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

Acta Math. Hungar. 68 (1-2) (1995), 1-6. ON THE INTEGRAL OF THE LEBESGUE FUNCTION OF INTERPOLATION. II P. ERDŐS, member of the Academy, J. SZABADOS and P. VÉRTESI (Budapest)1 To Professor K. Tandori on his seventieth birthday Let (1) xk = cos tk {k = l,...,n+ 1; 0 й Í1 < ... < tn+i ^ ж) be an arbitrary system of nodes of interpolation, and let n+1 \n(x) := |k(*)| k=i (where lk{x) are the fundamental functions of Lagrange interpolation) be the corresponding Lebesgue function. In [1], we gave a lower estimate for the integral of the Lebesgue function with respect to an arbitrary set of nodes (1) over a fixed interval [a, b] C [-1,1], for n’s sufficiently large depending on the interval [a, 6]. In this paper we extend this result to intervals depending on n, and for all n’s (Theorem 1). The method of proof is the same as in [1], with a slight modification. We also prove that, apart form a multiplicative constant, our result is best possible. (In fact, Theorem 2 is slightly stronger than that, since it estimates the maximum of the Lebesgue function in the interval in question.) Theorem 1. There exists an absolute constant c > 0 such that for an arbitrary system of nodes (1) and arbitrary intervals [an,bn] ^ [—1,1] we have fbr. / Xn(x)dx ^ c(bn-an) log (n(an-/?„) + 2) (a„ = cosara, = cos/3n). J dn Proof. Assume first that log (n(an — /?„) +2) log2 n(an-ßn) 2Ű7T ' 1 Research supported by Hungarian National Science Foundation Grant No. 1910 (second and third authors) and No. T7570 (third author). 0236-5294/95/$4.00 © 1995 Akadémiai Kiadó, Budapest

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