Abstract :
The main result of this paper characterizes generalizationsof Zolotarev polynomials as extremal functions in the Kolmogorov–Landauproblemf(m)(0)→sup,f∈WrHω[0, 1], ‖f‖C[0, 1]⩽B, ((★))whereω(t) is a concave modulus of continuity,r, m: 1⩽m⩽r,are integers, andB⩾B0(r, m, ω). We show that theextremal functionsZBhaver+1 points of alternance andthe full modulus of continuity ofZ(r)B: ω(Z(r)B; t)=ω(t) for allt∈[0, 1]. This generalizesthe Karlinʹs result on the extremality of classical Zolotarevpolynomials in the problem (★) forω(t)=tand allB⩾Br.
