Approximation by Boolean sums of positive linear operators. II: Gopengauz-type estimates
A theorem of Gopengauz says that for every \(f\in C[-1,1]\) there exists an algebraic polynomial \(p_ n\) of degree n (depending on f) such that there holds \[ | f(x)-p_ n(x)| \leq c\omega_ 2(f,\sqrt{1-x^ 2}\cdot n^{-1}),\quad | x| \leq 1, \] where \(\omega_ 2\) denotes the classical second order modulus of continuity and c is an absolute constant independent of n, f and x. The main result of the present paper gives a construction for algebraic polynomials of degree sn-s, \(s\geq 3\), furnishing an estimate of the same type. This construction is based on the Boolean sum of positive linear and polynomial operators introduced by Pichugov and Lehnhoff respectively and certain interpolation operators. [For part I see Rend. Mat. Appl. VII., Ser. 6, No.4, 525-546 (1986; Zbl 0689.41019).]
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