Bernstein-type inequalities for linear combinations of shifted gaussians

Let P-n 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 parallel to U(n)((m))parallel to(Lq) ((R)) <= (c(1+1/q) m)(m/2) n(m/2) parallel to U(n)parallel to(Lq (R)) for all U-n is an element of (G) over tilde (n), q is an element of (0 infinity], and m = 1, 2, . . . , where c is an absolute constant and (G) over tilde (n) = {f : f(t) = Sigma(j=1)(N) P-m (j) (t)e(-(t-lambda j)2) , lambda(j) is an element of R, P-mj is an element of P-mj, Sigma(j-1)(N) (m(j) + 1) <= n}. Some related inequalities and direct and inverse theorems about the approximation by elements of (G) over tilde (n) in L-q(R) are also discussed.