By V. S. Vladimirov, Yu. N. Drozzinov, B. I. Zavialov (auth.)

ISBN-10: 9400928319

ISBN-13: 9789400928312

ISBN-10: 9401077746

ISBN-13: 9789401077743

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B ) X ~ where ~ is an open set in the space Rn -1 of variables (x 2 ..... x n ) and let 1 ED '((1)). 0) on the variable x 1 of the distribution 1 is defined as a solution of the differential equation Xo E (a. b) and w. •0) = 1 . •0) E D'((1)). 7. Let F be the closure of a regular domain. 3): (jOlt = j-1fjot(jJ ). 10) Some facts on the theory of distributions 17 5. Change of variables. 4). Let f (y) be from 0·(el 2 ). 1) Since the operation ¢l -+ ¢l(x (y )) 1J «x (y ))/y) 1 is an isomorphism from O(el) onto 0«(01)' then the operation f (y) -+ f (y (x)) is an isomorphism from O'Cel l ) onto O'(el).

If 1 (n II (t) = "" = J 1 ('T) d 'T • E e" (Rl) then (_1)10 1 (nl(t). We shall use the following criterion of existence and some properties of the operations introduced. Lemma. Let cp E e""(Rl) satisfy the conditions j ~(j l(t ) : I t < I k dt 00. k. j = O. 1. 2) -00 "" J cp(t )t k dt = o. 1. N. N ~ 1. 3) Further, let 01 and 13 be real numbers which satisfy the conditions M = N - 2 + [01] + [a) ~ 0, N + [(1) ~ 1 and N + [13) ~ 1. (Illcp j r l = (Illcp(o

34 Chapter 1 Example 1. e irz I = e-(it)/(4z), Imz > O. ei'x'2'I(JTEI)=e-(it)/(4Z), t = = lEI. S. Vladimirov (1979, p. 11). 5) Proof. p(T 2)](P). p(t). 5) shows that the B-transform is an isomorphism from SS+ onto SS+. p E SS+. 6) The B -transform provides the isomorphism from SS ~ onto $ ~ and the inversion formula valid (cf. 7) ~. Remark. On functions 1 Example 2. B [8] Example 3. 35 E (1(0,00) the B -transform takes the form = 8. ). Imz > o. 8) Proof. )dt. +. Properties of the B -transform. (a)B[f(kT)](t)= k\B[llltj.

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Tauberian Theorems for Generalized Functions by V. S. Vladimirov, Yu. N. Drozzinov, B. I. Zavialov (auth.)


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