Preprint: Two Papers on Easily-generated Function Analysis

[wei (wei@mail.login.com.cn)]

#6 Preprint: Two Papers on Easily-generated Function Analysis

1. Square wave analysis
Abstract

This paper concerns square wave analysis, which is in no small part a
generalization of the Fourier analysis based on sine and cosine
functions. Following sine and cosine functions square waves become
another frequently used and easily generated waveform in electronics,
so it is an urgent practical problem to study the basic properties of
square waves and the fundamental theory of square wave analysis. Like
the sine-cosine function system, the square wave system, i.e., the
function system of square waves with different frequencies ( or
periods), is linearly independent and complete in the real Hilbert
spaces $L^{2}(-pi,pi)$, so square waves can approximate an arbitrary
quadratically integrable function with a vanishing mean-square
error. Since the square wave system is non-orthogonal, its
biorthogonal functions are presented so that a function with a certain
condition can be expanded as a square wave series easily. Meanwhile
the orthogonalization of the square wave system is discussed so that a
quadratically integrable function can be approximated best by a
superposition of finite square waves. These results form the
theoretical basis of square wave analysis technique in modern
electronics.

2. Triangular wave Analysis
Abstract

This paper concerns triangular wave analysis including triangular wave
series and triangular wave transformation, which is very similar to
Fourier analysis based on sine and cosine functions. Besides
sine-cosine functions, triangular waves are frequently-used and
easily-generated periodic functions in electronics as well, so it is
an urgent practical problem to study the basic properties of
triangular waves and the fundamental theory of triangular wave
analysis. We show that triangular waves and sine-cosine functions not
only have the similar graphs, but also possess similar analysis
properties. Any continuous periodic function may be approximated
uniformly by linear combinations of triangular waves as well as
trigonometric functions, and every function $f(x)in L^{2}[-pi,pi]$
has a triangular wave series as well as a Fourier series. Since the
triangular waves are nonorthogonal in $L^{2}[-pi,pi]$, the
orthonormalization is discussed so that a function $f(x)in
L^{2}[-pi,pi]$ can be approximated best by a superposition of given
finite triangular waves. Finally, we introduce the theory of the
triangular wave transformation in $L^{2}(-infty,infty)$, which has a
close relation with Fourier transformation. These results form the
theoretical foundation of the technique of triangular wave analysis in
modern electronics.

Sumary

Easily-generated function analysis is a new and practical
generalization of Fourier analysis based on sine-cosine function
analysis, and a new theory and approach of signal processing in
electronics. As frequency analysis based on general periodic function,
it will become a new branch of harmonic analysis. Because it has a
close relation with number theory, it is a new field of applications
of number theory. A lot of interesting and important questions are
unsolved still. We are willing to coorprate with any organization or
person in any form. Dr Yuchuan wei wishes to be a visiting scholar in
an English-spoken country.

See The Wavelet Digest, Volume 7, Issue 8, (7),(14).

Yours sincerely,
Dr. Yuchuan Wei
wei@mail.login.com.cn

(or weiyuch@bltda.com.bta.net.cn)