On the sharpness of a Jackson estimate by Ditzian-Totik (Q1925088)

Let \(C[-1,1]\) be the space of functions \(f\) continuous on \([-1,1]\) with uniform norm \(|\cdot |\) and let \(\varepsilon_n (f,x)\) be the best approximation to \(f\) from the set \(P_n\) of algebraic polynomials of degree \(\leq n\). The weighted modulus of smoothness of order \(r\) according to Ditzian and Totik is denoted by \[ \omega^r_\varphi (f;t): =\sup_{0\leq h\leq t} \Delta^r_{h\varphi(x)} f(x), \quad \varphi(x): =(1-x^2)^{1/2} \] where \[ \Delta^r_h f(x)= \begin{cases} \sum^r_{k=0} (-1)^k {r\choose k} f\Bigl(x+{rh \over 2} -kh\Bigr) \quad & \text{if } \Bigl|x\pm {rh\over 2} \Bigr|\leq 1 \\ 0 & \text{otherwise}. \end{cases} \] Ditzian and Totik had shown that for \(f\in C[-1,1]\) \(|\varepsilon_n(f;x) - f(x) |\leq C \omega^r_\varphi (f; {1\over n})\), \(x\in [-1,1]\). Here the author shows that this result is sharp. Consider a function \(\omega_r(t)\) satisfying the following four properties: 1) \(0= \omega_r(0) < \omega_r(t) \leq \omega_r (T)\), \(0<t<T\), 2) \(\omega_r(t)\) is continuous for \(t\geq 0\), 3) \({T^r \over \omega_r(T)} \geq {t^r\over \omega_r(t)}\) if \(T\geq t>0\), 4) \(\lim_{t\to 0+} {t^r\over \omega_r (t)} = 0\). Here the author proves the following theorem: For each \(\omega_r(t)\) satisfying (1)--(4), there exists a function \(f\in C[-1,1]\) such that \(\omega_\varphi^r(f;t) = O(\omega_r(t))\) and \[ \limsup_{n\to\infty} {\bigl|\varepsilon_n(f;x)-f(x) \bigr|\over\omega_r (1/n)} \geq \varepsilon >0 \] for each \(x\in[-1,1]\). This shows the pointwise sharpness of Ditzian Totik's result. The author used ideas of Grünwald, Marcinkiewicz and Vertesi on divergence of interpolation processes to prove his pointwise result.