International Journal of Mathematics and Mathematical Sciences 
Volume 28 (2001), Issue 8, Pages 461-467
doi:10.1155/S0161171201006056

On the number of zeros of iterated operators on analytic Legendre expansions

Carl Prather

 Department of Mathematics, Virginia Tech, Blacksburg 24061-0123, VA, USA
Received 10 September 2000

Abstract

Let L=(1−z2)D2−2zD, D=d/dz and f(z)=∑n=0∞cnpn(z), with Pn being the nth Legendre polynomial and f analytic in an ellipse with foci ±1. Set Lk=L(Lk−1), k≥2. Then the number of zeros of Lkf(z) in this ellipse is O(klnk).