부산대학교 도서관

This paper is a study of orthogonal polynomials with respect to aSobolev-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 thereal line. When the two measures satisfy a coherence condition, thecorresponding orthogonal polynomials, denoted $\Cal S_n(\nu_0, \nu_1,s; x)$, have explicit and simple connection formulas with the monicorthogonal polynomials corresponding to $\nu_0, \nu_1$, denoted $\CalP_n(\nu_0; x), \Cal P_n(\nu_1; x)$, respectively. In this paper, theauthors study coherent pairs of positive measures of the second kind(which they abbreviate CPPM2K); a pair $(\nu_0, \nu_1)$ is CPPM2K ifand 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 andthe coherence condition. It is these simple connection formulae thatmotivate the study of coherent pairs. It is worth pointing out that thenotion 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 ofmeasures to be coherent if the monic orthogonal polynomials insteadsatisfied$$\Cal P_n(\nu_1; x) = \dfrac{1}{n+1} \left( \Cal P_{n+1}'(\nu_0; x) -\rho_n(\nu_0, \nu_1) P_{n}'(\nu_0; x) \right), \quad \rho_n(\nu_0,\nu_1)\neq 0, \ n \geq 1.$$This original notion of coherent pairs also implies simple connectionformulas for the Sobolev orthogonal polynomials similar to (1), and Iserleset al. [op. cit.] exploited this to efficiently obtain expansions of functions $f \inW_2^1(\Bbb{R}; \nu_0, \nu_1)$ (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 beused to develop fast algorithms to achieve the same goal, though thisaspect is left for future study.\par The paper under review can be divided into two halves. The first isconcerned with general properties of CPPM2K, including general formulasfor the expansion of $\Cal S_n(\nu_0, \nu_1, s; x)$ in terms of $\{\CalP_n(\nu_0; x)\}$, 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, inspecific examples, allow one to compute $\gamma_n$ in terms of otherknown sequences of polynomials. This connection is used to prove thatthe zeros of $\Cal S_n$ are eigenvalues of a particular modification ofa tridiagonal matrix. The second half of the paper is devoted to a detailedinvestigation of a particular example where one of the measures is theclassical 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 themeasures, 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, thefunctions $\gamma_n(s)$ are identified as Wilson polynomials.