Pointwise Approximation On Baskakov Type Operators

In this paper, we will introduce modulus of smoothness ωφλ2r(f,t) and weighted modulus of smoothness ωφλ2r(f ,t)w to discuss pointwise approximation on Baskakov-type operators. In the following we denote the linear combinations of Baskakov, Baskakov-Kantorovich and Baskakov-Durrmeyer operators by Vn,r(f,x), Kn,r(f,x)and Mn,r(f,x). Now we state our main work: (1). When 1 -1/r -a/r<λ≤1, by ωφλ2r(f ,t)w we study pointwise approximation on Vn,r(f,x) and obtain a direct theorem and an equivalent theorem and a local saturation result. Moreover, if 0 < α<2r, then when 1 -1/r ≤λ≤1, we give an equivalent theorem by ωφλ2r(f,t), when 0 ≤λ<1-1/r it is not true. (2). If 0 ≤λ≤1, for Vn,r(f,x) we obtain an equivalent theorem by ωφλ2r(f,t) when 0 < α min {2 (r +1)/(2?λ),2r}, it is not true. For Kn,r(f,x)and Mn,r(f,x), we can also obtain similar results. These results include some previous ones. (3). On discussing equivalent theorem we also obtain the following results. For Vn,r(f,x), when 1 -1/r ≤λ≤1, φ(x) can replace δn(x), when 0 ≤λ<1-1/r, r ≥2, it can not. For Kn,r(f,x) and Mn,r(f,x), when 0 ≤λ<1, r ≥1, δn(x) can not be replaced by ? (x), when λ=1, it is true. (4). To solve the inverse part we introduce a new K -functional. And we prove a weak inverse inequality, which gives a general way to solve the inverse theorem on studying pointwise approximation by ω?2λr (f ,t)w.