The Degree of Convergence of Sequences of Linear Positive Operators

Abstract

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 <S IjfjI -IlLn(l) -1|| + |lLn(l) + 111W(An)1 (1) where (see Remark b) An = 1 1 (Ln( [t-X]2)) (X)II 1/2(2) and II stands for the sup norm over [a,b]. In particular, if Ln(1) = 1, as is often the case, (1) reduces to 1f -LX(f)JI 2w(Mn).(3)Remarks: (a) In forming Ln([tx]2), the x is held fixed, while t forms the functions t,t2 on which L, operates. A similar convention is used throughout. 1196

ON A SEQUENCE OF KANTOROVICH TYPE OPERATORS VIA RIEMANN TYPE q-INTEGRAL

“…Using the fact L n,qn (e 0 ; x) = 1, from the well-known result of Shisha and Mond [29], it follows that…”

On certain GBS-Durrmeyer operators based on $q$-integers

“…Shisha and Mond [28] established an important result meant to evaluate the approximation degree of continuous and differentiable univariate real valued functions by linear positive operators using the modulus of continuity. Following the ideas from [28], Badea et al in [7] proved the following Shisha-Mond-type theorem to evaluate the approximation degree for B -continuous functions using GBS operators.…”

“…Following the ideas from [28], Badea et al in [7] proved the following Shisha-Mond-type theorem to evaluate the approximation degree for B -continuous functions using GBS operators. …”

The degree of approximation by positive operators on compact connected abelian groups

“…Censor (see [3], Theorem 2 and the remarks immediately preceding it) gave a version of this result for the multidimensional torus T". For other results along these lines, including the case for algebraic polynomials on the unit interval, see [4], [6], [7], [8], [10] and [11]. In a new direction Nishishiraho [9] has given a quantitative version of Korovkin's theorem for compact subsets of a locally convex Hausdorff space, which includes the case where the underlying space is real Euclidean space.…”

“…[21 Approximation by positive linear operators 365 in a quantitative form by Shisha and Mond (see [10], Theorem 3 and [11]) in which the rate of convergence of (T n f) is estimated in terms of that of (T n l) and {T n e x ) and the modulus of continuity of/. Censor (see [3], Theorem 2 and the remarks immediately preceding it) gave a version of this result for the multidimensional torus T".…”