Milman′s modulus of smoothness and normal structure

By using of the equivalent relation between the DitzianTotik modulus of smoothness ω2φλ(f,t)(0λ1) and the Kfunctional K2φλ(f,t2),both the direct and inverse approximation theorems of modified LupasBaskakov operators are discussed,and the equivalent results are obtained,which combine the pointwise(λ=1 )and global(λ=0 )approximation equivalent theorems.

Using the relation between the weighted modulus of smoothness and the weighted main-part modulus of smoothness,we discuss the pointwise direct and equivalent approximation theorem with Jacobi weight for Beta operator.

We use the theory of modulus of smoothness of Besov space and construct a smooth discriminant function in time domain.

In this paper, we obtain an equivalent form of the convex modulus and the smooth modulus.

In this paper, we give a relation of convexity characteristic and smooth modulus in the Banach space.

In this paper,we introduce a new moduli of smoothness ω~r_(φ~λ)(f,r) to give the pointwise results of approximation with a new integrable Bemstein operator to supplement and perfect the previous results.

To study the operators[2] which don t satisfy Ln(t,x)=x,we introduce a new K~ functional in this paper,and give the relate between it and moduli of smoothness.