Let (u, v) be a pair of quasidefinite and symmetric linear functionals with {Pn}n≥0 and {Qn}n≥0 as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials {Rn}n≥0 as follows:
\(\begin{array}{cc}\frac{{P}_{n+2}^{\mathrm{^{\prime}}}\left(x\right)}{n+2}+{b}_{n}\frac{{P}_{n}^{\mathrm{^{\prime}}}\left(x\right)}{n}-{Q}_{n+1}\left(x\right)={d}_{n-1}\left(x\right),& n\ge 1.\end{array}\)
We present necessary and sufficient conditions for {Rn}n≥0 to be orthogonal with respect to a quasidefinite linear functional w. In addition, we consider the case where {Pn}n≥0 and {Qn}n≥0 are monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product
\(\langle p,q\rangle s=\underset{-1}{\overset{1}{\int }}pq{\left(1-{x}^{2}\right)}^{-1/2}dx+{\uplambda }_{1}\underset{-1}{\overset{1}{\int }}{p}^{\mathrm{^{\prime}}}{q}^{\mathrm{^{\prime}}}{\left(1-{x}^{2}\right)}^{1/2}dx+{\uplambda }_{2}\underset{-1}{\overset{1}{\int }}{p}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}{q}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}d\mu \left(x\right),\)
where μ is a positive Borel measure associated with w and λ1, λ2 > 0; λ2 is a linear polynomial of λ1.
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 10, pp. 1411–1428, October, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i10.7293.
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Molano, L. Inverse Problems, Sobolev–Chebyshev Polynomials, and Asymptotics. Ukr Math J 75, 1601–1620 (2024). https://doi.org/10.1007/s11253-024-02281-3
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DOI: https://doi.org/10.1007/s11253-024-02281-3