Weak Convergence of Varying Measures and Hermite—Padé Orthogonal Polynomials

Abstract

It is known that the common denominator of the Hermite-Padé approximants of a mixed Angelesco-Nikishin system shares orthogonality relations with respect to each function in the system. It is less known that they also satisfy full orthogonality with respect to a varying measure. This problem motivates our interest in extending the class of varying measures with respect to which weak asymptotics of orthogonal polynomials takes place. In particular, for the case of a Nikishin system, we prove weak asymptotics of the corresponding varying measures.

Ratio asymptotics of Hermite-Padé polynomials for Nikishin systems

“…. , l. On the other hand, since deg Q n l ,k = deg Q * n,k + 1 and both of these polynomials are orthogonal with respect to the same varying weight, then by the ratio asymptotic theorem for varying measures (see Theorem 6 on page 567 of [4]) we have…”

“…By the proved Theorem 1.2, we have limit in (20) along the whole sequence Λ. Reasoning as in the deduction of formulas (22) and (27), but now in connection to orthonormal polynomials (see [4] and [5]), it follows that…”

Ratio and relative asymptotics of polynomials orthogonal with respect to varying Denisov-type measures

“…Following the same scheme as in the proof of Theorem 9 in [13] (see also [5,Theorem 8]), from Theorem 1 one obtains Corollary 3. Suppose that, for each k ∈ Z, ({d n }, {w 2n }, k) is strongly admissible on S and { n } is a Denisov-type sequence on S. Then, for each k ∈ Z, and any function f continuous on S, we have…”

“…The existence of such normalizing constants is clearly indicated in [1] and is based in the present situation on Corollary 3. In [1], instead of Theorems 1, 2 and Corollary 3, the authors make use of similar results for orthogonal polynomials with respect to varying measures without mass points outside of k developed earlier by B. de la Calle and G. López contained in [5,6]. To conclude, you show that the system of boundary-value problems has a unique solution which may be expressed by means of the algebraic functions defined above.…”

“…Such results are relevant for the proof of asymptotic properties of orthogonal polynomials with respect to fixed measures as well (see [1][2][3]5,12,14]). …”