Abstract. Revisiting the results in [L1, L2], we consider the polynomial approximation on $(-1,1)$ with the weight $w(x)={\rm e}^{-(1-x^2)^{-\alpha }}$, $\alpha >0$. We introduce new moduli of smoothness, equivalent to suitable $K$-functionals, and we prove the Jackson theorem, also in its weaker form. Moreover, we state a new Bernstein inequality, which allows us to prove the Salem--Stechkin inequality. Finally, also the behaviour of the derivatives of the polynomials of best approximation is discussed.

AMS Subject Classification (1991): 41A10, 41A17, 41A25, 41A27

Keyword(s): Jackson theorems, Markoff--Bernstein inequalities, orthogonal polynomials, approximation by polynomials, one-sided approximation

Received November 10, 2009, and in revised form February 3, 2010. (Registered under 6175/2009.)