从伯恩斯坦到泽尔尼克的基础转换
Change of Basis from Bernstein to Zernike
摘要：
我们增加了以前的工作范围的变化之间的基础有限基地的多项式通过定义上升和下降基地，并介绍了三种技术，从已知的定义。在升基中的多项式的最小次数可以增加，例如Bernstein和Zernike径向多项式的基。它们在计算机辅助设计和光学中有应用。给出了从单项式到伯恩斯坦多项式的降基和Zernike径向多项式的升基的映射的系数函数，并证明了它们的正确性。考虑到奇偶性，我们定义了八个一般的基变换矩阵和组成它们的系数函数的有关方程。一个主要的例子是由R Farouki [7]考虑的从移位勒让德多项式到伯恩斯坦多项式的基的变化。通过分析，我们可以找到一个更一般的系数函数的超几何函数，并通过矩阵的行和列找到系数函数的递推公式。证明了列系数函数等价于其元素的Lagrange插值多项式。我们还提供了一个公开问题的解决方案[7]，通过显示没有一般的封闭形式的表达式的系数函数从戈恩的算法。截断、交替和叠加进一步扩大了范围。例如，交替使得能够在Bernstein和Zernike径向多项式之间改变基。总结了基变换矩阵广群是一个小范畴，三角基变换矩阵和交错基变换矩阵是它的满子范畴中的态射，截断是一个共变函子.
英文摘要：
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.
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