An analysis of moving least squares (m.l.s.) methods for smoothing and interpolating scattered data is presented. In particular, theorems are proved concerning the smoothness of interpolants and the description of m.l.s. processes as projection methods. Some properties of compositions of the m.l.s. projector, with projectors associated with finiteelement schemes, are also considered. The analysis is accompanied by examples of univariate and bivariate problems.
Companion matrices of matrix polynomials L(λ) (with possibly singular leading coefficient) are a familiar tool in matrix theory and numerical practice leading to so-called "linearizations" λB − A of the polynomials. Matrix polynomials as approximations to more general matrix functions lead to the study of matrix polynomials represented in a variety of classical systems of polynomials, including orthogonal systems and Lagrange polynomials, for example. For several such representations, it is shown how to construct (strong) linearizations via analogous companion matrix pencils. In case L(λ) has Hermitian or alternatively complex symmetric coefficients, the determination of linearizations λB − A with A and B Hermitian or complex symmetric is also discussed.
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