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.