论文标题
锥的单一和罗德里格斯正交多项式
Monomial and Rodrigues orthogonal polynomials on the cone
论文作者
论文摘要
我们研究了两个正交多项式的家族，相对于重量函数$ W（t）（t^2- \ | x \ |^2） \ le t，\，x \ in \ mathbb {r}^d，t> 0 \} $ in $ \ mathbb {r}^{d+1} $。第一个家庭由单一多项式组成$ \ MATHSF {V} _ {\ MATHBF {K}，N}，N}（X，T）= T^{n- | \ MathBf {K} |} |} x^\ s^\ Mathbf {K}使用$ | \ Mathbf {K} | \ le n $，在所有多项式中具有最低$ l^2 $规范，$ t^{n- | \ Mathbf {k} |} X^\ Mathbf {K} + \ Mathsf {P Mathsf {p} $都带有$°\ Mathsf {p} $ \ mathsf {v} _ {\ mathbf {k}，n} $。第二个家庭由Rodrigues类型公式定义的正交多项式组成，当时$ W $是Laguerre的重量或Jacobi重量，在两种情况下都满足生成功能。多项式的两个家族是部分生物表达的。
We study two families of orthogonal polynomials with respect to the weight function $w(t)(t^2-\|x\|^2)^{μ-\frac12}$, $μ> -\frac 12$, on the cone $\{(x,t): \|x\| \le t, \, x \in \mathbb{R}^d, t >0\}$ in $\mathbb{R}^{d+1}$. The first family consists of monomial polynomials $\mathsf{V}_{\mathbf{k},n}(x,t) = t^{n-|\mathbf{k}|} x^\mathbf{k} + \cdots$ for $\mathbf{k} \in \mathbb{N}_0^d$ with $|\mathbf{k}| \le n$, which has the least $L^2$ norm among all polynomials of the form $t^{n-|\mathbf{k}|} x^\mathbf{k} + \mathsf{P}$ with $°\mathsf{P} \le n-1$, and we will provide an explicit construction for $\mathsf{V}_{\mathbf{k},n}$. The second family consists of orthogonal polynomials defined by the Rodrigues type formulas when $w$ is either the Laguerre weight or the Jacobi weight, which satisfies a generating function in both cases. The two families of polynomials are partially biorthogonal.
