Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and [formula omitted]-minimal polynomials, [formula omitted

Abstract

Listen

Translate article

Let ( P 2 n * ( z ) ) be a sequence of polynomials with real coefficients such that lim n P 2 n * ( e i ϕ ) = G ( e i ϕ ) uniformly for ϕ ∈ [ α - δ , β + δ ] with G ( e i ϕ ) ≠ 0 on [ α , β ] , where 0 ⩽ α < β ⩽ π and δ > 0 . First it is shown that the zeros of p n ( cos ϕ ) = Re { e - in ϕ P 2 n * ( e i ϕ ) } are dense in [ α , β ] , have spacing of precise order π / n and are interlacing with the zeros of p n + 1 ( cos ϕ ) on [ α , β ] for every n ⩾ n 0 . Let ( P ˜ 2 n * ( z ) ) be another sequence of real polynomials with lim n P ˜ 2 n * ( e i ϕ ) = G ˜ ( e i ϕ ) uniformly on [ α - δ , β + δ ] and G ˜ ( e i ϕ ) ≠ 0 on [ α , β ] . It is demonstrated that for all sufficiently large n the zeros of p n ( cos ϕ ) and p ˜ n ( cos ϕ ) strictly interlace on [ α , β ] if Im { G ˜ ( e i ϕ ) / G ( e i ϕ ) } ≠ 0 on [ α , β ] . If the last expression is zero then a weaker kind of interlacing holds. These interlacing properties of the zeros are new for orthogonal polynomials also. For instance, for large n a simple criteria for interlacing of zeros of Jacobi polynomials on [ - 1 + ɛ , 1 - ɛ ] , ɛ > 0 , is obtained. Finally it is shown that the results hold for wide classes of weighted L q -minimal polynomials, q ∈ [ 1 , ∞ ] , linear combinations and products of orthogonal polynomials, etc.