ESTIMATES FOR FUNCTIONALS WITH A KNOWN, FINITE SET OF MOMENTS, IN TERMS OF MODULI OF CONTINUITY, AND BEHAVIOR OF CONSTANTS, IN THE JACKSON-TYPE INEQUALITIES

A new technique is developed for estimating functionals by moduli of continuity. The generalized Jackson inequality A(sigma-0)(f) <= {1/((2m)(m)) Sigma(m-1)(k=0) K-2k/(gamma pi)(2k) nu(k)(m) + K-2m/(gamma pi)(2m) nu(m)(m)/2(2m) } omega(2m) (f, gamma pi/sigma) is an example of such an estimate. Here r, m is an element of N, sigma, gamma > 0, a function f is uniformly continuous and bounded on R, A(sigma-0) is the best uniform approximation by entire functions of type less than sigma, omega(2m) is a uniform modulus of continuity of order 2m, K-s are the Favard constants, and nu(m) = 8/((2m)(m)) Sigma(l=0) (left perpendicular(m-1)/2right perpendicular) ((2m)(m-2l-1))/(2l + 1)(2), where left perpendicularxright perpendicular is the entire part of x. Similar inequalities are obtained for best approximations of periodic functions by splines. In some cases, the constants in inequalities are close to optimal.