Download Analytic Functions Kozubnik 1979: Proceedings of a by Lars V. Ahlfors (auth.), Julian Ławrynowicz (eds.) PDF

Analytic

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

ISBN-10: 3540099859

ISBN-13: 9783540099857

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Read Online or Download Analytic Functions Kozubnik 1979: Proceedings of a Conference Held in Kozubnik, Poland, April 19–25, 1979 PDF

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

Sample text

Paley - Wiener - Zygmund) integral [4] which is defined as follows. Definition. N. real functions of bounded variation on n Vn(t)=j~=l [a,b] b ~J(t) f v(s)~j(s)ds a (complete orthonormal) . Z. integral is defined by b b a for all Note. b n ~ a x£C[a,b] for which the above limit exists. It was shown in [5] that this integral exists for almost all and is essentially independent of the choice of v is bounded variation, integral. ~°i' ~2' .... xEC[a,b] Moreover if it is essentially equivalent to the Riemann-Stieltjes Clearly "almost all" may be replaced by "s-almost all" in this statement.

5 . 6). It follows from I~II is uniquely determined by fo__r n = 1,2~ ... I_< ~ k= i Proof. N. Z. 1)]. 25) up to a scale-invariant Thus G Wiener null set. ~n!! 3. N. V. Z. integrals be based on it. , ~). 36) ... ,2 k ; k=l,2, ... 35) J(~) < m 2~k~ , m = - 2 k + l , lira ~ k -~ ~) L2 VEL~ Hence J k ( V ) d ~ n ( V ) = G (~') n = 1,2~ ... H. A. ]~n II and E ll~nl! converges. [" J(v)d~,(v)= ~ G (~)= G(~) n=l " n L2 for all ~ for which v-almost all ~: i n s-alraost all ~. ,v. l' b ^" which Ja vj(t)dx B u t b y Le~ana 2 .

N. 3) F(~) Moreover Corollary 2. O N. {gn]n= 1,2 .... N. V. on where x' on . 1 is s-almost e v e r y - equal to the old one. Z. 3). then be defined up to an equivalence class. be- The integral will 24 Some Algebras If F s S, A > O, and G(x)= F(Ix), we have G ~ S. Note. l< ~ For all EEG, n=l Then b6~ Proof. < ~ Hence ~6~ and the lemma is proved. 2 can be extended by omitting the restriction that bl' b2 ' "'" be real and non-negative. 4 Proof. Let The space S F I, F 2 E S is a linear space. H. A.

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