Acta Mathematica Academiae Scientiarum Hungaricae 68. (1995)

1995 / 1-2. szám - Erdős P. - Szabados J. - Vértesi P.: On the integral of the Lebesgue function of interpolation. II

P. ERDOS, J. SZABADOS and P. VERTES! (2) namely and Then n(an — ßn) < c\ with an absolute constant C\ > 0, and the statement of the theorem trivially holds, since Ап(ж) ^ 1 (|ar| ^ 1). So from now on we assume that Also, without loss of generality we may assume that [ßn,an] can be considered as the union of subintervals obtained by consider­ing the partition points tk G [ßn,an]. Among'these subintervals, let [<$„,7„] be of maximal length. (If there are no f*’s in [/3n,an], then let -yn = an, Sn = ßn.) We distinguish two cases. Case 1: ln - Sn > ёЪ • ЫЧ°п-Рп)+2). Then let log 2 a polynomial formed from the roots of the Chebyshev polynomial 71 Tn(x) := cos(u arccos x) = 2”_1 (a; — Zk). k=1 Since in this case there exists an absolute constant c2 > 0 and a set Hn C [cn, dn\ such that and (3) log (n(an - ßn) + 2) < log2 ßn) 207Г an s Q S' s ^ T -A = *-2' Or COS cos a, — bn ön­37n + cos dn := cos27 n + Зй„ Pn(x) := П (x - zk), Zk£[cn4n] \тп(у)\г\ (у 6 нп) \Hn\ = ^2 (Ттг ^п)? Mathematica Hungarica 68, 1995

Next