On real and complex zeros of orthogonal polynomials in a discrete Sobolev space

Abstract

Let { S n ( x; c, N)} denote a set of polynomials orthogonal with respect to the discrete Sobolev inner product 〈f,g〉 = ʃ ∞ -∞f(x)g(x)dψ(x)+Nf ′(c)g ′(c) , where N ⩾ 0, c ϵ R . For N = 0, put K n ( x) = S n ( x; ·, 0). Then S n ( x; c, N) has at least n − 2 different real zeros; their position with respect to the zeros of K n can be determined using the tangent to the graph of y = K n ( x) in ( c, K n ( c)). On the other hand, if n ⩾ 3, then c can be chosen such that S n ( x; c, N) has two complex zeros if N is sufficiently large.