Download Analytic Functions Błażejewko 1982: Proceedings of a by Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.) PDF

By Boris Nikolaevič Apanasov (auth.), Julian Ławrynowicz (eds.)

ISBN-10: 3540127127

ISBN-13: 9783540127123

ISBN-10: 3540386971

ISBN-13: 9783540386971

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Additional resources for Analytic Functions Błażejewko 1982: Proceedings of a Conference held in Błażejewko, Poland, August 19–27, 1982

Sample text

E. in fl. :J. :J. = 0 and then, a 1 = PI (E uE ) no = O. e. in (Eo UE ) no. Yet, u* is continuous in 0 U Eo U E , and then, in 1 1 particular, in 0 n Eo and in 0 n E 1 , so that u* lEo =0 and u* IE1 = 1 imply u * lEo n D =0 and u * IE1 no = 1, respectively. e. e. e. e. in the sets Eo no and E1 no which are measurable, so that almost all their points are of linear density in the direction of the coordinate axes (see Saks [21], p. e. in Eo no E1 no. e. in (Eo UE ) no which, 1 together with (16), allows us to conclude that (15) holds.

1 have Yet, all the points the form (11), i . e. th zero. Thus, the hypotheses of the preceding theorem, are satisfied what, by (23), yields o. 54 Petru Caraman Hence (24) cap [E',E'(r ,oo) ,D) 1 P = = inf u JlvulPdm D capp[E',E'(r1,r2),DnE'(r2}] = = inf J IVulPdm u DnE'(r ) 2 O. If we denote by U(E ,E ,D} the class of admissible functions for o 1 cap (E ,E ,D), then for r

Choosing A=l and A=i, suc- 29 Positive Definiteness and Holomorphy cessively, we conclude that is a Cauchy-sequence for any weakly complete, that weakly in U,WEU. So (Zm) , U to an operator for any S(~)=So(~) ~EDo' S(z)EB(u:u), s(u) (z)EQ, ZED, S(·) ZED. n(u:u) Thus - we conclude that is, in fact, in S(·)EH(D;B(U:U». ace). ~ ~:Q+n. and clearly UEU, the function Finally, since also and the proof is complete. 2) should be modified by using any ho1omorphic cover map Riemann mapping converges lim (So(zm)u,u)U = (S(z)u,u)U The last theorem can be extended somewhat by letting domain in B(u:U), S(')E(D;B(U:U» Moreover, since for any unit vector s (u) (z) is ho1omorphic in This implies, since every Hilbert space is being an element of n:n+Q We shall not pursue this point here.

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