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Characterization of smooth, compact algebraic curves in \(\mathbb{R}^ 2\). (English) Zbl 0834.41012

Jakóbczak, Piotr (ed.) et al., Topics in complex analysis. Proceedings of the semester on complex analysis, held in autumn of 1992 at the International Banach Center in Warsaw, Poland. Warszawa: Polish Academy of Sciences, Institute of Mathematics, Banach Cent. Publ. 31, 125-134 (1995).
The purpose of this paper is to give the following characterization of a smooth compact connected curve \(K\) in \(\mathbb{R}^2\): (1) \(K\) is algebraic; (2) \(K\) satisfies a tangential Markov inequality with exponent one, i.e. there exists \(M= M(K) >0\) such that \(|D_T p|_K\leq M(\deg p) |p|_K\) for all polynomials \(p\) where \(D_T\) denotes the unit tangential derivative along \(K\); (3) For some \(0< \alpha<1\), \(K\) satisfies a Bernstein theorem: there exists \(B>0\) such that for \(f\in {\mathcal C} (K)\), if \(\text{dist}_K (f, {\mathcal P}_n )\leq n^{- \alpha}\), then \(f\in \text{Lip} (\alpha)\) and \(|f|_\alpha \leq B\) where \({\mathcal P}_n\) is the space of all polynomials of degree at most \(n\) in two variables and \(|f|_\alpha\) denotes the \(\text{Lip} (\alpha)\) norm of \(f\).
Reviewer’s remark. Here smooth should be read \(({\mathcal C}^\infty)\) and not \(({\mathcal C}^1)\) as is misprinted in the paper. An extension of the above result to a smooth compact \(m\)-dimensional submanifold of \(\mathbb{R}^n\) without boundary \((1\leq m\leq N-1)\) has been given in L. Bos, N. Levenberg, P. Milman and B. A. Taylor [Indiana Univ. Math. J. 44, No. 1, 115-138 (1995; Zbl 0824.41015)].
For the entire collection see [Zbl 0816.00022].

MSC:

41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
14P05 Real algebraic sets

Citations:

Zbl 0824.41015
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