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

Sorin G. Gal ()
Additional contact information
Sorin G. Gal: University of Oradea

A chapter in Progress in Approximation Theory and Applicable Complex Analysis, 2017, pp 401-428 from Springer

Abstract: Abstract By considering a multiply connected compact set G ⊂ ℂ $$G \subset \mathbb{C}$$ 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 1 n $$\frac{1} {n}$$ if q = 1 and 1 q n $$\frac{1} {q^{n}}$$ 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.

Keywords: Conformal mapping; Compact sets of several components; Faber–Walsh polynomials; q-Bernstein–Faber–Walsh polynomials with q ≥ 1; Voronovskaja type formula; Exact order of approximation; Szász–Mirakjan–Faber–Walsh type operators; Truncated Szász–Mirakjan operators; Weighted approximation; Szegö domain; Szegö curve; Zeroes distribution; 30E10; 41A25 (search for similar items in EconPapers)
Date: 2017
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:spochp:978-3-319-49242-1_19

Ordering information: This item can be ordered from
http://www.springer.com/9783319492421

DOI: 10.1007/978-3-319-49242-1_19

Access Statistics for this chapter

More chapters in Springer Optimization and Its Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().