THE LEBESGUE FUNCTION AND LEBESGUE CONSTANT OF LAGRANGE INTERPOLATIONFOR ERDOS WEIGHTS

We establish pointwise as well as uniform estimates for Lebesgue funct ions associated with a large class of Erdos weights on the real line. An Erdos weight is of the form W := exp(-Q), where Q:R --> R is even a nd is of Faster than polynomial growth at infinity. The archetypal exa mples are W-k,W- (alpha)(x) := exp(-Q(k,) (alpha)(x)), (i) where Q(k, alpha) (x) := exp(k) (\x\(alpha)), alpha > 1, k greater than or equal to 1 Here exp(k) := exp(exp(exp(...))) denotes the kth iterated expone ntial. W-A,W- B(x) := exp(-Q(A,B)(x)), (ii) where Q(A,B)(x) := exp(log (A +x(2)))(B), B >1 and A > A(0). For a carefully chosen system of nod es chi(n) := {xi(1), xi 2, ..., xi(n)}, n greater than or equal to 1, our result imply in particular, that the Lebesgue constant \\Lambda(n) (W-k,W- alpha chi(n))\\L-infinity(R) := sup(x is an element of R) \Lam bda(n)(W-k,W- (alpha), chi(n))\ (x) satisfies uniformly for n greater than or equal to N-0, \\Lambda(n)(W-k,W- alpha, chi(n))\\L-infinity(R) similar to log n. Moreover, we show that this choice of nodes is opti mal with respect to the zeros of the orthonormal polynomials generated by W-2. Indeed, let U-n := {x(j,n): 1 less than or equal to j less th an or equal to n}, n greater than or equal to 1, where the x(k, n) are the zeros of the orthogonal polynomials p(n)(W-2, .) generated by W-2 . Then in particular, we have uniformly for n greater than or equal to N, \\Lambda(n),(W-k,W- alpha, U-n)\\L-infinity(R) similar to n(1/6)(P i(j=1)(k) login)(1/6) Here, log(j) :=log(log(log(...))) denotes the jt h iterated logarithm. We deduce sharp theorems of uniform convergence of weighted Lagrange interpolation together with rates of convergence. In particular, these results apply to W-k,W- alpha, and W-A,W- (B). ( C) 1998 Academic Press.