Preprint: Papers on Easily-generated Function Analysis

[Yuchuan Wei (weiyuch@bltda.com.bta.net.cn)]

#2 Preprint: Papers on Easily-generated Function Analysis

Title: Frequency analysis based on general periodic functions

Abstract: Following the sine-cosine function, sawtooth wave, square
wave, triangular wave, trapezoidal wave and so on become new
easily-generated periodic functions in modern electronics. Similar to
Fourier's idea, a natural question is whether a signal can be
considered as a superposition of easily-generated functions with
different frequencies. Therefore it is necessary to generalize
Fourier analysis based on sine-cosine functions into frequency
analysis based on general periodic functions. In this paper, we
introduce the frequency series and frequency transformation based on
general periodic functions. We discuss when a frequency system is a
complete system or an unconditional basis in $L^{2}[-pi,pi]$, and
when a frequency transformation can be carried out in
$L^{2}(-infty,+infty)$. For practical convenience almost all
easily-generated functions in electronics are considered carefully as
examples. As a new and practical generalization of classical Fourier
analysis, these results will become a theoretical foundation for the
technique of easily generated function analysis in signal
processing. A lot of questions in this paper are unsolved. ( to appear
in J. Math. Phys. 40, 3654-3684, July issue, 1999.)

Easily-generated function analysis is
* Frequency analysis based on Easily-generated periodic functions;
* a new and practical generalization of classic Fourier analysis;
* a new direction of Harmonic analysis;
* a new theory of electrical waveforms;
* a new technique in signal processing;
* a new application of number theory in electronics.

See Back issues: Wavelet Digest 7.8 #7, #14 and WD 7.11 #6

A list of papers on easily-generated function analysis:
1. Square wave analysis,
J. Math. Phys., 39(1999), Augest Issue, 4226-4245,
2. Triangular wave analysis,
Computers and Mathematics with Applications, 37, 1999, No. 6, 37-56.
3. Frequency analysis based on general periodic functions
J. Math. Phys. 40 (1999), 3654-3684. July issue, (to be published )
4. Basic vibrations and anharmonic analysis of vibrations,
Inter. J. of Non-linear Mechanics, 34(1999),No.6,1061-1069(to be published)
5. Frequency analysis based on Easily-generated functions,
Letter-to-editor, Applied and computational harmonic analysis. (accepted)
6. Dirichlet Multiplication and Easily-generated Function Analysis,
submitted
7. Square wave series and its applications, submitted.
8. New methods for Fourier analysis and Fourier sinthesis, submitted
9. Transmission of information by biorthogonal functions, submitted

Dr. Yuchuan Wei is with Department of Electrical Engineering, Box 202,
Beijing University of Aeronautics and Astronautics, Beijing 100083,
China. He would like to be a visiting scholar in the fields of signal
processing, applied mathematics. Please reply to: Dr. Yuchuan Wei,
Dept. Physics, Beijing Univ. of Sci. and Tech., Beijing 100083, China
Email: weiyuch@bltda.com.bta.net.cn