Lagrange interpolation of bandlimited functions on slowly increasing sequences

Let A = {z(n,)k : 1 <= k <= n, n is an element of N} be a triangular array of distinct complex numbers and let f be an entire function. Suppose L(n-1)f is the unique polynomial of degree at most n-1 which interpolates f at z(n,k) for k is an element of is an element of {1, ..., n}, i.e., L(n)f (z(n,k)) = f (z(n,k)). In this note, we show that L-nf converges to f uniformly on compact subsets of the complex plane provided Lambda is bounded. We next consider the case when z(n,k) = z(k) where {vertical bar z(k)vertical bar}(k is an element of N) is a slowly increasing unbounded sequence in the sense that for some alpha is an element of 0, 1[, (k - 1)(alpha) <= vertical bar z(k) <= k(alpha) for each k is an element of N. If f is bandlimited, we prove as well that L-nf converges uniformly (and rapidly) to f on compact subsets of the complex plane. The rate of convergence that we obtain is optimal to some extent.