论文标题
从伯恩斯坦变成Zernike的基础
Change of Basis from Bernstein to Zernike
论文作者
论文摘要
我们通过定义上升和下降基础，并引入三种将其定义为已知的技术，从而提高了多项式有限基础之间的基础工作范围。 在上升的基础上，多项式的最小程度可能会增加，例如伯恩斯坦和Zernike径向多项式。他们在计算机辅助设计和光学方面具有应用。 我们提供了从单体映射到伯恩斯坦多项式降低基础的系数函数，以及Zernike radial radial多项式的上升，并证明了它们的正确性。 允许奇偶校验，我们定义了八个基础矩阵的一般更改和组成其系数函数的相关方程。 一个主要的例子是从转移的Legendre多项式转变为R Farouki [7]考虑的Bernstein多项式。该分析使我们能够为系数函数找到更一般的高几何函数，并通过矩阵行和列查找系数函数的复发。我们显示列系数函数等于其元素的拉格朗日插值多项式。我们还通过证明Gosper算法的系数函数没有一般的闭合形式表达式来解决开放问题[7]。 截断，交替和叠加进一步增加了范围。交替实现，例如，伯恩斯坦和Zernike径向多项式之间的基础变化。一个摘要表明，基矩阵的变化类是一个小类别，而基矩阵的三角形和交替的变化是完全子类别中的形态。截断是协变函数。
We increase the scope of previous work on change of basis between finite bases of polynomials by defining ascending and descending bases and introducing three techniques for defining them from known ones. The minimum degrees of polynomials in an ascending basis can increase such as with bases of Bernstein and Zernike Radial polynomials. They have applications in computer-aided design and optics. We give coefficient functions for mappings from the monomials to descending bases of Bernstein polynomials, and ascending ones of Zernike Radial polynomials and prove their correctness. Allowing for parity, we define eight general change of basis matrices and the related equations for composing their coefficient functions. A main example is the change of basis from shifted Legendre polynomials to Bernstein polynomials considered by R Farouki [7]. The analysis enables us to find a more general hypergeometric function for the coefficient function, and recurrences for finding coefficient functions by matrix row and column. We show the column coefficient functions are equivalent to the Lagrange interpolation polynomials of their elements. We also provide a solution to an open problem [7] by showing there is no general closed form expression for the coefficient function from Gosper's Algorithm. Truncation, alternation and superposition increase the scope further. Alternation enables, for example, a change of basis between Bernstein and Zernike Radial polynomials. A summary shows that the groupoid of change of basis matrices is a small category, and triangular and alternating change of basis matrices are morphisms in full subcategories of it. Truncation is a covariant functor.
