Pointwise Remez- and Nikolskii-type inequalities for exponential sums
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Let (formula presented) So En is the collection of all n + 1 term exponential sums with constant first term. We prove the following two theorems. Theorem 1 (Remez-type inequality for En at 0). Let s ∈ (0, 1/2]. There are absolute constants C1 > 0 and c2 > 0 such that (formula presented) where the supremum is taken for all f ∈ En satisfying (formula presented) Theorem 2 (Nikolskii-type inequality for En). There are absolute constants c1 > 0 and c2 > 0 such that (formula presented) for every a < y < b and q > 0. It is quite remarkable that, in the above Remez-and Nikolskii-type inequalities, En behaves like Pn, where Pn denotes the collection of all algebraic polynomials of degree at most n with real coefficients.
author list (cited authors)
Borwein, P., & Erdélyi, T.