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Search: MSC category 41A17 ( Inequalities in approximation (Bernstein, Jackson, Nikol'skiii-type inequalities) )

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1. CJM 2015 (vol 68 pp. 109)

Kopotun, Kirill; Leviatan, Dany; Shevchuk, Igor
 Constrained Approximation with Jacobi Weights In this paper, we prove that, for $\ell=1$ or $2$, the rate of best $\ell$-monotone polynomial approximation in the $L_p$ norm ($1\leq p \leq \infty$) weighted by the Jacobi weight $w_{\alpha,\beta}(x) :=(1+x)^\alpha(1-x)^\beta$ with $\alpha,\beta\gt -1/p$ if $p\lt \infty$, or $\alpha,\beta\geq 0$ if $p=\infty$, is bounded by an appropriate $(\ell+1)$st modulus of smoothness with the same weight, and that this rate cannot be bounded by the $(\ell+2)$nd modulus. Related results on constrained weighted spline approximation and applications of our estimates are also given. Keywords:constrained approximation, Jacobi weights, weighted moduli of smoothness, exact estimates, exact ordersCategories:41A29, 41A10, 41A15, 41A17, 41A25

2. CJM 2007 (vol 59 pp. 730)

Erdélyi, T.; Lubinsky, D. S.
 Large Sieve Inequalities via Subharmonic Methods and the Mahler Measure of the Fekete Polynomials We investigate large sieve inequalities such as $\frac{1}{m}\sum_{j=1}^{m}\psi ( \log | P( e^{i\tau _{j}}) | ) \leq \frac{C}{2\pi }\int_{0}^{2\pi }\psi \left( \log [ e| P( e^{i\tau }) | ] \right) \,d\tau ,$ where $\psi$ is convex and increasing, $P$ is a polynomial or an exponential of a potential, and the constant $C$ depends on the degree of $P$, and the distribution of the points $0\leq \tau _{1}<\tau _{2}<\dots<\tau _{m}\leq 2\pi$. The method allows greater generality and is in some ways simpler than earlier ones. We apply our results to estimate the Mahler measure of Fekete polynomials. Category:41A17

3. CJM 1998 (vol 50 pp. 152)

Min, G.
 Inequalities for rational functions with prescribed poles This paper considers the rational system ${\cal P}_n (a_1,a_2,\ldots,a_n):= \bigl\{ {P(x) \over \prod_{k=1}^n (x-a_k)}, P\in {\cal P}_n\bigr\}$ with nonreal elements in $\{a_k\}_{k=1}^{n}\subset\Bbb{C}\setminus [-1,1]$ paired by complex conjugation. It gives a sharp (to constant) Markov-type inequality for real rational functions in ${\cal P}_n (a_1,a_2,\ldots,a_n)$. The corresponding Markov-type inequality for high derivatives is established, as well as Nikolskii-type inequalities. Some sharp Markov- and Bernstein-type inequalities with curved majorants for rational functions in ${\cal P}_n(a_1,a_2,\ldots,a_n)$ are obtained, which generalize some results for the classical polynomials. A sharp Schur-type inequality is also proved and plays a key role in the proofs of our main results. Keywords:Markov-type inequality, Bernstein-type inequality, Nikolskii-type inequality, Schur-type inequality, rational functions with prescribed poles, curved majorants, Chebyshev polynomialsCategories:41A17, 26D07, 26C15
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