The polynomial-Fourier transform with minimum mean square error for noisy data (with Y. Wang), Journal of Computational and Applied Mathematics, vol.234, no.5, pp.1586-1610, 2010.


In 2006, Naoki Saito proposed a Polyharmonic Local Fourier Transform (PHLFT) to decompose a signal f ∈ L2(Ω) into the sum of a polyharmonic component u and a residual v, where Ω is a bounded and open domain in Rd. The solution presented in PHLFT in general does not have an error with minimal energy. In resolving this issue, we propose the least squares approximant to a given signal in L2([-1, 1]) using the combination of a set of algebraic polynomials and a set of trigonometric polynomials. The maximum degree of the algebraic polynomials is chosen to be small and fixed. We show in this paper that the least squares approximant converges uniformly for a Holder continuous function. Therefore Gibbs phenomenon will not occur around the boundary for such a function. We also show that the PHLFT converges uniformly and is a near-least squares approximation in the sense that it is arbitrarily close to the least squares approximant in the L2 norm as the dimension of the approximation space increases. Our experiments show the proposed method is robust in approximating a highly oscillating signal. Even when the signal is corrupted by noise, the method is still robust. The experiments also reveal that an optimum degree of trigonometric polynomial is needed in order to attain minimal l2 error of the approximation when there is noise present in the data set. This optimum degree is shown to be determined by the intrinsic frequency of the signal. We also discuss the energy compaction of the solution vector and give an explanation to it.

Keywords: algebraic polynomials, trigonometric polynomials, least squares approximation, mean square error, noisy data.

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