Weak copositive and interwining approximation
The authors continue their research into the question of the degree of approximation of functions in \(L_p[0,1]\), by polynomials satisfying various constraints. If a function changes its sign finitely many times in \([0,1]\), the approximation by polynomials which keep the sign of the function throughout the interval is called copositive approximation. For a general function \(f\), the concept of one-sided approximation is that of approximating \(f\) by pairs of polynomials \(p_n\) and \(q_n\) so that \(p_n\leq f\leq q_n\) throughout the interval. Recently [Can. J. Math. 49, No. 1, 74-99 (1997; Zbl 0879.41006)], the authors have introduced a concept which is strongly related to the two notions described above. Namely, the concept of intertwining approximation which for a prescribed finite collection of points in \([0,1]\), considers the approximation of \(f\) by pairs of polynomials \(p_n\) and \(q_n\) so that both interpolate \(f\) at the prescribed points and both alternate above and below \(f\) as they pass through these points and keep \(f\) in between them at all time. That is, the polynomials \(p_n\) and \(q_n\) are such that, say, \[ \text{sgn}(-1)^j(p_n(x)-f(x))= \text{sgn}(-1)^j(f(x)-q_n(x)),\tag{*} \] in the \(j\)th subinterval formed by the prescribed collection of points. Here, following recent work of \textit{I. A. Shevchuk} and the reviewer [J. Approximation Theory 95, 53-81 (1998; Zbl 0911.41003)], they investigate the improvement that can be achieved in the degree of copositive and of intertwining approximation if we relax a bit the constraints by requiring in the case of copositive approximation, that the polynomials keep the sign of \(f\), except in small neighborhoods of the points where \(f\) changes its sign; and in the case of intertwining approximation, that the pairs of polynomials satisfy (*) except in small neighborhoods of the prescribed points. Yet another possibility which is a bit more restrictive is that the pairs will be intertwining but not with respect to the original collection of points, rather with respect to some points which are close to the original ones.
scientific article; zbMATH DE number 1271290
scientific article