Comparing the degrees of unconstrained and constrained approximation

יום ב', 21/03/2016 - 14:00

It is quite obvious that one should expect that the degree of constrained approximation
be worse than the degree of unconstrained approximation. However, it turns out that in certain cases
we can deduce the behavior of the degrees of the former from information about the latter.

Let $E_n(f)$ denote the degree of approximation of $f\in C[-1,1]$,
by algebraic polynomials of degree $<n$, and assume that we know
that for some $\alpha>0$ and $\Cal N\ge1$,
$$n^\alpha E_n(f)\leq1,\quad n\geq\Cal N.$$
Suppose that $f\in C[-1,1]$, changes its monotonicity or convexity $s\ge0$ times in $[-1,1]$ ($s=0$ means that $f$
is monotone or convex, respectively). We are interested in what may be said about its degree of
approximation by polynomials of degree $<n$ that are comonotone or coconvex with
$f$. Specifically, if $f$ changes its monotonicity or convexity at
$Y_s:=\{y_1,\dots,y_s\}$ ($Y_0=\emptyset$) and the degrees of comonotone and coconvex approximation
are denoted by $E^{(q)}_n(f,Y_s)$, $q=1,2$, respectively. We investigate when can one say that
$$n^\alpha E^{(q)}_n(f,Y_s)\le c(\alpha,s,\Cal N),\quad n\ge\Cal N^*,$$
for some $\Cal N^*$. Clearly, $\Cal N^*$, if it exists at all (we prove it
always does), depends on $\alpha$, $s$ and $\Cal N$. However, it turns
out that for certain values of $\alpha$, $s$ and $\Cal N$, $\Cal N^*$ depends also
on $Y_s$, and in some cases even on $f$ itself, and this dependence is essential.