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Bernstein’s inequality on algebraic curves. (English) Zbl 0842.26013
In this paper, authors estimate the growth of polynomials on a smooth algebraic curve \(\Gamma= \{(x, y)\in \mathbb{R}^2: y= \psi(x),|x|\leq 1\}\), where \(Q(x, \psi(x))\equiv 0\), \(Q\) being a polynomial with real coefficients. Under the assumptions that (I) \(|\psi(x)|\leq 1\) for \(|x|\leq 1\), (II) \(Q\) has degree at most \(D\), (III) \(|Q(x, y)|\leq C\) for \(|x|,|y|\leq 1\) and (IV) \(|{\partial Q\over \partial y} (x, y)|\geq c> 0\) for \((x, y)\in \Gamma\), they show that, given a polynomial \(P\) of degree \(d\) and \(f(x)= P(x, \psi(x))\), there exists a constant \(C_*> 0\) depending only on \(d\), \(D\), \(C\), \(c\) such that \[ (\text{i})\quad \max_{|x|\leq 1} |f(x)|\leq C_*\max_{|x|\leq {1\over 2}}|f(x)|,\qquad (\text{ii})\quad \max_{|x|\leq 1}|f'(x)|\leq C_* \max_{|x|\leq 1} |f(x)|, \]
\[ \text{(iii)}\qquad \max_{|x|\leq 1} |f(x)|\leq C_* \int^1_{- 1} |f(x)|dx. \] Here, the point is that \(C_*\) does not depend on \(\Gamma\). The authors’ main tool is an extension theorem that permits to find, to every polynomial \(P\) of degree \(d\), a rational function \(F/G\) of degree \(d_*\) such that \(P= F/G\) on \(\Gamma\), \(c_*< G< C_*\) on \(\{|x|, |y|\leq 1\}\) and \(\max_{|x|, |y|\leq 1} |F|\leq C_*\max_\Gamma |P|\). One should add that such an extension with \(F/G\) replaced by a polynomial is, in general, impossible which is shown by the authors by an elementary counterexample.

MSC:
26D05 Inequalities for trigonometric functions and polynomials
26C05 Real polynomials: analytic properties, etc.
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
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References:
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