New Bases for Polynomial-Based Spaces

Since it is well-known that the Vandermonde matrix is ill-conditioned, while the interpolation itself is not unstable in function space, this paper surveys the choices of other new bases. These bases are data-dependent and are categorized into discretely l2-orthonormal and continuously L2-orthonormal bases. The first one construct a unitary Gramian matrix in the space l2(X) while the later construct a unitary Gramian matrix in the space L2[-1,1]. The first one is defined via a factorization of Vandermonde matrix while the latter is given by a factorization of the Gramian matrix corresponding to monomial bases. A discussion of various matrix factorization (e.g. Cholesky, QR, SVD) provides a variety of different bases with different properties. Numerical results show that matrices of values of the new bases have smaller condition number rather that the common monomial bases. It can also be pointed out that the new introduced bases are good candidates for interpolation.