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 (m) n ‖Lq(R) ≤ (cm)n‖Un‖Lq(R) for all Un ∈ e Gn, q ∈ (0,∞], and m = 1, 2, . . . , where c is an absolute constant and e Gn := ( f : f(t) = N X j=1 Pmj (t)e −(t−λj )2 , λj ∈ R , Pmj ∈ Pmj , N X j=1 (mj + 1) ≤ n ) . Some related inequalities and direct and inverse theorems about the approximation by elements of e Gn in Lq(R) are also discussed.