A Sharp Bernstein-type Inequality for Exponential Sums

A subtle Bernstein-type extremal problem is solved by establishing the equality sup 06=f∈e E2n |f ′(0)| ‖f‖[−1,1] = 2n − 1 , where e E2n := ( f : f(t) = a0 + n X j=1 aje λjt + bje −λjt , aj , bj , λj ∈ R ) . This settles a conjecture of Lorentz and others and it is surprising to be able to provide a sharp solution. It follows fairly simply from the above that 1 e − 1 n − 1 min{y − a, b − y} ≤ sup 06=f∈En |f ′(y)| ‖f‖[a,b] ≤ 2n − 1 min{y − a, b − y} for every y ∈ (a, b), where En := ( f : f(t) = a0 + n X j=1 aje λjt , aj , λj ∈ R ) . The proof relies on properties of the particular Descartes system (sinh λ0t , sinhλ1t , . . . , sinhλnt) , 0 < λ0 < λ1 < · · · < λn for which certain comparison theorems can be proved. Essentially sharp Nikolskii-type inequalities are also proved for En. 1991 Mathematics Subject Classification. Primary: 41A17.