We prove that, for comonotone approximation, an analog of the second Jackson inequality with generalized Ditzian-Totik modulus of continuity ω ϕ k r , is not true for ( k , r ) = ( 2, 2 ) even in the case where the constant depends on a function.

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function f \in W^r [0,1] by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function f \in C^r [0,1] \Wedge \Delta^0, where \Delta^0 is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider r \in N, r>2. In [8] is consider r \in R, r>2. It was proved that for monotone approximation estimates of the form (1) are fails for r \in R, r>2. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6],[11]). In [5] is consider r \in N, r>2. It was proved that for convex approximation estimates of the form (1) are fails for r \in N, r>2. In [6] is consider r \in R, r\in(2;3). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(2;3). In [11] is consider r \in R, r\in(3;4). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(3;4). In [9] is consider r \in R, r>4. It was proved that for f \in W^r [0,1] \Wedge \Delta^2, r>4 estimate (1) is not true. In this paper the question of approximation of function f \in W^r [0,1] \Wedge \Delta^2, r>4 by algebraic polynomial p_n \in \Pi_n \Wedge \Delta^2 is consider. It is proved, that for f \in W^r [0,1] \Wedge \Delta^2, r>4, estimate (1) can be improved, generally speaking.

We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function $f є W^r [0,1]$ by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function $f є C^r [0,1] \cap \Delta^0$ where $\Delta^0$ is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider $r є , r > 2$. In [8] is consider $r є , r > 2$. It was proved that for monotone approximation estimates of the form (1) are fails for $r є , r > 2$. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6]). In [5] is consider $r є , r > 2$. In [6] is consider $r є , r > 2$. It was proved that for convex approximation estimates of the form (1) are fails for $r є , r > 2$. In this paper the question of approximation of function $f є W^r \cap \Delta^1, r є (3,4)$ by algebraic polynomial $p_n є \Pi_n \cap \Delta^1$ is consider. The main result of the work generalize the result of work [8] for $r є (3,4)$.

No abstract