  An adaptive procedure for constructing a series of biorthogonal polynomials
to a basis of monomials spanning the same finite-dimensional inner product
space is proposed. By taking advantage of the orthogonality of the original
basis, our procedure circumvents the well-known instability problem arising
from the matrix inversion involved in classical polynomial regression.
Moreover, the recurrent generation of biorthogonal polynomials in our framework
facilitates the upgrading of all polynomials to include one additional element
in the set whilst also allowing for a natural downgrading of the polynomial
regression approximation. This is achieved by the posterior removal of any
basis element leading to a straightforward approach for reducing the
approximation order. We illustrate the usefulness of this approach through a
series of examples where we derive the resulting biorthogonal polynomials from
Legendre, Laguerre, and Chebyshev orthogonal bases.
