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By G. I. Eskin

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Denote T+ = |ϕs P+0 ∂ϕs ∂ϕs + P+1 , ∂n ∂n |ϕs P−1 ∂ϕs , ∂n λs ∈Δ T− = λs ∈Δ and consider the rational approximation of DN DN = DN(Δ) + K : DN++ := DN 00 ++ DN 10 ++ DN 01 ++ DN 11 ++ = T++ DN+− := DN 00 +− DN 10 +− DN 01 +− DN 11 +− = T++ DN−+ := DN 10 −+ DN 10 −+ DN 11 −+ DN 11 −+ = T−+ 01 K++ 11 K++ I T+ + λI Δ − LΔ 00 K++ 10 K++ λI Δ I T− + − LΔ 01 K+− 11 K+− λI Δ I 10 11 . T+ + K−+ , K−+ − LΔ , , ˜ Consider the Krein formula for DN ˜ = DN++ − DN+− DN I DN−+ . 5). Introduce T+− I T + := Q(λ) : EΔ → EΔ , K−− + K− −+ and P+ − 01 K+− 11 K+− I 11 + K 1 P− := J (λ).

Dijksma and H. Langer 1. Introduction In the papers [4] and [5] the Schur transformation for generalized Nevanlinna functions with a reference point z1 in the open upper half-plane was considered. 6], see [3]. This transformation or a simple modification of it we call here the Schur transformation for Nevanlinna functions, and it is the starting point for the present paper. To give more details, we consider a Nevanlinna function n which has for some integer p ≥ 1 an asymptotic expansion of order 2p + 1 at ∞, for example n(z) = − s0 s1 s2p − 2 − · · · − 2p+1 + o z z z 1 z 2p+1 , z = iy, y → ±∞.

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