On the Bernstein's constant in convex approximation
aa r X i v : . [ m a t h . C A ] F e b ON THE BERNSTEIN’S CONSTANT IN CONVEX APPROXIMATION
SORIN G. GAL
Abstract.
Denoting by E (+2) n ( f ) the best uniform approximation of f by convex polyno-mials of degree ≤ n , there is an open question if there exists the limit lim n →∞ n λ E (+2) n ( | x | λ )for λ ≥ Introduction
A famous result of Bernstein [1], [2] states that for any λ > λ not even integer, thereexists finite the limit lim n →∞ n λ E n ( | x | λ ) >
0, where E n ( | x | λ ) = inf { max {| P ( x ) − | x | λ | ; x ∈ [ − , } ; P ∈ P n ( R ) } , and P n ( R ) denotes the set of all real polynomials (that is with real coefficients) of degree ≤ n .It is worth noting that the exact values of the above limits are not known. Details andgeneralizations of these results, can be found in the papers [4]-[6], [10] and in the book [7].In this note we consider a similar problem in the case of the best approximation by convexpolynomials, which was first mentioned in [3], p. 325.2. Convex Approximation
Taking into account that | x | λ is convex for λ ≥
1, denoting E (+2) n ( f ) = inf {k f − P k ∞ ; P ∈ P n ( R ) , P ′′ ( x ) ≥ , ∀ x ∈ [ − , } , it is natural to consider the following open question. Open Question 1.
There exists finite the limit lim n →∞ n λ E (+2) n ( | x | λ ) for λ ≥ ? Note that since by [8], for f convex on [ − ,
1] and λ > E n ( f ) = O ( n − λ ) iff E (+2) n ( f ) = O ( n − λ ) , n → ∞ , it easily follows that the sequence ( n λ E (+2) n ( | x | λ )) n ∈ N for λ ≥ Open Question 2. If f is convex on R and the sequence ( λ n ) n satisfies the conditionsin [7] , p. 3, then it is valid that we have lim n →∞ E (+2) n (cid:18) f ; L ∞ (cid:20) − n (1 − λ n ) σ , n (1 − λ n ) σ (cid:21)(cid:19) < ∞ ( or ≤ A (+2) σ ( f, L ∞ ( R )) < ∞ ) ?Here A (+2) σ ( f, L ∞ ( R ) = inf {k f − g k L ∞ ( R ) ; g ∈ B σ , g convex on R } and B σ denotes the classof all entire functions of exponential type σ .This open question is partially supported by the fact that by [9], for a convex function f we have E (+2) n ( f ) L p ≤ E n − ( f ′′ ) L p , for all n ≥ p = ∞ and this estimate cannot beimproved. References [1] S. N. Bernstein,
Sur la meilleure approximation de | x | par les polynomes des degr´es donn´es , ActaMathematica, (1914), 1-57.[2] S. N. Bernstein, On the best approximation of | x | p by polynomials of very high degree , Izv. Math. Nauk,SSSR, 1938, 169-180.[3] S. G. Gal, Approximation by Complex Bernstein and Convolution Type Operators , World ScientificPubl., New Jersey, London, Singapore, Beijing, Shanghai, Hong Kong, 2009.[4] M. I. Ganzburg,
Limit theorems for the best polynomial approximations in the L ∞ metric , UkrainianMath. J., (1991), 299-305.[5] M. I. Ganzburg, Limit theorems and best constants of approximation theory , In : Handbook on AnalyticComputational Methods in Applied Mathematics (G. Anastassiou, ed.), Boca Raton, FL: CRC Press,2000.[6] M. I. Ganzburg,
The Bernstein constant and polynomial interpolation at the Chebyshev nodes , J. Approx.Theory, (2002), 193-213.[7] M. I. Ganzburg,
Limit Theorems of Polynomial Approximation with Exponential Weights , Memoirs ofthe American Mathematical Society, vol. , No. , Providence, Rhode Island, 2008.[8] K. A. Kopotun, D. Leviatan and I. A. Shevchuk,
Coconvex approximation in the uniform norm-the finalfrontier , Acta Math. Hungarica, (2006), No. 1-2, 117-151.[9] D. Leviatan and I. A. Shevchuk,
Counterexamples in convex and higher order constrained approximation ,East J. Approx., (1995), 391-398.[10] D. S. Lubinsky, On the Bernstein constants of polynomial approximation , Constr. Approx., (2007),303-366. Department of Mathematics and Computer Science, University of Oradea, 410087 Oradea,Romania
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