Lipschitz-Nikolskiimath constants and asymptotic simultaneous approximation on theM n -operatorsconstants and asymptotic simultaneous approximation on theM n -operators

This note is a study of approximation of classes of functions and asymptotic simultaneous approximation of functions by theM _n -operators of Meyer-König and Zeller which are defined by $$(M_n f)(x) = (1 - x)^{n + 1} sumlimits_{k = 0}^infty {fleft( {frac{k}{{n + k}}} right)} left( begin{array}{l} n + k k end{array} right)x^k , n = 1,2,....$$ Among other results it is proved that for 0<α≤1 $$mathop {lim }limits_{n to infty } n^{alpha /2} mathop {sup }limits_{f in Lip_1 alpha } left| {(M_n f)(x) - f(x)} right| = frac{{Gamma left( {frac{{alpha + 1}}{2}} right)}}{{pi ^{1/2} }}left{ {2x(1 - x)^2 } right}^{alpha /2} $$ and if for a functionf, the derivativeD ^m+2 f exist at a pointx∈(0, 1), then $$mathop {lim }limits_{n to infty } 2n[D^m (M_n f) - D^m f] = Omega f,$$ where Ω is the linear differential operator given by $$Omega = x(1 - x)^2 D^{m + 2} + m(3x - 1)(x - 1)D^{m + 1} + m(m - 1)(3x - 2)D^m + m(m - 1)(m - 2)D^{m - 1} .$$