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In this paper, we study a class of approximation problems, appearing in dataapproximation and signal processing. The approximations are constructed ascombinations of polynomial splines (piecewise polynomials), whose parametersare subject to optimisation, and so called prototype functions, whose choice isbased on the application, rather than optimisation. The correspondingoptimisation problems can be formulated as Linear Least Squares Problems(LLSPs). If the system matrix is non-singular, then the corresponding problemcan be solved using the normal equations method, while for singular casesslower (but more robust) methods have to be used. In this paper we developsufficient conditions for non-singularity. These conditions can be verifiedmuch faster than the direct singularity verification of the system matrices.Therefore, the algorithm efficiency can be improved by choosing a better suitedmethod for solving the corresponding LLSPs.

Linear least squares problems involving fixed knots polynomial splines and their singularity study