Best Approximation by Logarithmically Concave Classes of Functions

Abstract

The paper contains results on best approximation by logarithmically concave classes of functions. For example, we prove the following: Let $\mathcal{P}_{c}$ denote the class of real polynomials, having − 1 and 1 as consecutive zeros, and whose zeros z _k  = x _k +i y _k , i ² = −1, satisfy the inequality | y _k  | ≤ | x _k  | − 1. Let i(x) = 1, x ∈ [−1, 1] be the unit function on the interval [−1, 1] and 1 ≤ p < ∞. Then, there exists a unique constant c _p such that

$$\displaystyle{ \inf _{q\in \mathcal{P}_{c}}\int _{-1}^{1}\vert i(x) - q(x)\vert ^{p}dx =\int _{ -1}^{1}\vert i(x) - c_{ p}(1 - x^{2})\vert ^{p}dx. }$$

The exact values of the best approximation are found in the particular cases p = 1 and p = 2.