The distribution of the zeros of the Hermite-Padé polynomials for a pair of functions forming a Nikishin system

The Gonchar-Stahl $ \rho^2$-theorem and associated directions in the theory of rational approximations of analytic functions

“…In the paper [48] it was assumed that the ratio of two jumps f (x) := ∆f 2 (x)/∆f 1 (x), x ∈ E, is an analytic complex-valued function on E and f has an analytic continuation from each E j along any path in C avoiding the finite set e f of the branch points of f . It was also assumed, that the set e f is symmetric with respect to real axis.…”

On a New Approach to the Problem of Distribution of Zeros of Hermite—Padé Polynomials for a Nikishin System

“…We note the papers [40], [46] and [24], in which the equilibrium problem for a mixed Green-logarithmic potential was employed for the study of the limit distribution of the zeros of Hermite-Padé polynomials for a tuple [1, f 1 , f 2 ], where a pair of functions f 1 , f 2 forms a generalized (complex) Nikishin system (see also [12], [47], [32], [41]). The method of investigation proposed in the present paper is different from that of [40], [46] and [24]. Some precursor considerations and results that eventually culminated in the statement of the potential theory equilibrium problem on the Riemann surface w 2 = z 2 − 1 were obtained by the author in [48].…”

On the convergence of type I Hermite–Padé approximants

“…Suetin sent us [19] and [20]. The first one of these papers announces the results contained in the second one.…”

“…The authors obtain the logarithmic asymptotic of the sequences of Hermite-Padé polynomials a n,j , j = 1, 2, and an analogue of (1.8) for j = 1. Convergence is proved in capacity (see [19,Theorem 1] and [20,Corollary 1].…”