Polynomial Approximation in Lt , ( 0 < p < 1 )

We are interested in the approximation of functions f ~ Lp(I), 0 < p < 1, I = [ 1 , 1], by algebraic polynomials. Such approximation has previously been studied by other authors, most notably, Storozhenko, Krotov, and Oswald [-S-K-O] and Khodak [K]. Our main departure from these previous works is that we shall prove direct estimates for the error of polynomial approximation in terms of the Ditzian-Totik modulus of smoothness. This modulus measures smoothness differently at the endpoints of I than in the interior. Such dependence on the position of the point is crucial if we wish to characterize functions with a certain error of polynomial approximation (see [D-T]). We, however, do not in this paper discuss inverse estimates in terms of this modulus. A second variant of our work is to consider the approximation of monotone functions by monotone algebraic polynomials in Lp, 0 < p < 1. We establish the same estimates as for the unconstrained case but only for the firstand secondorder moduli. There is a result of Shvedov I-S] which says that such estimates cannot hold for smoothness order greater than 2. The usual estimates for approximating f ~ Lp(I) by algebraic polynomials are described in terms of the ordinary kth order modulus-of smoothness of f. If J is