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A discrete weighted Markov-Bernstein inequality for sequences and polynomials. (English) Zbl 07265514
Authors’ abstract: “For parameters \(c \in(0, 1)\) and \(\beta > 0\), let \(\ell_2(c, \beta)\) be the Hilbert space of real functions defined on \(\mathbb{N} \) (i.e., real sequences), for which \[ \| f \|_{c , \beta}^2 : = \sum\limits_{k = 0}^\infty \frac{ ( \beta )_k}{ k !} c^k [ f ( k ) ]^2 < \infty . \] We study the best (i.e., the smallest possible) constant \(\gamma_n(c, \beta)\) in the discrete Markov-Bernstein inequality \[ \| {\Delta} P \|_{c , \beta} \leq \gamma_n(c, \beta) \| P \|_{c , \beta}, \;\;\; P \in \mathcal{P}_n, \] where \(\mathcal{P}_n\) is the set of real algebraic polynomials of degree at most \(n\) and \({\Delta} f(x) : = f(x + 1) - f(x)\).
We prove that
(i)
\(\gamma_n(c, 1) \leq 1 + \frac{ 1}{ \sqrt{ c}}\) for every \(n \in \mathbb{N}\) and \(\lim\limits_{n \to \infty} \gamma_n(c, 1) = 1 + \frac{ 1}{ \sqrt{ c}} \);
(ii)
For every fixed \(c \in(0, 1), \gamma_n(c, \beta)\) is a monotonically decreasing function of \(\beta\) in \((0, \infty)\);
(iii)
For every fixed \(c \in(0, 1)\) and \(\beta > 0\), the best Markov-Bernstein constants \(\gamma_n(c, \beta)\) are bounded uniformly with respect to \(n\).
A similar Markov-Bernstein inequality is proved for sequences, and a relation between the best Markov-Bernstein constants \(\gamma_n(c, \beta)\) and the smallest eigenvalues of certain explicitly given Jacobi matrices is established.”
Added by reviewer: There is an interesting section “Comments” where the authors indicate several related unsolved problems and discuss directions for further investigations.
MSC:
41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities)
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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References:
[1] Aleksov, D.; Nikolov, G., Markov \(L_2\) inequality with the Gegenbauer weight, J. Approx. Theory, 225, 224-241 (2018) · Zbl 1380.41006
[2] Aleksov, D.; Nikolov, G.; Shadrin, A., On the Markov inequality in the \(L_2\) norm with the Gegenbauer weight, J. Approx. Theory, 208, 9-20 (2016) · Zbl 06588614
[3] Arestov, V. V., On integral inequalities for trigonometric polynomials and their derivatives, Math. USSR, Izv., 18, 1-17 (1982) · Zbl 0517.42001
[4] Boas, R. B., Entire Functions (1954), Academic Press: Academic Press New York
[5] Carleson, L., Bernstein’s approximation problem, Proc. Am. Math. Soc., 2, 953-961 (1951) · Zbl 0044.07002
[6] Dörfler, P., New inequalities of Markov type, SIAM J. Math. Anal., 18, 490-494 (1987) · Zbl 0612.41020
[7] Dörfler, P., Über die bestmögliche Konstante in Markov-Ungleichungen mit Laguerre Gewicht, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II, 200, 13-20 (1991) · Zbl 0748.41009
[8] Dörfler, P., Asymptotics of the best constant in a certain Markov-type inequality, J. Approx. Theory, 114, 84-97 (2002) · Zbl 1120.41015
[9] Dzrbasjan, M. M., On metrical criteria of completeness of systems of polynomials in unbounded domains, Dokl. Akad. Nauk Armen. SSR, 7, 3-10 (1947)
[10] Erdelyi, T., Notes on inequalities with doubling weights, J. Approx. Theory, 100, 60-72 (1999) · Zbl 0985.41009
[11] Feynman, R. P., Forces in molecules, Phys. Rev., 56, 340-343 (1939) · Zbl 0022.42302
[12] Freud, G., Orthogonal Polynomials (1971), Akadémiai Kiadó/Pergamon Press: Akadémiai Kiadó/Pergamon Press Budapest · Zbl 0226.33014
[13] Ganzburg, M. I., Limit Theorems for Polynomial Approximation with Exponential Weights, Mem. Amer. Math. Soc., vol. 897 (2008) · Zbl 1142.30011
[14] Hellmann, H. G.A., Zur rolle der kinetischen Elektronenenergie für die zweischen-atomaren Kräfte, Z. Phys., 85, 180-190 (1933) · JFM 59.1554.03
[15] Hille, E.; Szegő, G.; Tamarkin, J. D., On some generalizations of a theorem of A. Markoff, Duke Math. J., 3, 729-739 (1937) · JFM 63.0314.03
[16] Ismail, M. E.H., The variation of zeros of certain orthogonal polynomials, Adv. Appl. Math., 8, 111-118 (1987) · Zbl 0628.33001
[17] Ismail, M. E.H., Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and Its Applications, vol. 98 (2005), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1082.42016
[18] Ismail, M. E.H.; Muldoon, M. E., A discrete approach to monotonicity of zeros of orthogonal polynomials, Trans. Am. Math. Soc., 323, 65-78 (1991) · Zbl 0718.33004
[19] Ismail, M. E.H.; Zhang, R., On the Hellmann-Feynman theorem and the variation of zeros of certain special functions, Adv. Appl. Math., 9, 439-446 (1988) · Zbl 0684.33004
[20] Izumi, S.; Kawata, T., Quasi-analytic class and closure of \(\{ t^n \}\) in the interval \((- \infty, \infty))\), Tohoku Math. J., 43, 267-273 (1937) · JFM 63.0197.03
[21] Koekoek, R.; Swarttouw, R. F., The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue (1998), Delft University of Technology, Report 98-17
[22] Koosis, P., The Logarithmic Integral I (1988), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0665.30038
[23] Lubinsky, D., A survey of weighted polynomial approximation with exponential weights, Surv. Approx. Theory, 3, 1-105 (2007) · Zbl 1181.41004
[24] Lubinsky, D., Weighted Markov-Bernstein inequalities for entire functions of exponential type, Publ. Inst. Math. (Beograd) (N.S.), 96, 110, 181-192 (2014) · Zbl 1349.42054
[25] Markov, A. A., On a question of D.I. Mendeleev, Zap. Petersb. Akad. Nauk, 62, 1-24 (1889), (in Russian). Available also at:
[26] Mastroianni, G.; Totik, V., Weighted polynomial inequalities with doubling and \(A_\infty\) weights, Constr. Approx., 16, 37-71 (2000) · Zbl 0956.42001
[27] Nevai, P.; The anonymous referee, The Bernstein inequality and the Schur inequalities are equivalent, J. Approx. Theory, 182, 103-109 (2014) · Zbl 1290.41007
[28] Nikolov, G., Markov-type inequalities in the \(L_2\)-norms induced by the Tchebycheff weights, Arch. Inequal. Appl., 1, 361-376 (2003) · Zbl 1062.41010
[29] Nikolov, G.; Shadrin, A., Markov \(L_2\)-inequality with the Laguerre weight, (Ivanov, K.; Nikolov, G.; Uluchev, R., Constructive Theory of Functions, Sozopol 2018 (2018), Professor Marin Drinov Academic Publishing House: Professor Marin Drinov Academic Publishing House Sofia), 207-221 · Zbl 1445.41004
[30] Nikolov, G.; Shadrin, A., On the Markov inequality in the \(L_2\)-norm with the Gegenbauer weight, Constr. Approx., 49, 1, 1-27 (2019) · Zbl 1443.41008
[31] Rahman, Q. I.; Schmeisser, G., \( L^p\) inequalities for entire functions of exponential type, Trans. Am. Math. Soc., 320, 91-103 (1990) · Zbl 0699.30022
[32] Riesz, M., Formule d’interpolation pour la derivée d’un polynome trigonométrique, C. R. Math. Acad. Sci. Paris, 158, 1152-1154 (1914) · JFM 45.0404.02
[33] Riesz, M., Sur le problème des moments et le théorème de Parseval correspondent, Acta Litt. ac Scient. Univ. Hung., 1, 209-225 (1922-23) · JFM 49.0708.02
[34] Schmidt, E., Über die nebst ihren Ableitungen orthogonalen Polynomensysteme und das zugehörige Extremum, Math. Anal., 119, 165-204 (1944) · Zbl 0028.39402
[35] Turán, P., Remark on a theorem of Ehrhard Schmidt, Mathematica (Cluj), 2, 373-378 (1960) · Zbl 0101.04702
[36] Wall, H. S.; Wetzel, M., Quadratic forms and convergence regions for continued fractions, Duke Math. J., 11, 89-102 (1944) · Zbl 0060.16504
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