Polynomial inequalities in L^p norms with generalized Jacobi weights
Abstract: We give concrete estimates of Schur-and Nikolskii-type inequalities with the best exponent of polynomial degree in L p norms with generalized Jacobi weights. In particular, we obtain these inequalities with the Chebyshev weight, with the Gegenbauer weights and with the classical Jacobi ones.
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Constants in Markov’s and Bernstein inequality on a finite interval in $${\mathbb {R}}$$
In this paper we demonstrate the constants in the pointwise Bernstein inequality $$\begin{aligned} |P^{(\alpha )}(x)|\le \left( \frac{2n}{\sqrt{(x-a)(b-x)}}\right) ^{\alpha }||P||_{[a,b]}, \end{aligned}$$ | P ( α ) ( x ) | ≤ 2 n ( x - a ) ( b - x ) α | | P | | [ a , b ] , for the $$\alpha -$$ α - th derivative of an algebraic polynomial in $$L^{\infty }-$$ L ∞ - norms on an interval in $${\mathbb {R}}$$ R , where $$\alpha \ge 3$$ α ≥ 3 . This result was obtained using the tools of theory of pluripotential and we apply it to get the main result which is a new generalization of V. A. Markov’s type inequalities $$\begin{aligned} ||P^{(\alpha )}||_p\le C^{1/{p}}\left( \frac{2}{b-a}\right) ^{\alpha }||T^{(\alpha )}_{n}||_{[-1,1]}n^{2/p}||P||_{p}, \end{aligned}$$ | | P ( α ) | | p ≤ C 1 / p 2 b - a α | | T n ( α ) | | [ - 1 , 1 ] n 2 / p | | P | | p , for the $$\alpha -$$ α - th derivative of an algebraic polynomial in $$L^{p}$$ L p norms, where $$p\ge 1$$ p ≥ 1 . In particular, we show that for any $$\alpha \ge 3$$ α ≥ 3 the constant C in the V. A. Markov inequality satisfies the condition $$C\le 8\left( \frac{32\cdot 3,94741\cdot \pi M\alpha ^2}{3\sqrt{3}}\right) ^{1/p}$$ C ≤ 8 32 · 3 , 94741 · π M α 2 3 3 1 / p .
Constants in Markov’s and Bernstein inequality on a finite interval in $${\mathbb {R}}$$
In this paper we demonstrate the constants in the pointwise Bernstein inequality $$\begin{aligned} |P^{(\alpha )}(x)|\le \left( \frac{2n}{\sqrt{(x-a)(b-x)}}\right) ^{\alpha }||P||_{[a,b]}, \end{aligned}$$ | P ( α ) ( x ) | ≤ 2 n ( x - a ) ( b - x ) α | | P | | [ a , b ] , for the $$\alpha -$$ α - th derivative of an algebraic polynomial in $$L^{\infty }-$$ L ∞ - norms on an interval in $${\mathbb {R}}$$ R , where $$\alpha \ge 3$$ α ≥ 3 . This result was obtained using the tools of theory of pluripotential and we apply it to get the main result which is a new generalization of V. A. Markov’s type inequalities $$\begin{aligned} ||P^{(\alpha )}||_p\le C^{1/{p}}\left( \frac{2}{b-a}\right) ^{\alpha }||T^{(\alpha )}_{n}||_{[-1,1]}n^{2/p}||P||_{p}, \end{aligned}$$ | | P ( α ) | | p ≤ C 1 / p 2 b - a α | | T n ( α ) | | [ - 1 , 1 ] n 2 / p | | P | | p , for the $$\alpha -$$ α - th derivative of an algebraic polynomial in $$L^{p}$$ L p norms, where $$p\ge 1$$ p ≥ 1 . In particular, we show that for any $$\alpha \ge 3$$ α ≥ 3 the constant C in the V. A. Markov inequality satisfies the condition $$C\le 8\left( \frac{32\cdot 3,94741\cdot \pi M\alpha ^2}{3\sqrt{3}}\right) ^{1/p}$$ C ≤ 8 32 · 3 , 94741 · π M α 2 3 3 1 / p .
In this note we give sharp Schur type inequalities for univariate polynomials with convex weights. Our approach will rely on application of two-dimensional Markov type inequalities, and also certain properties of Jacobi polynomials in order to prove sharpness.
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