An analogue of the Bernstein-Walsh lemma in Jordan regions of the complex plane
© Abdullayev and Özkartepe; licensee Springer. 2013
Received: 17 May 2013
Accepted: 7 November 2013
Published: 2 December 2013
In this paper we continue to study two-dimensional analogues of Bernstein-Walsh estimates for arbitrary Jordan domains.
MSC:Primary 30A10; 30C10; secondary 41A17.
Keywordsalgebraic polynomials conformal mapping Bernstein lemma
1 Introduction and main results
Let be a finite region, with , bounded by a Jordan curve , , (with respect to ). Let be the univalent conformal mapping of Ω onto the Δ normalized by , , and .
Let denote the class of arbitrary algebraic polynomials of degree at most .
where σ denotes a two-dimensional Lebesgue measure.
Hence, setting , according to (1.2), we see that the C-norm of a polynomial in and is equivalent, i.e., the norm increases with no more than a constant with respect to .
where and , are constants. Therefore, if we choose , then (1.4) we can see that the -norm of polynomials in and G is equivalent.
In this work, we study a problem similar to (1.4) in , , for regions with arbitrary Jordan boundary.
Now we can state our new result.
where , .
The sharpness of (1.5) can be seen from the following remark:
2 Some auxiliary results
Let be a finite region bounded by the Jordan curve L. Let , , .
We note that, throughout this paper, (in general, different in different relations) are positive constants.
and we see that (2.1) is true. □
where , .
From (2.8) and (2.11) we complete the proof. □
where , .
where , .
3 Proof of the theorem
Now, let us make a proof of (3.1).
Now, from (3.4) and (3.5) we complete the proof. □
3.1 Proof of the remark
- Walsh JL: Interpolation and Approximation by Rational Functions in the Complex Domain. Am. Math. Soc., Providence; 1960.MATHGoogle Scholar
- Hille E, Szegö G, Tamarkin JD: On some generalization of a theorem of A Markoff. Duke Math. J. 1937, 3: 729–739. 10.1215/S0012-7094-37-00361-2MathSciNetView ArticleGoogle Scholar
- Lehto O, Virtanen KI: Quasiconformal Mapping in the Plane. Springer, Berlin; 1973.View ArticleGoogle Scholar
- Abdullayev FG: On the some properties of the orthogonal polynomials over the region of the complex plane (Part III). Ukr. Math. J. 2001, 53(12):1934–1948. 10.1023/A:1015419521005View ArticleGoogle Scholar
- Duren PL: Theory of Hp Spaces. Academic Press, San Diego; 1970.Google Scholar
- Polya G, Szegö G: Problems and Theorems in Analysis I. Nauka, Moscow; 1978. (Russian edition)Google Scholar
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