By Z. Ciesielski (auth.), Juan Alfredo Gómez-Fernandez, Francisco Guerra-Vázquez, Guillermo López-Lagomasino, Miguel A. Jiménez-Pozo (eds.)

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**Additional info for Approximation and Optimization: Proceedings of the International Seminar held in Havana, Cuba, Jan. 12–16, 1987**

**Example text**

IIE i s to F in o p p o s i t e t h e c o n d e n s e r (E,F,~> s y m m e t r y and t h i s 2 . T h e main r e s u l t . compact E. exact nevertheless, alternative derivatives of f to liE, n sup norm on E. L e t E be a c o n t i n u u m on ~ , c u r v e 1 y i n q i__nnC \ E F a , an d { ~ n } a s e q u e n c e o f a neighborhood ~ of t h e compact F. 1 1 2n l o O s e n ( Z ) ~ and (E,F,~)cS then the sequence of rectifiable holomorphic functions l_~f ~ ~(z), functions Jordan zC~, on 28 @n ( t ) f n (z) satisfies = ~ F z - t z~E ' (2) ' relation 1/n l~m p n ( f n , E ) We formulated configuration; it theorem is also = e 1 true number o f c o n t i n u u m s and F finite number o f rectifiable true for , w=w(E,F,~).

From (22) of region of points a the cut follows, the relation f dt tIIP ('£')' - 0. (23) E. in 38 From h e r e , we o b t a i n t h a t g(z) the formula z = ni I ~ de~ines a (singlevalued) holomorphic t y p e i n d i c a t e d a b o v e . N o w , we f i x r e s p e c t t o ~) dt , z~D function in t ~ an (analytic any region D of and s y m m e t r i c a l the with m a r c I" f r o m a t o a using the condition Z (g (z ) - g (a) ) dz Re at the same time the =O region ~ Im ~ ro " d t dz t ~-CE ) D=G\F is also = ~, z~F; fixed.

Of which the S-property the in zeros and t h e c o n v e r g e n c e of Pad~ a p p r o x i m a n t s p r o b l e m s on t h e r a t e distribution of proof ~n and can a condenser papers introduced under ~_=~). 2; this applications observation Let indicated the E is points {~n,k ), the ~n ( z ) associated the hE the corresponding = 1 ~ ~ k-I 6 to relation point (E~F,~) last P /Q n n of rational [n-l,n] ! P n their of means t h a t and Q n ratio function of in a the triangular compact We c h o o s e t h e t a b l e set {~n,k } (4) the n.