Sobolev Orthogonal Polynomials with a Small Number of Real Zeros Original Research Article

Let {Sλn} denote a set of polynomials orthogonal with respect to the Sobolev inner product 〈f, g 〉 = ∫3−1f(x) g(x) dx + λ ∫1−1) f′(x) g′(x) dx + ∫31f′(x) g′(x) dx, where λ ≥ 0. If n is odd and λ sufficiently large, then Sλn has exactly one real zero. If n is even, n ≥ 2, and λ sufficiently large, then Sλn has exactly two real zeros. This result can be generalized to a more general inner product.