  We study approximation of functions by algebraic polynomials in the H\"older
spaces corresponding to the generalized Jacobi translation and the
Ditzian-Totik moduli of smoothness. By using modifications of the classical
moduli of smoothness, we give improvements of the direct and inverse theorems
of approximation and prove the criteria of the precise order of decrease of the
best approximation in these spaces. Moreover, we obtain strong converse
inequalities for some methods of approximation of functions. As an example, we
consider approximation by the Durrmeyer-Bernstein polynomial operators.
