We give direct approximation theorems for some linear operators in certain weighted spaces. The results are given in terms of some DitzianTotik moduli of smoothness.

We prove that given a bialgebra surjection π : E → H with nilpotent kernel such that H is a Hopf algebra which is formally smooth as a K-algebra, then π has a section which is a right H-colinear algebra homomorphism. Moreover, if H is also endowed with an ad-invariant integral, then the section can be chosen to be H-bicolinear. We also deal with the dual case. As an application, we prove that e...

stabilizers have been used to improve the texture of ice cream for several decades. they added it to ice cream to control ice and lactose crystal growth during hardening and storage, to give body and stiffness during freezing for air incorporation and to impart smoothness in body and texture. nowadays, most of the industrial stabilizers are provided from abroad, so, extraction, determination of...