Upper Bounds for Bernstein Basis Functions
From Markov’s bounds for binomial coefficients (for which a short proof is given) upper bounds are derived for Bernstein basis functions of approximation operators and their maximum. Some related inequalities used in approximation theory and those for con
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Abstract From Markov’s bounds for binomial coefficients (for which a short proof is given) upper bounds are derived for Bernstein basis functions of approximation operators and their maximum. Some related inequalities used in approximation theory and those for concentration functions are discussed. Keywords Bernstein basis functions for approximation operators • Markov bounds for binomial coefficients • Zeng’s upper bounds for binomial probabilities • Extension of upper bounds for binomial probabilities via discretization of the argument. Rogozin’s and some other inequalities for concentration functions
Mathematics Subject Classification (2010): 41A36, 41A44, 60E15, 60G50
V. Gupta () School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi-110078, India e-mail: [email protected] T. Shervashidze A. Razmadze Mathematical Institute, 1, M. Aleksidze St., Tbilisi 0193, Georgia I. Vekua Institute of Applied Mathematics, I. Javakhishvili Tbilisi State University, 2, University St., Tbilisi 0186, Georgia e-mail: [email protected] A.N. Shiryaev et al. (eds.), Prokhorov and Contemporary Probability Theory, Springer Proceedings in Mathematics & Statistics 33, DOI 10.1007/978-3-642-33549-5 17, © Springer-Verlag Berlin Heidelberg 2013
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V. Gupta and T. Shervashidze
1 Markov’s Bounds for Binomial Coefficients. Preliminaries One can get upper bounds for Bernstein basis functions of approximation operators, i.e., binomial probabilities b.kI n; p/ D Cnk p k .1 p/nk ;
p 2 Œ0; 1;
k D 0; 1; : : : ; n;
using direct analytic or probabilistic methods. First estimates of b.kI n; p/ can be found in “Ars Conjectandi” by J. Bernoulli, see [3] and commentary by Yu.V. Prokhorov “Law of Large Numbers and Estimates for Probabilities of Large Deviations” on pp. 116–155 in the same [3]. Using an additional argument together with one to obtain the Stirling formula Markov proved the double inequality for binomial coefficients Cnk which we prefer to write in the form of bounds for b.kI n; p/ (see [12], pp. 72, 73 or formula (16) on p. 135 in above mentioned commentary in [3]; cf. formula (135) in Chap. IV “The rate of approximation of functions by linear positive operators” of [11]): Theorem A. Let n 1, k 1, n k 1 and p 2 .0; 1/. Then np k n.1 p/ nk n 2k.n k/ k nk r np k n.1 p/ nk n < b.kI n; p/ < DW Ma.kI n; p/: 2k.n k/ k nk
1
1
1
e 12n 12k 12.nk/
r
(1)
Let us give a short proof of (1) with 1=.12n C 1/ instead of 1=.12n/ in the exponent in the left-hand side. Proof. The proof is based on the double inequality which refines Stirling asymptotics (2) .2/1=2 nnC1=2 e nC1=.12nC1/ < nŠ < .2/1=2 nnC1=2 e nC1=.12n/ (see Feller’s book [5], Chap. II, and Robbins’ paper [15] referred therein). Due to (2) we have Cnk D nŠ=ŒkŠ.n k/Š < Œn=.2 k.n k//1=2 nn k k .n k/.nk/ expŒ1=.12n/ 1=.12k C 1/ 1=.12.n k/ C 1/:
(3)
The nominator of the latter exponent equals to .12k C 1/.12.n k/ C 1/ 12n.12n C 2/ D 144Œk.n k/ .1=4/n2 108n
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