Summary.
Let w α,beta (t)=(1−t)α(1+t)β, α,β>−1, denote the Jacobi weight function. For 0≤α,β<5/2 it is shown that on compact subintervals of (−1,1), the corresponding Stieltjes polynomials and their derivatives are asymptotically equal to certain Jacobi polynomials. This also leads to asymptotic representations of those weights of Gauss-Kronrod quadrature formulae, which correspond to nodes in a compact subinterval of (−1,1). On the other hand, it is demonstrated that for the parameters satisfying min(α,β)≥0 and max(α,β)>5/2 the Stieltjes polynomial has only few real zeros and that Gauss-Kronrod quadrature is not possible. So far, results of the above type have been known only for ultraspherical weight functions \({{w_\lambda(t) = \big(1-t^2\big)^{{\lambda-1/2}}}}\) for 0≤λ≤2. For λ>3, the impossibility of Gauss-Kronrod quadrature has been proved by the authors recently.
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Mathematics Subject Classification (1991): 33C10, 33C45, 42C05, 65D32
The second author is sponsored by a Heisenberg scholarship of the Deutsche Forschungsgemeinschaft.
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Peherstorfer, F., Petras, K. Stieltjes Polynomials and Gauss-Kronrod Quadrature for Jacobi Weight Functions. Numer. Math. 95, 689–706 (2003). https://doi.org/10.1007/s00211-002-0412-2
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DOI: https://doi.org/10.1007/s00211-002-0412-2