L_p空间中的代数多项式逼近

Algebraic Polynimial Approximation in L_p (0<p<1)

对于任意的ｆ∈Ｌｐ，０〈ｐ〈１Ｋ为正整数，存在ｎ阶代数多项式Ｐｎ，使得‖ｆ－Ｐｎ‖ｐ≤ｃω＾ψｋ（ｆ，１／ｎ）这里ω＾ψｋ（ｆ，ｔ）是ｆ在Ｌｐ中的ｐｉｔｚｉａｎ－Ｔｏｔｉｋ光滑模，Ｃ是与Ｋ，Ｐ有关的常数，如果ｆ是不减的，Ｋ≤２，则Ｐｎ也是不减的，文中证明了Ｋ≥３不成立，如果ｆ是凸的，Ｋ≤２，则Ｐｎ是凸的，文中证明了Ｋ≥４不成立。

That for f∈L_p0<p<1 and k a positive integer there exists an algebraic polynomial P_n of degree≤n such that whereω_K~ψ (f, t)is the Ditzian-Totik modulus of smoothness of in L_p and C is a constant depending only on k and p. 2f f is nondecreasing and k≤2 then the polynomial P_n can also be taken to be nondecreasing we prove this cannot hold for k≥3,If f is convex and k≤2,then the polynimial P_n can also be taken to be convex we prove this cannot hold for k ≥4.