Nearly Comonotone Approximation Ii

When we approximate a continuous function f which changes its monotonicity nitely many, say s times, in ?1; 1], we wish sometimes that the approximating polynomials follow these changes in monotonicity. However, it is well known that this requirement restricts very much the degree of approximation that the polynomials can achieve, namely, only the rate of ! 2 (f; 1=n) and even this not with a constant (dependent only on s), rather with a constant which depends on the location of the interior extrema. Recently, we proved that relaxing the comonotonicity requirement in intervals of length proportional to 1=n about the interior extrema of the function and in intervals of length 1=n 2 near the endpoints, what we called nearly comonotone approximation, allows the polynomials to achieve a pointwise approximation rate of ! 3 (moreover, with a constant which depends only on s). We show here that even when we relax the requirement of monotonicity of the polynomials on sets of measures approaching 0 (no matter how slowly or how fast), ! 4 is not reachable. We conclude the paper with results on the sizes of the deleted sets which allow the ! 3 degree of approximation in the norm; and when f is diierentiable, allow estimates involving the kth modulus of smoothness of f 0. x1. Introduction Let f 2 C ?1; 1] change monotonicity nitely many, say s 1 times, on I := ?1; 1]. Speciically, let Y s be the set of all collections Y := fy i g s i=1 of points, ?1 < y s < ::: < y 1 < 1. In the proof it will be convenient to put y s+1 := ?1 and y 0 := 1. For Y 2 Y s we set (x) = (x; Y) := s Y i=1 (x ? y i); and denote by (1) (Y) the collection of functions f 2 C ?1; 1], which change monotonicity at the points y i , 1 i s, and which are nondecreasing in (y 1 ; 1), that is, f is nonde-creasing in the intervals (y 2j+1 ; y 2j) and it is nonincreasing in (y 2j ; y 2j?1). A polynomial p n 2 P n , the space of polynomials of degree not exceeding n, is said to be comonotone with f, on a set E I, if and only if p 0