Inverse Problems, Sobolev–Chebyshev Polynomials, and Asymptotics

Let (u, v) be a pair of quasidefinite and symmetric linear functionals with {P_n}_n≥0 and {Q_n}_n≥0 as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials {R_n}_n≥0 as follows:

\(\begin{array}{cc}\frac{{P}_{n+2}^{\mathrm{^{\prime}}}\left(x\right)}{n+2}+{b}_{n}\frac{{P}_{n}^{\mathrm{^{\prime}}}\left(x\right)}{n}-{Q}_{n+1}\left(x\right)={d}_{n-1}\left(x\right),& n\ge 1.\end{array}\) $\begin{array}{cc}\frac{P_{n+2}^{{}^{\prime }}\left(x\right)}{n+2}+b_n\frac{P_n^{{}^{\prime }}\left(x\right)}{n}-Q_{n+1}\left(x\right)=d_{n-1}\left(x\right),& n\ge 1.\end{array}$

We present necessary and sufficient conditions for {R_n}_n≥0 to be orthogonal with respect to a quasidefinite linear functional w. In addition, we consider the case where {P_n}_n≥0 and {Q_n}_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

\(\langle p,q\rangle s=\underset{-1}{\overset{1}{\int }}pq{\left(1-{x}^{2}\right)}^{-1/2}dx+{\uplambda }_{1}\underset{-1}{\overset{1}{\int }}{p}^{\mathrm{^{\prime}}}{q}^{\mathrm{^{\prime}}}{\left(1-{x}^{2}\right)}^{1/2}dx+{\uplambda }_{2}\underset{-1}{\overset{1}{\int }}{p}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}{q}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}d\mu \left(x\right),\) $⟨p,q⟩s=\underset{-1}{\stackrel{1}{\int }}pq{\left(1,-,x^2\right)}^{-1/2}dx+{\mathrm{\lambda }}_1\underset{-1}{\stackrel{1}{\int }}{p}^{{}^{\prime }}{q}^{{}^{\prime }}{\left(1,-,x^2\right)}^{1/2}dx+{\mathrm{\lambda }}_2\underset{-1}{\stackrel{1}{\int }}{p}^{{}^{\prime }{}^{\prime }}{q}^{{}^{\prime }{}^{\prime }}d\mu \left(x\right),$

where μ is a positive Borel measure associated with w and λ₁, λ₂ > 0; λ₂ is a linear polynomial of λ₁.