Bernstein‐Type Inequalities for Linear Combinations of Shifted Gaussians Academic Article uri icon


  • 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 ∥Un(m)| Lq(ℝ) ≤ (c1+1/qm)m/2 n m/2∥Un∥Lq(ℝ) for all U n ∈ G̃n, q ∈ (0, ∞], and m = 1, 2,..., where c is an absolute constant and G̃n := {f : f(t) = ∑j=1N Pmj (t)e-(t-λj)2, λj ∈ ℝ, Pmj, ∑j=1N (mj + 1) ≤ n}. Some related inequalities and direct and inverse theorems about the approximation by elements of G̃n in Lq(ℝ) are also discussed. © 2006 London Mathematical Society.

author list (cited authors)

  • Erdélyi, T.

citation count

  • 6

publication date

  • January 2006