부산대학교 도서관

This paper is a study of orthogonal polynomials with respect to a Sobolev-type inner product $$ \langle f, g \rangle_\germ{s} = \int f(x) g(x) \roman{d} \nu_0 (x) + s \int f'(x) g'(x) \roman{d} \nu_1(x), \quad s >0, $$ where $\nu_0, \nu_1$ are two positive Borel measures supported on the real line. When the two measures satisfy a coherence condition, the corresponding orthogonal polynomials, denoted $\Cal S_n(\nu_0, \nu_1, s; x)$, have explicit and simple connection formulas with the monic orthogonal polynomials corresponding to $\nu_0, \nu_1$, denoted $\Cal P_n(\nu_0; x), \Cal P_n(\nu_1; x)$, respectively. In this paper, the authors study coherent pairs of positive measures of the second kind (which they abbreviate CPPM2K); a pair $(\nu_0, \nu_1)$ is CPPM2K if and only if $$ \dfrac{1}{n+1} \Cal P_{n+1}'(\nu_0; x) = \Cal P_n(\nu_1; x) - \tau_n(\nu_0, \nu_1) \Cal P_{n-1}(\nu_1; x), \quad \tau_n (\nu_0, \nu_1) \neq 0, \ n \geq 1. $$ The notion of CPPM2K was first introduced in [M.~H. Suni et al., J. Math. Anal. Appl. {\bf 525} (2023), no.~1, Paper No. 127118; MR4552386], where it was observed that if $(\nu_0, \nu_1)$ is CPPM2K, then there exist $\{\gamma_n(s)\}_{n = 1}^\infty$ such that $$ \aligned \Cal S_{n+1}(\nu_0, \nu_1, s; x) - \gamma_n(s) \Cal S_{n}(\nu_0, \nu_1, s; x) &= \Cal P_{n+1}(\nu_0; x),\\ \Cal S'_{n+1}(\nu_0, \nu_1, s; x) - \gamma_n(s) \Cal S'_{n}(\nu_0, \nu_1, s; x) &= (n+1)\left(\Cal P_n(\nu_1; x) - \tau_n(\nu_0, \nu_1) \Cal P_{n-1}(\nu_1; x)\right), \endaligned \tag1 $$ where the second equality follows from the first by differentiation and the coherence condition. It is these simple connection formulae that motivate the study of coherent pairs. It is worth pointing out that the notion of coherent pairs of measures goes back to [J. Approx. Theory {\bf 65} (1991), no.~2, 151--175; MR1104157], where A. Iserles et al. dubbed a pair of measures to be coherent if the monic orthogonal polynomials instead satisfied $$ \Cal P_n(\nu_1; x) = \dfrac{1}{n+1} \left( \Cal P_{n+1}'(\nu_0; x) - \rho_n(\nu_0, \nu_1) P_{n}'(\nu_0; x) \right), \quad \rho_n(\nu_0, \nu_1)\neq 0, \ n \geq 1. $$ This original notion of coherent pairs also implies simple connection formulas for the Sobolev orthogonal polynomials similar to (1), and Iserles et al. [op. cit.] exploited this to efficiently obtain expansions of functions $f \in W_2^1(\Bbb{R}; \nu_0, \nu_1)$ (with the inner product $\langle \cdot , \cdot \rangle _\germ{s}$ above) in terms of $\Cal S_n$. The authors of the work under review indicate that CPPM2K can also be used to develop fast algorithms to achieve the same goal, though this aspect is left for future study. \par The paper under review can be divided into two halves. The first is concerned with general properties of CPPM2K, including general formulas for the expansion of $\Cal S_n(\nu_0, \nu_1, s; x)$ in terms of $\{\Cal P_n(\nu_0; x)\}$, a proof of (1) and various formulas for $\gamma_n(s)$, and general properties (e.g., estimates, monotonicity) and recurrence relations satisfied by $\gamma_n(s)$ which can, in specific examples, allow one to compute $\gamma_n$ in terms of other known sequences of polynomials. This connection is used to prove that the zeros of $\Cal S_n$ are eigenvalues of a particular modification of a tridiagonal matrix. The second half of the paper is devoted to a detailed investigation of a particular example where one of the measures is the classical Jacobi weight. Precisely, let $$ \omega^{(\alpha, \beta)}(x) = \frac{\Gamma\left( \alpha + \beta + 2\right)}{2^{\alpha + \beta + 1} \Gamma(\alpha + 1) \Gamma(\beta + 1)} (1 - x)^\alpha (1 + x)^\beta; $$ the measures are $$ \aligned \roman{d} \nu_1(x) &= \omega^{(\alpha+1, \beta+1)}(x) \roman{d} x, \quad x \in [-1, 1],\\ \roman{d} \nu_0(x) & = (1 - \epsilon )\eta^{(\alpha, \beta, q)} (x - q)^{-1} \omega^{(\alpha, \beta)}(x) \roman{d} x + \epsilon \delta_{q}, \quad |q|\geq 1, \ 0 \leq \epsilon < 1,\ x \in \Cal I(q), \endaligned $$ and $\Cal I(q) = [-1, 1] \cup \{q\}$ (unless $\epsilon = 0$ of course). These measures are CPPM2K and given the explicit nature of the measures, the connection formulas $\gamma_n$ and polynomials $\Cal S_n$ are studied and various asymptotic formulas for them as the parameters $q, s, n$ are varied. Finally, for a particular choice of parameters, the functions $\gamma_n(s)$ are identified as Wilson polynomials.