Generalized Monotone Approximation in L_p Space

【摘要】 <正> Let f（x）∈L_p[0,1],1≤p≤∞. We shall say that function f（x）∈△^k（integer k≥1） if for anyh∈[0, 1/k]and x∈[0,1-kh], we have △_h^kf（x）≥0. Denote by ∏_n the space of algebraic polynomials of degreenot exceeding n and defineE_n,k（f）_p:=inf||f（x）-P_n（x）||L.[0,1] We prove that for any positive integer k, if f（x）∈△^k∩L_p[0,1], 1≤p≤∞, then we haveE_n,k（f）_p≤Cω₂（f,1/n）_pwhere C is a constant only depending on k.更多还原

【Abstract】 Let f（x）∈L_p[0,1],1≤p≤∞. We shall say that function f（x）∈△^k（integer k≥1） if for any h∈[0, 1/k]and x∈[0,1-kh], we have △_h^kf（x）≥0. Denote by ∏_n the space of algebraic polynomials of degree not exceeding n and define E_n,k（f）_p:=inf||f（x）-P_n（x）||L.[0,1] We prove that for any positive integer k, if f（x）∈△^k∩L_p[0,1], 1≤p≤∞, then we have E_n,k（f）_p≤Cω₂（f,1/n）_p where C is a constant only depending on k.更多还原