The polynomial Fourier transform with minimum mean square error for noisy data

In 2006, Naoki Saito proposed a Polyharmonic Local Fourier Transform (PHLFT) to decompose a signal f∈L2(Ω)f∈L2(Ω) into the sum of a polyharmonic component  uu and a residual  vv, where ΩΩ is a bounded and open domain in RdRd. 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])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 Hölder 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 L2L2 norm as the dimension of the approximation space increases. Our experiments show that 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 the trigonometric polynomial is needed in order to attain the minimal l2l2 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.