Abstract. When we approximate a continuous function $f$ which changes its monotonicity finitely 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 $\omega_2(f,1/n)$ 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 $\omega_3$ (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), $\omega_4$ is not reachable. We conclude the paper with results on the sizes of the deleted sets which allow the $\omega_3$ degree of approximation in the norm; and when $f$ is differentiable, allow estimates involving the $k$th modulus of smoothness of $f'$.

AMS Subject Classification (1991): 41A10, 41A25, 41A29

Keyword(s): Approximation by polynomials, Shape preserving approximation

Received February 15, 1999. (Registered under 2725/2009.)