Approximation by Bernstein–Faber–Walsh and Szász–Mirakjan–Faber–Walsh Operators in Multiply Connected Compact Sets of $\mathbb{C}$

Abstract

By considering a multiply connected compact set $G \subset \mathbb{C}$ and an analytic function on G, we attach the q-Bernstein–Faber–Walsh polynomials with q ≥ 1, for which Voronovskaja-type results with quantitative upper estimates are given and the exact orders of approximation in G for these polynomials, namely $\frac{1} {n}$ if q = 1 and $\frac{1} {q^{n}}$ if q > 1, are obtained. Also, given a sequence with the property λ _n  ↘ 0 as fast as we want, a type of Szász–Mirakjan–Faber–Walsh operator is attached to G, for which the approximation order O(λ _n ) is proved. The results are generalizations of those previously obtained by the author for the q-Bernstein–Faber polynomials and Szász–Faber type operators attached to simply connected compact sets of the complex plane. The proof of existence for the Faber–Walsh polynomials used in our constructions is strongly based on some results on the location of critical points obtained in the book of Walsh (The location of critical points of analytic and harmonic functions, vol 34. American Mathematical Society, New York, 1950), which is also used in the major book of Rahman–Schmeisser (Analytic theory of polynomials, vol 26. Oxford University Press Inc, New York, 2002). At the end of the chapter, we present and motivate a conjecture and an open question concerning the use of truncated classical Szász–Mirakjan operators in weighted approximation and in solving a generalization of the Szegó’s problem concerning the zeroes distribution for the partial sums of the exponential function, respectively. Concerning the open question, the extensions of Eneström–Kakeya Theorem in Govil–Rahman (Tôhoku Math J 20(2):126–136, 1968) and other results on the location of the zeroes of polynomials in the Rahman–Schmeisser’s book (Analytic theory of polynomials, vol 26. Oxford University Press Inc, New York, 2002) are of interest.