论文标题
在修改的内核多项式和经典类型Sobolev正交多项式上
On modified kernel polynomials and classical type Sobolev orthogonal polynomials
论文作者
论文摘要
在本文中，我们研究了修改的内核多项式：$ u_n（x）= \ sum_ {k = 0}^n c_k g_k（x）$，具体取决于参数$ c_k> 0 $，其中$ \ {g_k \} _ 0^_ 0^\ suftty $是矫正$是矫正线上的真实线。除了$ c_k = g_k（t_0）> 0 $的内核多项式外，可以选择$ c_k $作为$ g_k $的相应二阶差异方程的其他解决方案。结果表明，所有这些多项式都满足$ 4 $的订单复发关系。 $ g_k $的案件是雅各比或拉瓜雷多项式的特殊兴趣。参数的合适选择$ c_k $暗示$ u_n $作为$（3 \ times 3）$矩阵度量的sobolev正交多项式。此外，进一步的参数可为$ u_n $提供微分方程。在后一种情况下，多项式$ u_n（x）$都是$ x $和$ n $中的广义特征值问题的解决方案。
In this paper we study modified kernel polynomials: $u_n(x) = \sum_{k=0}^n c_k g_k(x)$, depending on parameters $c_k>0$, where $\{ g_k \}_0^\infty$ are orthonormal polynomials on the real line. Besides kernel polynomials with $c_k = g_k(t_0)>0$, for example, $c_k$ may be chosen to be some other solutions of the corresponding second-order difference equation of $g_k$. It is shown that all these polynomials satisfy a $4$-th order recurrence relation. The cases with $g_k$ being Jacobi or Laguerre polynomials are of a special interest. Suitable choices of parameters $c_k$ imply $u_n$ to be Sobolev orthogonal polynomials with a $(3\times 3)$ matrix measure. Moreover, a further selection of parameters gives differential equations for $u_n$. In the latter case, polynomials $u_n(x)$ are solutions to a generalized eigenvalue problems both in $x$ and in $n$.
