Distribution of the zeros of Padé polynomials and analytic continuation

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

“…for some infinite subsequence Λ ⊂ N; besides, S(µ) ⊂ F , µ ∈ M 1 (F ), µ(1) = 1 by the above properties of the polynomial Q n,2 . We claim that relation (46) and the orthogonality relation (44) contradict each other. Setting…”

“…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].…”

Representation of conformal maps by rational functions

“…In the upper-left plot, we see chains of poles of r with diminishing residues approaching all six prevertices (two pairs of which are quite close together). This is a familiar effect in rational approximation [38,41], and it occurs because all six points correspond to branch point singularities of f , with local behavior of type (z − z j ) 1/2 for the five salient corners and (z − z j ) 3/2 at the reentrant corner. The degree 58 marked in the title indicates that r is a rational function of type (58, 58), that is, with 58 poles (some of which are not in the picture).…”

Approximations of algebraic functions by rational ones – functional analogues of diophantine approximants

“…Подробнее о гипотезе Колчина ее связи § с современным развитием теории аппроксимаций Паде см. обзор С. П. Суетина [20].…”