Sequentially-ordered Sobolev inner product and Laguerre-Sobolev polynomials

Abstract

We study the sequence of polynomials $\{S_n{\}}_{n\ge 0}$ that are orthogonal with respect to the general discrete Sobolev-type inner product $$⟨f,g{⟩}_{\mathsf{s}}=\int f(x)g(x)d\mu (x)+\sum _{j=1}^N\sum _{k=0}^{d_j}{\lambda }_{j,k}{f}^{(k)}(c_j)g^{(k)}(c_j),$$ where $\mu$ is a finite Borel measure whose support $\suppμ$ is an infinite set of the real line, ${\lambda }_{j,k}\ge 0$, and the mass points $c_i$, $i=1,\dots ,N$ are real values outside the interior of the convex hull of $\suppμ$ ($c_i\in\RR\setminus\inter{\ch{\suppμ}}$). Under some restriction of order in the discrete part of $⟨\cdot ,\cdot {⟩}_{\mathsf{s}}$, we prove that $S_n$ has at least $n-d^{\ast }$ zeros on $\inter{\ch{\suppμ}}$, being $d^{\ast }$ the number of terms in the discrete part of $⟨\cdot ,\cdot {⟩}_{\mathsf{s}}$. Finally, we obtain the outer relative asymptotic for $\{S_n\}$ in the case that the measure $\mu$ is the classical Laguerre measure, and for each mass point, only one order derivative appears in the discrete part of $⟨\cdot ,\cdot {⟩}_{\mathsf{s}}$.