Bernstein-Type Inequalities for Linear Combinations of Shifted Gaussians

Let Pn be the collection of all polynomials of degree at most n with real coefficients. A subtle Bernstein-type extremal problem is solved by establishing the inequality ‖ U n ( m ) ‖ L q ( R ) ⩽ ( c 1 + 1 / q m ) m / 2 n m / 2 ‖ U n ‖ L q ( R ) for all U n ∈ G ˜ n , q ∈ ( 0 , ∞ ] , and m = 1, 2, …, where c is an absolute constant and G ˜ n ≔ { f : f ( t ) = ∑ j = 1 N P m j ( t ) e − ( t − λ j ) 2 , λ j ∈ R , P m j ∈ P m j , ∑ j = 1 N ( m j + 1 ) ⩽ n } Some related inequalities and direct and inverse theorems about the approximation by elements of G ˜ n in Lq (R) are also discussed. 2000 Mathematics Subject Classification 41A17 (primary).