论文标题
Sobolev在圆锥表面上的正交多项式
Sobolev orthogonal polynomials on the conic surface
论文作者
论文摘要
关于重量函数的正交多项式$ w_ {β，γ}（t）= t^β（1 -t）^γ$，$γ> -1 $，在圆锥表面$ \ {（x，t）上：\ | x \ | = t，\，x \ in \ mathbb {r}^d，\，\，t \ le 1 \} $最近进行了研究，当$β= -1 $β= -1 $时，它们被证明是二阶差异操作员$ \ mathcal {d}_γ$的二阶差异操作员的特征。我们将设置扩展到Sobolev Inner产品，该产品定义为$ t $变量中$ s $ the部分衍生物的集成，相对于$ w_ {β+s，0} $在圆锥表面上，加上圆锥轮辋上的积分总和。我们的主要结果为正交投影算子提供了明确的构造和公式；后者用于利用差分运算符和投影操作员的相互作用，这使我们能够研究傅立叶正交系列的收敛性。对于正整数$ s $，这项研究可以视为重量函数的正交结构$ w_ {β，-s} $的扩展。它尤其表明，sobolev正交多项式是$ \ mathcal {d}_γ$当$γ= -1 $时的特征。
Orthogonal polynomials with respect to the weight function $w_{β,γ}(t) = t^β(1-t)^γ$, $γ> -1$, on the conic surface $\{(x,t): \|x\| = t, \, x \in \mathbb{R}^d, \, t \le 1\}$ are studied recently, and they are shown to be eigenfunctions of a second order differential operator $\mathcal{D}_γ$ when $β=-1$. We extend the setting to the Sobolev inner product, defined as the integration of the $s$-th partial derivatives in $t$ variable with respect to $w_{β+s,0}$ over the conic surface plus a sum of integrals over the rim of the cone. Our main results provide an explicit construction of an orthogonal basis and a formula for the orthogonal projection operators; the latter is used to exploit the interaction of differential operators and the projection operator, which allows us to study the convergence of the Fourier orthogonal series. The study can be regarded as an extension of the orthogonal structure to the weight function $w_{β, -s}$ for a positive integer $s$. It shows, in particular, that the Sobolev orthogonal polynomials are eigenfunctions of $\mathcal{D}_γ$ when $γ= -1$.
