By William B. Jones, W. J. Thron, Haakon Waadeland (eds.)
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Extra info for Analytic Theory of Continued Fractions: Proceedings of a Seminar-Workshop held at Loen, Norway, 1981
Except for two values of z is reduced by one. 1. 2) p2m_l(Z) = It is also true that S2m(Z,Zm) 6 T2m_l(Z) and hence Cnk(Z, ~ k ) Dnk(Z'Zk) 6T2m_l(Z) Proof. S2m_l(Z,~) transformation  , for is the image of It n k > 2m-I , under a linear fractional and hence is a circle (or straight llne). 2) . 9) . C. 1) ~ Let C T-fractlon. 6) . 2. 2) for has a unique solution. z = 0 38 We hope to be able to prove that this is the case if and only if pn(Z) § 0 References I. Berg, Christian, written communication, 2.
2 . It is the analogue of a theorem of Markoff  . 3. Let ~(t) E~(a,b) with 0 < a < b < +~ . 18) (B) coefficients zS b d~(t) a z+t The positive every compact subset by the integral Proof. 4. 18) (A): = defined ... 18) converges to the holomorphic uniformly function on defined . 18) . subset of z E K, This D . 1]) theorem the sequence The coefficients Fn,G n and T-fraction be an arbitrary 6(K) for for all such that, j = 1,2 .... 4) is a normal by the Stieltjes-Vitali [8, Theorem of the positive (B) -J=iiz+t(n) I -- 6(K) G I B n(z) It follows .
11) A similar argument holds if is odd. 6) and vanishes by orthogonality since we obtain fbF(t)d+(t ) = n~ F(t~n))faLn, j(t)d+(t)b a j=l = (n)~,L(n) L w. j=l which proves (A) . 13) n (B) r~. j ) J we first show that Pn(Z) = zf b Qn(Z)~n(t)d~(t) a z-t , n = 0,1,2,... 2) . 2) . 3) . lb) . 15) that Bn(Z ) denote the positive nth T-fractlon m = 0,1,2,... 5) . 16) Vn(Z) and of the corresponding J=l z + ~ n) has the same form as , An(Z) n = 1,2,3,... 1 and Pn(t~n)) = w~ n) . 2. 1 . 17) F(t) for which the Riemann-Stieltjes integral on the exists.
Analytic Theory of Continued Fractions: Proceedings of a Seminar-Workshop held at Loen, Norway, 1981 by William B. Jones, W. J. Thron, Haakon Waadeland (eds.)