S-Coherent measures pairs and Orthogonal polynomials with respect to Sobolev products. Applications (2019)
In this dissertation, the concept of s-coherence, or symmetric $(1,1)$-coherence, of pairs of quasi-definite linear functionals, and the polynomials orthogonal with respect to certain Sobolev inner product type play a preponderant role. The concept of symmetric $(1,1)-$ coherent pair is defined as follows. Let $u$ and $v$ denote two symmetric quasi-definite linear functionals and $ \left\{ P_{n}\right\} _{n\geq 0}$ and $\left\{ R_{n}\right\} _{n\geq 0}$ will denote their respective sequences of monic orthogonal polynomials, (SMOP in short). Suppose that there exist sequences of non-zero real numbers $\left\{ a_{n}\right\} _{n\geq 0}$ and $\left\{b_{n}\right\} _{n\geq 0},$ with $a_{n}b_{n}\neq 0,$ such that \begin{equation} \frac{P_{n+3}^{\prime }(x)}{n+3}+a_{n}\frac{P_{n+1}^{\prime }(x)}{n+1}=R_{n+2}(x)+b_{n}R_{n}(x),\text{ \ \ }n\geq 0, \label{simm} \end{equation} holds. Then the pair $\left\{ u,v\right\} $ is said to be a \textit{ Symmetric }$(1,1)-$\textit{Coherent Pair}. This concept is a natural extension of the concept of symmetric coherent pairs of quasi-definite linear functionals. The structure of this work is as follows. First, a classification of symmetric $(1,1)-$coherent pairs is stated by using a symmetrization process. In addition, we study how from (\ref{simm}), and using the symmetrization process, we can arrive to a non-coherence algebraic relation. Then, the corresponding inverse problem is analyzed exhaustively. After this, we consider the Sobolev inner product \begin{equation} \left\langle p,q\right\rangle _{S}=\int_{ \mathbb{R} }p(x)q(x)d\mu _{0}(x)+\lambda \int_{ \mathbb{R} }p^{\prime }(x)q^{\prime }(x)d\mu _{1}(x),\text{ \ }\lambda 0, \label{innersobolev44} \end{equation} where we assume that $u$ and $v$ are positive-definite, with $\mu _{0}$ and $\mu _{1}$ as the respective positive Borel measures and $\left\{ S_{n}^{\lambda }\right\} _{n\geq 0}$ as the Sobolev orthogonal polynomials associated with (\ref{innersobolev44}). So, the algebraic relation \begin{equation} S_{n+3}^{\lambda }(x)+\eta _{n}(\lambda )S_{n+1}^{\lambda }(x)=P_{n+3}(x)+% \widetilde{a}_{n}P_{n+1}(x), \end{equation} is considered, where special attention is placed on the so called \textit{Sobolev coefficients} $\left\{ \eta _{n}(\lambda ) \right\} _{n\geq 0}$. Then, their recurrence properties as well as those of the corresponding Sobolev norms $\left\{ \left\Vert S_{n}^{\lambda }\right\Vert _{S}^{2}\right\} _{n\geq 0}$ are studied. On the other hand, the particular symmetric $(1,1)-$coherent pair $\left\{ \mu _{0},\mu _{1}\right\} ,$ $d\mu _{0}=e^{-x^{2}}dx,$ $d\mu _{1}=\frac{x^{2}+a}{x^{2}+b}e^{-x^{2}}dx,$ is taken into account. In this way, limit behavior of Soboleb coefficients and the asymptotic properties of Sobolev polynomials are deeply studied. Finally, we exhibit an algorithm to compute Fourier coefficients in expansions of functions that belong to the Sobolev space $W_{2}^{1}\left[ \mathbb{R} ,\mu _{0},\mu _{1}\right] $ by using Sobolev polynomials.