Constructive approximation by monotonous polynomial sequences in \(Lip_ M\alpha\), with a\(\in (0,1]\) (Q582493)

The principal result of this paper is the following theorem: If \(f\in lip_{M^{\alpha}}\) (where (0,1]), then for each p, a fixed integer \(\geq 3/\alpha\), the polynomial sequence defined by \(P_ n(x)=B_{n^ p}(f;x)+a_ n\), where \(a_ n=2K_ 0M\sum^{\infty}_{j=n}1/j^ 2\), \(B_ n(f;x)\) the Bernstein polynomials and \(K_ 0\) is Sikkema's constant, satisfies \(P_ n\to f\), uniformly on [0,1] and \[ f(x)<P_{n+1}(x)<P_ n(x),\quad \forall x\in [0,1], \] \(\forall n=1,2,... \).