Title: Ladder operators for generalized Zernike or disk polynomials Show Chinese title

Title: 广义Zernike或圆多项式的阶梯算子

Abstract: The aim of this work is to report on several ladder operators for generalized Zernike polynomials which are orthogonal polynomials on the unit disk $\mathbf{D}\,=\,\{(x,y)\in \mathbb{R}^2: \; x^2+y^2\leqslant 1\}$ with respect to the weight function $W_{\mu}(x,y)\,=\,(1-x^2-y^2)^{\mu}$ where $\mu>-1$. These polynomials can be expressed in terms of the univariate Jacobi polynomials and, thus, we start by deducing several ladder operators for the Jacobi polynomials. Due to the symmetry of the disk and the weight function $W_{\mu}$, it turns out that it is more convenient to use complex variables $z\,=\, x+iy$ and $\bar{z}\,=\,x-iy$. Indeed, this allows us to systematically use the univariate ladder operators to deduce analogous ones for the complex generalized Zernike polynomials. Some of these univariate and bivariate ladder operators already appear in the literature. However, to the best of our knowledge, the proofs presented here are new. Lastly, we illustrate the use of ladder operators in the study of the orthogonal structure of some Sobolev spaces. Show Chinese abstract

Abstract: 本文的目的是报告一些广义Zernike多项式的阶梯算子，这些多项式是在单位圆盘$\mathbf{D}\,=\,\{(x,y)\in \mathbb{R}^2: \; x^2+y^2\leqslant 1\}$上关于权函数$W_{\mu}(x,y)\,=\,(1-x^2-y^2)^{\mu}$的正交多项式，其中$\mu>-1$。这些多项式可以用单变量Jacobi多项式来表示，因此我们首先推导出一些Jacobi多项式的阶梯算子。由于圆盘和权函数$W_{\mu}$的对称性，发现使用复变量$z\,=\, x+iy$和$\bar{z}\,=\,x-iy$更为方便。确实，这使我们能够系统地使用单变量阶梯算子来推导复数广义Zernike多项式的类似算子。其中一些单变量和双变量阶梯算子已经在文献中出现过。然而，据我们所知，这里提供的证明是新的。最后，我们展示了阶梯算子在研究某些Sobolev空间正交结构中的应用。