Markov–Nikolskii type inequalities for exponential sums on finite intervals

AbstractLet Λn:={λ0<λ1<⋯<λn} be a set of real numbers. The collection of all linear combinations of eλ0t,eλ1t,…,eλnt over R will be denoted byE(Λn):=span{eλ0t,eλ1t,…,eλnt}. Motivated by a question of Michel Weber (Strasbourg) we prove the following couple of theorems.Theorem 1Let0<q⩽p⩽∞,a,b∈R, anda<b. There are constantsc1=c1(p,q,a,b)>0andc2=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 2Let0<q⩽p⩽∞,a,b∈R, anda<b. There are constantsc1=c1(p,q,a,b)>0andc2=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 all0<q⩽p⩽∞, while the upper bound holds whenp⩾1and0