 By Golinskii L., Totik V.

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In – H. Stahl obtained asymptotics for non-Hermitian orthogonal polynomials even for varying measures and gave several applications of them to Pad´e approximation. 2)) was proved by A. I. Aptekarev and W. Van Assche via the Riemann-Hilbert approach. A similar result holds on the support of the measure, as well as for the case of varying weights, see . 12 Multiple orthogonality Multiple orthogonality comes from simultaneous Pad´e approximation. It is a relatively new area where we have to mention the names of E.

There are two types of multiple orthogonality corresponding to the appropriate Hermite-Pad´e approximation. In type I we are looking for polynomials Qn,j of degree nj − 1 for each j = 1, . . , r such that r xk Qn,j (x)dµj (x) = 0, k = 0, 1, . . , |n| − 2. j=1 These orthogonality relations give |n| − 1 homogeneous linear equations for the |n| coefficients of the r polynomials Qn,j , so there is a non-trivial solution. If the rank of the system is |n| − 1, then the solution is unique up to a multiplicative factor, in which case the index n is called normal.

There exists µ ∈ P such that Φmk (zj , µ) = 0 for j = mk−1 + 1, . . , mk . The following consequence of this result may seem kind of amazing. Let a measure µ belong to the class of nontrivial probability measures P. 1) supp νn = Zn , νn {zjn } = n 46 ¯ be a space of with l(zjn ) equal to the multiplicity of the zero zjn . Let M+ (D) ¯ probability measures on D endowed with the weak* topology. A measure µ ∈ P ¯ there is a sequence of indices nj is said to be universal if for each σ ∈ M+ (D) such that νnj (µ) converges to σ as j → ∞ in weak* topology.