Let {φi(z;α)}i=0∞, corresponding to α∈(−1,1), be orthonormal Geronimus polynomials. We study asymptotic behavior of the expected number real zeros, say

One of the main results of this paper is an upper bound for the total number of real isolated zeros of complete Abelian integrals, exponential in the degree of the form (Theorem 1 below). This result improves a previously obtained in [IY1] double exponential estimate for the number of real isolated zeros on a positive distance from the singular locus. In fact, the theorem on zeros of Abelian in...

Abstract. Littlewood in his 1968 monograph “Some Problems in Real and Complex Analysis” [9, problem 22] poses the following research problem, which appears to still be open: “If the nm are integral and all different, what is the lower bound on the number of real zeros of PN m=1 cos(nmθ)? Possibly N − 1, or not much less.” Here real zeros are counted in a period. In fact no progress appears to h...