Treffer: Optimal Constant in Approximation by Bernstein Operators: Optimal constant in approximation by Bernstein operators

Let \(B[0,1]\) be the space of bounded real-valued functions defined on \([0,1]\) endowed with the sup-norm denoted by \(\|\cdot\|\), \(C[0,1]\) be the subspace of continuous functions, \(\|\cdot\|= \|\cdot\|_{C[0,1]}\), and \(\Pi_k\) be the spaces of algebraic polynomials of degree at most \(k\). The Bernstein polynomial of degree \(n\) is given by \[ B_n(f, x)= \sum^n_{k=0} f\Biggl({k\over n}\Biggr) {n\choose k} x^k(1- x)^{n-k},\qquad x\in [0,1]. \] The second-order modulus of continuity of \(f\) is given by \[ \omega_2(f, h)= \max\{|\Delta^2_\rho f(x)|,\, x,x+2\rho\in [0,1],\,0< \rho\leq h\},\quad h> 0. \] The author proves the following \[ \sup_{f\in B[0,1]\setminus\Pi_1} {\| B_n(f)- f\|\over \omega_2(f,{1\over \sqrt{n}})}= \sup_{f\in C[0,1]\setminus\Pi_1} {\| B_n(f)- f\|\over \omega_2(f,{1\over \sqrt{n}})}= 1. \] From this result it follows that \(C= 1\) is the smallest constant for which the inequality \[ \| B_n(f)- f\|\leq C\omega_2\Biggl(f, {1\over\sqrt{n}}\Biggr) \] holds on the class of continuous functions \(f\), as well as on the class of bounded functions \(f\).