By Lars V. Ahlfors (auth.), Julian Ławrynowicz (eds.)

ISBN-10: 3540099859

ISBN-13: 9783540099857

ISBN-10: 3540392475

ISBN-13: 9783540392477

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Additional resources for Analytic Functions Kozubnik 1979: Proceedings of a Conference Held in Kozubnik, Poland, April 19–25, 1979

Example text

F vj(t) dxj(t)~d~(~) j=l a L2 Clearly FE S and thus S is a linear space. We next prove a uniqueness theorem. 6) provides a i-i correspondence _> between ~ and -> S . Finally, i f F, G g S a n d F(x) = G(x) for almost all x, then F = G. Proof. 10) < ~+ 6 e -u2/2 26 Some B a n a e h Then g*~8~8 Algebras is a continuous bounded function of class LI on ~i having a piecewise continuous derivative of bounded variation and vanishing outside a finite interval. Z. Z. A. 15) vj(t) = pjmj(t)+ fj(t)/ vj(s) fj(s)ds a and b b b b /.

6) v q F(x)dx = ~ e x p [ ~ i }] A X5 nv k=l C n n ~] (tj -tj_l)[gZ juk, g]2]d~(t'×u) j=l n 1 = JX where Rn v v exp{~ n 6 k Z= 1 6 =Zl jZ= l (2 - 6j ~) Uk, j Uk, & (tj- a)] d~(~X~) t O =- a . Proof. 8). 2] and equation (4 6) we have anf q F(x)dx= ~ C~[a,b] 8~ exp[~q i v :~ ~ ~ [vk(t)]2dt}da(v ) k= i a n tj n expC ~ii k:lZ j~ l ~ I~z__j u~,Jdt}d~(~Xul AnX 'En~ tj-i and the theorem is proved. h Let F 6S n (5-7] Z n=l l~nll<~ for n = 1 , 2 , ... 9) q Fn ( n = i C [a,b] Cv [ a , b ] Proof. For each related to bn a measure ~ n = i~2~ by equation on anf ....

Finally, i f F, G g S a n d F(x) = G(x) for almost all x, then F = G. Proof. 10) < ~+ 6 e -u2/2 26 Some B a n a e h Then g*~8~8 Algebras is a continuous bounded function of class LI on ~i having a piecewise continuous derivative of bounded variation and vanishing outside a finite interval. Z. Z. A. 15) vj(t) = pjmj(t)+ fj(t)/ vj(s) fj(s)ds a and b b b b /. 16) ~ { Go:,~j, ~ [- 0 . f . y. ]~ e~{~j]~a(~)~j(s)~ i~2%) a %) exp[i Z wjpj ] exp[- { j=l j = %) ~,,~j u a v i j= i vj(s) b =f n Cg~ i=l w~.

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Analytic Functions Kozubnik 1979: Proceedings of a Conference Held in Kozubnik, Poland, April 19–25, 1979 by Lars V. Ahlfors (auth.), Julian Ławrynowicz (eds.)


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