On Certain Orthogonal Polynomials, Nikolski- and Turán-Type Inequalities, and Interpolatory Properties of Best Approximants Original Research Article

For f ∈ C[−1, 1] denote by Bn,p(f) its best Lp-approximant by polynomials of degree at most n (1 ≤ p ≤ ∞). The following statement is the main result of the paper: Let 1 < p ≤ ∞, f ∈ C[−:1, 1], and assume that for a given (a,b) ⊂ [−:1, 1] there exists a sequence of integers n1 < n2 < ··· < nj < ··· such that f − Bnj, p(f) is zero free on (a, b). Then lim supj → ∞nj + 1/nj< 1.