Markov–Nikolskii type inequalities for exponential sums on finite intervals

Let Λn:={λ0<λ1<⋯<λn}Λn:={λ0<λ1<⋯<λn} be a set of real numbers. The collection of all linear combinations of eλ0t,eλ1t,…,eλnteλ0t,eλ1t,…,eλnt over RR will be denoted byE(Λn):=span{eλ0t,eλ1t,…,eλnt}.E(Λn):=span{eλ0t,eλ1t,…,eλnt}. Motivated by a question of Michel Weber (Strasbourg) we prove the following couple of theorems.Theorem 1. Let  00c1=c1(p,q,a,b)>0and  c2=c2(p,q,a,b)c2=c2(p,q,a,b)depending only on p, q, a, and b such thatc1(n2+∑j=0n|λj|)1q−1p⩽sup0≠P∈E(Λn)‖P‖Lp[a,b]‖P‖Lq[a,b]⩽c2(n2+∑j=0n|λj|)1q−1p.Theorem 2. Let  00c1=c1(p,q,a,b)>0and  c2=c2(p,q,a,b)c2=c2(p,q,a,b)depending only on p, q, a, and b such thatc1(n2+∑j=0n|λj|)1+1q−1p⩽sup0≠P∈E(Λn)‖P′‖Lp[a,b]‖P‖Lq[a,b]⩽c2(n2+∑j=0n|λj|)1+1q−1p,where the lower bound holds for all  0