1. Withagivenrealfunctionf, onefrequently associates a sequence (Ln(f))n= of real functions (usually algebraic or trigonometric polynomials), used to approximate f. Quite often, for every given n, such an Ln is a linear operator that is positive, i.e., 5) 0 implies Ln(0) ) 0. For such operators, P. P. Korovkin' has recently proved some remarkable results. Thus, if Ljf converges uniformly to f in the particular cases f(t)-1, f(t)_t, f(t)t2 then it does so for every continuous, real f. Similarly, if Ln(f) converges uniformly tof forf(t)=l, cos t, sint it does so for every continuous, 27r-periodic real f. More generally, LX(f) converges uniformly to f, for every continuous real f, provided such a convergence holds when f = fo, f = fi, f = f2, where (fofi,f2) is a unisolvent (Tchebycheff) sequence of real continuous functions.2. Our purpose is to recast Korovkin's results in a quantitative form. Thus, for example, we shall estimate the rapidity of convergence of LX(f) to f, in terms of the rapidities of convergence of Ln(l) to 1, Ln(X) to x, and Ln(X2) to x2.
A linearpositive operator is a function L having the following properties. (a) The domain D of L is a nonempty set of real functions, all having the same real domain T. (b) For every f e D, L(f) is again a real function with domain T. (c) If f and g belong to D, and if a and b are reals, then af + bg e D, and L(af + bg) = aL(f) + bL(g). (d) Iff fe D, and f(x) ) 0 for every x e T, then L(f) (x) ) 0 for every x e T.Thus, if L is a linear positive operator and f, g e D, then f < g throughout T implies Lf ( Lg there, and If g throughout T implies Lf Lg there. 4. THEOREM 1. Let -o < a < b < X, and let L1,L2,. . . be linear positive operators, all having the same domain D which contains the restrictions of 1,tt2 to [a,b]. For n = 1,2,..., suppose Ln(1) is bounded. Let f e D be continuous in [ab], with modulus of continuity w. Then for n = 1,2,. . ., 11f -LX~)I
