Littlewood-Type Problems on [0,1]

Abstract

We consider the problem of minimizing the uniform norm on [0, 1] over non-zero polynomials p of the form p ( x ) = ∑ j = 0 n a j x j with | a j | ⩽ 1 , a j ∈ C , where the modulus of the first non-zero coefficient is at least δ > 0. Essentially sharp bounds are given for this problem. An interesting related result states that there are absolute constants c1 > 0 and c2 > 0 such that exp ⁡ ( − c 1 n ) ⩽ inf 0 ≠ p ∈ F n ∥ p ∥ [ 0 , 1 ] ⩽ exp ⁡ ( − c 2 n ) , for every n ⩾ 2, where Fn denotes the set of polynomials of degree at most n with coefficients from {−1, 0, 1}. This Chebyshev-type problem is closely related to the question of how many zeros a polynomial from the above classes can have at 1. We also give essentially sharp bounds for this problem. Inter alia we prove that there is an absolute constant c > 0 such that every polynomial p of the form p ( x ) = ∑ j = 0 n a j x j , with | a j | ⩽ 1 , | a 0 | = | a n | = 1 , a j ∈ C , has at most c n real zeros. This improves the old bound c n log ⁡ n given by Schur in 1933, as well as more recent related bounds of Bombieri and Vaaler, and, up to the constant c, this is the best possible result. All the analysis rests critically on the key estimate stating that there are absolute constants c1 > 0 and c2 > 0 such that | f ( 0 ) | c 1 / a ⩽ exp ⁡ ( c 2 / a ) ∥ f ∥ [ 1 − a , 1 ] , for every f ∈ S and a ∈ (0, 1], where S denotes the collection of all analytic functions f on the open unit disk D:= {z ∈ C: ∣z∣ < 1} that satisfy | f ( z ) | ⩽ 1 1 − | z | for z ∈ D .