University of Guilan Journal of Mathematical Modeling 2345-394X 7 1 2019 03 01 Bases for polynomial-based spaces 21 34 3049 10.22124/jmm.2018.11242.1189 EN Maryam Mohammadi Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran Maryam Bahrkazemi School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran Journal Article 2018 09 08 Since it is well-known that the Vandermonde matrix is ill-conditioned, this paper surveys the choices of other bases. These bases are data-dependent and are categorized into discretely $\ell^2$-orthonormal  and continuously $L^2$-orthonormal bases. The first one is defined via a decomposition of the Vandermonde matrix while the latter is given by a decomposition of the Gramian matrix corresponding to monomial bases. A discussion of various matrix decomposition (e.g. Cholesky, QR and SVD) provides a variety of different bases with different properties. Special attention is given to duality. Numerical results show that the matrices of values of the new bases have smaller condition numbers than the common monomial bases. It can also be pointed out that the new introduced bases are good candidates for interpolation. https://jmm.guilan.ac.ir/article_3049_690e13a27bd207112d0b5f88eabeeaaa.pdf