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
ON THE INTEGRAL OF THE LEBESGUE FUNCTION OF INTERPOLATION. II 3 where \Hn\ denotes the measure of the set obtained from Hn by projecting it to the unit circle. Thus, if \Hn\ denotes the ordinary measure of the set Hn, then we get Now denoting Cn = cos7„, Dn = cos<$n we obtain for x G [— 1,1] \ \ (^m ^n')t ^k G [cn, dn] and У G Hn provided 7„ — bn is smaller than a properly chosen absolute constant. (If this fails to hold, then the statement of the theorem reduces to that of [1].) Hence since evidently, there are at least z*’s in the interval [cn,dn]. Hence using the reproducing property of Lagrange interpolation we get (4) \ и I 2 I £/ I • a >» ■ “»/ 10c2sin^ log (n(an ~ ßn) + 2) IHn\ й -\Hn\smßn г c2sm —('In — 6n) ^ —-——±------------------------------L 7Г 2 log 2 n У - Zk X - Zk dn Cn sin 1 sin (^±^- - = V2' Dn c-n 2 cosJlL \Pn(x)\ Tn(x)I П \x - zk I £ [cnidn] \Pn(y)\ Tn(x) Tn(y)П Zk C [Cn2n] У - zk x - zk< = 2|p„(i/)| • Yl ^ = \Рп{у)\ -21 Zk£[cn4n] n( 1n-6n) IOtt < . , loK(n(qn-gnH-2) = IPn(y)\ • 21 >°*22 u(on ßn) T 2«|pn(iO|, П+1 \Pn(y)\ ^ ' \Ш\ = k=\ d(o!n ßn)\pn{y)\K(y) (yeHn), since by construction, there are no x^'s in the interval (Cn, Dn). Thus kn(y) = n(an — ßn)/2 ^ —— (y £ Hn). 2sina„ Hence and by (4) Г bn [ K{y)dy ^ / An(y)dy ^ > 'x JHn 2 sin ar a4cia Mathematica Hungarica 68, 1995