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- W2034736124 abstract "Let (P2n*(z)) be a sequence of polynomials with real coefficients such that limnP2n*(eiϕ)=G(eiϕ) 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 pn(cosϕ)=Re{e-inϕP2n*(eiϕ)} 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 (P˜2n*(z)) be another sequence of real polynomials with limnP˜2n*(eiϕ)=G˜(eiϕ) uniformly on [α-δ,β+δ] and G˜(eiϕ)≠0 on [α,β]. It is demonstrated that for all sufficiently large n the zeros of pn(cosϕ) and p˜n(cosϕ) strictly interlace on [α,β] if Im{G˜(eiϕ)/G(eiϕ)}≠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 Lq-minimal polynomials, q∈[1,∞], linear combinations and products of orthogonal polynomials, etc." @default.
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- W2034736124 date "2009-09-01" @default.
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- W2034736124 title "Interlacing and spacing properties of zeros of polynomials, in particular of orthogonal and <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si1.gif overflow=scroll><mml:msub><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mi>q</mml:mi></mml:mrow></mml:msub></mml:math>-minimal polynomials, <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML altimg=si2.gif overflow=scroll><mml:mi>q</mml:mi><mml:mo>∈</mml:mo><mml:mo stretchy=false>[</mml:mo><mml:mn>1</mml:mn><…" @default.
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- W2034736124 doi "https://doi.org/10.1016/j.jat.2008.04.004" @default.
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