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A Perturbative-Based Generalized Series Expansion in Terms of Non-Orthogonal Component Functions
Applied Mathematics (2017)
  • Robert B Szlavik
  • Dana C. Paquin, California Polytechnic State University - San Luis Obispo
  • Galen E. Turner, III, Louisiana Tech University
Abstract
In this paper we present a generalized perturbative approximate series expansion
in terms of non-orthogonal component functions. The expansion is
based on a perturbative formulation where, in the non-orthogonal case, the
contribution of a given component function, at each point, in the time domain
or frequency in the Fourier domain, is assumed to be perturbed by contributions
from the other component functions in the set. In the case of orthogonal
basis functions, the formulation reduces to the non-perturbative case
approximate series expansion. Application of the series expansion is demonstrated
in the context of two non-orthogonal component function sets. The
technique is applied to a series of non-orthogonalized Bessel functions of the
first kind that are used to construct a compound function for which the coefficients
are determined utilizing the proposed approach. In a second application,
the technique is applied to an example associated with the inverse problem
in electrophysiology and is demonstrated through decomposition of a
compound evoked potential from a peripheral nerve trunk in terms of contributing
evoked potentials from individual nerve fibers of varying diameter.
An additional application of the perturbative approximation is illustrated in
the context of a trigonometric Fourier series representation of a continuous
time signal where the technique is used to compute an approximation of the
Fourier series coefficients. From these examples, it will be demonstrated that
in the case of non-orthogonal component functions, the technique performs
significantly better than the generalized Fourier series which can yield nonsensical
results.
Keywords
  • Non-Orthogonal Functions,
  • Series Expansion,
  • Approximate Series Expansion,
  • Perturbative-Based Approximate Expansion,
  • Numerical Approximations
Publication Date
Winter January 25, 2017
DOI
10.4236/am.2017.81009
Citation Information
Robert B Szlavik, Dana C. Paquin and Galen E. Turner, III. "A Perturbative-Based Generalized Series Expansion in Terms of Non-Orthogonal Component Functions" Applied Mathematics Vol. 8 Iss. 1 (2017) p. 106 - 116 ISSN: 2152-7393
Available at: http://works.bepress.com/rszlavik/20/