An adaptive procedure for constructing polynomials which are biorthogonal to the basis of monomials in the same finite-dimensional inner product space is proposed. By taking advantage of available orthogonal polynomials, the proposed methodology reduces the well-known instability problem arising from the matrix inversion involved in classical polynomial regression. The recurrent generation of the biorthogonal basis facilitates the upgrading of all its members to include an additional one. Moreover, it allows for a natural downgrading of the basis. This convenient feature leads to a straightforward approach for reducing the number of terms in the polynomial regression approximation. The merit of this approach is illustrated through a series of examples where the resulting biorthogonal basis is derived from Legendre, Laguerre, and Chebyshev orthogonal polynomials.

翻译：本文提出了一种自适应构造方法，用于生成与同一有限维内积空间中单项式基双正交的多项式。该方法通过利用已有的正交多项式，有效缓解了经典多项式回归中因矩阵求逆而引起的不稳定性问题。双正交基的递归生成机制便于将其所有成员升级以纳入新增基函数，同时支持自然的降基操作。这一便捷特性为减少多项式回归逼近项数提供了一种直接途径。通过一系列算例——其中双正交基分别由Legendre、Laguerre和Chebyshev正交多项式导出——验证了该方法的优越性。