Generalized monotone approximation inL _p Space

Abstract

Letf(x) ∈L _p[0,1], 1⩽p⩽ ∞. We shall say that functionf(x)∈Δ^k (integerk⩾1) if for anyh ∈ [0, 1/k] andx ∈ [0,1−kh], we have Δ ^k_h f(x)⩾0. Denote by ∏ _n the space of algebraic polynomials of degree not exceedingn and define

$$E_{n,k} (f)_p : = \mathop {\inf }\limits_{\mathop {P_n \in \prod _n }\limits_{P_n^{(\lambda )} \geqslant 0} } \parallel f(x) - P_n (x)\parallel _{L_p [0,1]} .$$

We prove that for any positive integerk, iff(x) ∈ Δ^k ∩ L _p[0, 1], 1⩽p⩽∞, then we have

$$E_{n,k} (f)_p \leqslant C\omega _2 \left( {f,\frac{1}{n}} \right)_p ,$$

whereC is a constant only depending onk.