Publication: A canonical family of multiple orthogonal polynomials for Nikishin systems
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2009-12
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Abstract
For any pair of compact intervals Δ1 and Δ2 of the real line such that Δ1∩Δ2 = ø we obtain two pairs of absolutely continuous probability measures (μ1,μ2) and (τ1,τ2) supported on Δ1 and Δ2, respectively, such that:
- for appropriate constants C1 and C2, (μ1,μ2) is the Nikishin system generated by (μ1,C1τ1) and (τ1,τ2) the Nikishin system generated by (τ1,C2μ1),
- the polynomials of multiple orthogonality with respect to the Nikishin system (μ1,μ2) and indices {..., (n,n), (n+1,n), ...} satisfy a recurrence relations with constant coe±cients of period 2,
- 1/hat-μ1(z) and 1/hat-μ2(z) are the functions which describe the ratio asymptotics of multiple orthogonal polynomials with respect to an arbitrary Nikishin system N(σ1,σ2) verifying supp(σi) = Δi, and σi' > 0, i = 1,2, almost everywhere on Δi. Analogously, 1/hat-τ1(z) and 1/hat-τ2(z) give the ratio asymptotics for N(σ1,σ2).
- for appropriate constants C1 and C2, (μ1,μ2) is the Nikishin system generated by (μ1,C1τ1) and (τ1,τ2) the Nikishin system generated by (τ1,C2μ1),
- the polynomials of multiple orthogonality with respect to the Nikishin system (μ1,μ2) and indices {..., (n,n), (n+1,n), ...} satisfy a recurrence relations with constant coe±cients of period 2,
- 1/hat-μ1(z) and 1/hat-μ2(z) are the functions which describe the ratio asymptotics of multiple orthogonal polynomials with respect to an arbitrary Nikishin system N(σ1,σ2) verifying supp(σi) = Δi, and σi' > 0, i = 1,2, almost everywhere on Δi. Analogously, 1/hat-τ1(z) and 1/hat-τ2(z) give the ratio asymptotics for N(σ1,σ2).
Description
16 pages.-- MSC2000 codes: Primary 42C05, 33C25; Secondary 41A21.-- Submitted paper.
Keywords
Hermite-Padé orthogonal polynomials, Multiple orthogonal polynomials, Nikishin system, Varying measures, Ratio asymptotics