Polynomials close to 0 resp. 1 on disjoint sets - Totik Vilmos (JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 2020)
For disjoint compact subsets I, J of a real interval [A, B] construction is given for polynomials P-n of degree n = 1,2,... that approximate 0 on I and 1 on J with geometric rate, vanish (in a given order) at finitely many given points of I, take the value 1 (in a given order) at finitely given points of J, and otherwise lie in between 0 and 1 on [A, B] . When I and J consist of alternating intervals, then P-n can also be monotone on each subinterval of [A, B] \ (I U J) . Some further consequences (like approximation of piecewise constant functions or the trigonometric variant) are also considered. (C) 2019 Elsevier Inc. All rights reserved.