) of and ft , U 2 *v supp(u*v) c supp u n supp v .

4) for the m +2 = H(x-ct) value of m. 6 below). 2. What is 6^(x)? Let us see what happens when we first regularize the Dirac measure zations, of 6(x), square the regulari­ and attempt to take the limit. 8). real-valued. V' { M) is We given by first assume In order to find the out x=0. I a^ (x) 1)^(x)dx sequences in = it follows that it is bounded in imply that squares converge of if the sequence converges 6EL^(]R) , which real-valued in a^ is converges ijj = 1 in in a I a^(x)dx Hence, which with Then in subsequence that a^ such that eventually.

Formulae as the one above generally depend on the nonstandard representation of distributions chosen. Systematic computations have been undertaken by Todorov [220] in the setting of his “asymptotic functions". Our main generalized concern functions in this book constructed as is with algebras of factor algebras of the infinite product space E[n] = (c” (n)y°' 00) Q czjr with respect to suitable ideals, open. A distribution uEP'(fi) can be represented as an element of of a suitable sequence an associative, (^e)e>o commutative regularizations.