Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and Lq-minimal polynomials, q∈[1,∞]

Let be a sequence of polynomials with real coefficients such that uniformly for ϕ∈[α-δ,β+δ] with G(eiϕ)≠0 on [α,β], where 0⩽α<β⩽π and δ>0. First it is shown that the zeros of are dense in [α,β], have spacing of precise order π/n and are interlacing with the zeros of pn+1(cosϕ) on [α,β] for every n⩾n0. Let be another sequence of real polynomials with uniformly on [α-δ,β+δ] and on [α,β]. It is demonstrated that for all sufficiently large n the zeros of pn(cosϕ) and strictly interlace on [α,β] if 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 Lq-minimal polynomials, q∈[1,∞], linear combinations and products of orthogonal polynomials, etc.