Fast decreasing and orthogonal polynomials

This paper reviews some aspects of fast decreasing polynomials and some of their recent use in the theory of orthogonal polynomials. 1. Fast decreasing polynomials Fast decreasing, or pin polynomials have been used in various situations. They imitate the ”Dirac delta” best among polynomials of a given degree. They are an indispensable tool to localize results and to create well localized ”partitions of unity” consisting of polynomials of a given degree. We use the setup for them as was done in [5], from where the results of this section are taken. Let Φ be an even function on [−1, 1], increasing on [0, 1], and suppose that Φ(0) ≤ 0. Consider e−Φ(x), and our aim is to find polynomials Pn of a given degree ≤ n such that (1.1) Pn(0) = 1, |Pn(x)| ≤ e−Φ(x), x ∈ [−1, 1]. Let nΦ = n be the minimal degree for which this is possible. The following theorem gives an explicitly computable bound for this minimal degree. Theorem 1.1. (Ivanov–Totik [5]) 1 6 NΦ ≤ nΦ ≤ 12NΦ, where NΦ = 2 sup Φ−1(0)≤x<Φ−1(1) √ Φ(x) x2