A study of positive linear operators by the method of moments

Abstract

Let Q a compact and convex subset of R('k), k (GREATERTHEQ) 1 and let {L(,j)}(,j(ELEM)(, )) be a sequence of positive linear operators from C('n)(Q), (n (ELEM) ('+)) to C(Q). The convergence of L(,j) to the identity operator I is closely related to the weak convergence of a sequence of finite measures (mu)(,j) to the unit (Dirac) measure (delta)(,x0), x(,0) (ELEM) Q.New estimates are given for the remainder (VBAR)(INT)(,Q)f d(mu)(,j) -- f(x(,0))(VBAR), where f (ELEM) C('n)(Q). Using moments methods, Shisha-Mond type best or nearly best upper-bounds are established for various choices of k, Q, n and given moments of (mu)(,j). Some of them lead to attainable inequalities. The optimal functions/measures are typically spline functions and finitely supported measures. The corresponding inequalities involve various measures of smoothness of f such as the first or second modulus of continuity of f('(n)), the Peetre K-functional of f or certain modifications and generalizations.Finally some miscellaneous sharp inequalities are obtained. These lead to Korovkin type convergence theorems relative to ratios of Fourier-Stieltjes hyperbolic coefficients.