A discrete weighted Markov–Bernstein inequality for sequences and polynomials

Abstract

For parameters c∈(0,1) and β>0, let ℓ2(c,β) be the Hilbert space of real functions defined on N (i.e., real sequences), for which‖f‖c,β2:=∑k=0∞(β)kk!ck[f(k)]2<∞. We study the best (i.e., the smallest possible) constant γn(c,β) in the discrete Markov-Bernstein inequality‖ΔP‖c,β≤γn(c,β)‖P‖c,β,P∈Pn, where Pn is the set of real algebraic polynomials of degree at most n and Δf(x):=f(x+1)−f(x).We prove that(i)γn(c,1)≤1+1c for every n∈N and limn→∞⁡γn(c,1)=1+1c;(ii)For every fixed c∈(0,1), γn(c,β) is a monotonically decreasing function of β in (0,∞);(iii)For every fixed c∈(0,1) and β>0, the best Markov-Bernstein constants γn(c,β) are bounded uniformly with respect to n. A similar Markov-Bernstein inequality is proved for sequences, and a relation between the best Markov-Bernstein constants γn(c,β) and the smallest eigenvalues of certain explicitly given Jacobi matrices is established.