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   -> Volume 7, Issue 8


Thesis: A new and practical generalization of Fourier analysis
 
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Yuchuan Wei (weiyuch@bltda.com.bta.net.cn)
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PostPosted: Mon Jul 27, 1998 8:36 am    
Subject: Thesis: A new and practical generalization of Fourier analysis
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#7 Thesis: A new and practical generalization of Fourier analysis

Title: Easily-generated Function Analysis

Author: Yuchuan Wei

Yuchuan Wei was born on September 1,1966 in China. He was awarded
Bachelar of Science at Physics Dept in Lanzhou U. in 1988, Master of
Engineering at Institute of Electronics in Chinese Academy in 1994,
and Doctor of Engineering at School of Applied Science in Beijing
U. of Sci. and Tech. in 1998. He is good at electronics, mathematics,
and physics. Now he wishes to be a visiting scholar or post-doctor in
the field of signal processing, applied mathematics, mathematical
physics in a university or institue. <weiyuch@bltda.com.bta.net.cn>

Abstract of the thesis:

In this thesis we introduce easily-generated function analysis
systematically, which is a new and practical generalization of Fourier
analysis based on sine-cosine functions.

After sine-cosine function, sawtooth wave, square wave, triangular
wave and trapezoidal wave become new easily-generated functions in
modern electronics. The following questions arise naturally: 1. Can a
signal be considered as a superposition of easily- generated functions
with different frequencies? (2) How to decompose a signal into a
series of easily-generated functions with different frequencies ? (3)
How to approximate a signal best with finite easily-generated
functions? We call this basic and important problem easily generated
function analysis.

Since easily-generated function analysis has a close relation with
number theory, Chapter 1 introduces some basic concepts and
propositions in number theory, which will be used in the following
chapters.

Since square wave is the most important easily-generated function in
electronics, we first study square wave analysis. We call the function
system of square waves with different frequencies square wave system,
which is linearly independent, nonorthogonal and complete in
L2[-pi,pi]. The biorthogonal functions and orthonormalized functions
of square waves are given explicitly. Since square wave system is
complete, any signal can be approximated by a linear combination of
finite square waves to an arbitray mean-square error. By means of the
biorthogonal functions, a signal can be decomposed into a square wave
series. With the help of the orthonormalized functions, a signal can
be approximated best by a linear combination of given finite square
waves. [Ref 1]

Triangular wave and trapezoidal wave are common easily-generated
functions as well, so we next study triangular wave analysis and
trapezoidal wave analysis. Triangular wave and trapezoidal wave are
similar to sine-cosine function not only in waveform but aslo in
analysis properties. A continous function of period 2pi can be
approximated uniformly by a linear combination of triangular waves or
trapezoidal waves with different frequencies. Triangular wave system,
I- trapezoidal wave system, and II- trapezoidal wave system are bases
of L2[-pi,pi]. Therefore any signal can be synthesized by or
decomposed into a series of triangular waves or trapezoidal waves.
Additionally, the close relations of triangular wave analysis,
trapezoidal wave analysis, as well as Fourier analysis are given
explicitly in matrix notation. [Ref 2]

Easily-generated functions may have various waveforms, so we finally
study frequency analysis based on general periodic functions. In
Chapter 4 we introduce the concepts of frequency system and frequency
series based on a general periodic function. A frequency system may be
an orthogonal basis, an unconditional basis, a complete system, or an
incomplete system in the odd function subspace of L2[-pi,pi]. We
discuss what properties a periodic function needs to possess so that
its frequency system is a complete system or an unconditional basis.
For practical conveniece almost every easily-generated function in
electronics is considered as examples. A number of complete frequency
systems or unconditional bases of practical importance are given
explicitly. [Ref 3]

This makes it possible to represent a signal by easily-generated
functions with different frequencies. The results in this thesis form
a theory foundation for the technique of easily-generated function
analysis in electronics. Also this is another important application of
number theory.

Chapter 1. ABC of Number Theory
Chapter 2. Square Wave Analysis
Chapter 3. Trangular wave analysis and Trapezoidal Wave Analysis
Chapter 4. Frequency Analysis Based on General Periodic Functions

[Ref.1] Square Wave Analysis, 39(1998) No.8, J. Math. Phys.
[Ref.2] Triangular Wave Analysis and Trapezoidal Wave Analysis.(submitted)
[Ref.3] Frequency Analysis based on general periodic functions.(submitted)

17 Figures, 120 pages, in English.

Further work and cooperations:

As a new and practical generalization of classical Fourier Analysis,
Easily-generated Function Analysis is a new branch of morden harmonic
analysis. As an interdisciplinary foundamental research, it has a
close relation with electronics, signal processing, functional
analysis, number theory, physics and so on. My research is nothing
than a beginning and of course there are a lot of work to do.

For a functional analysis, the unsolved questions are: What are the
necessary and sufficient conditions for the function system
1, X(x),Y(x),X(2x),Y(2x),...,X(nx),Y(nx),...
to be (1) a complete system; (2) a basis; (3) a unconditional basis in
L2[-pi,pi]? Here X(x) and Y(x) are an even and odd function of period
2pi. These are the basic questions in frequency analysis based on
general periodic functions. A more concrete and practical question is:
For the even square wave Xs(x)=pi/4sgn(cos x) and the odd square wave
Ys(x)=pi/4sgn(sin x), is the square wave system
1, Xs(x),Ys(x),Xs(2x),Ys(2x),...,Xs(nx),Ys(nx),...
a basis of L2[-pi,pi] ? [ Ref.1 ]
Here sgn is the sign function, i.e., sgn(x)=1 for x>1; sgn(x)=0 for x=0;
sgn(x)=-1 for x<0.

For a professor of number theory, the unsolved question is:
Does the series
sin x = Ys(x)- 1/3Ys(3x)-1/5Ys(5x)+...+u(2n-1)/(2n-1)Ys[(2n-1)x] +...
converge at every point? Here u denotes the Moebius
function. [ Ref.1 ]

For an engineer of electronics, a series of schematic diagram can
become circuits and products. Many of them can become patents.

For a signal processor, easily-generated function analysis is a new
theory and idea, and it will bring you new methods. For a software
engineer a lot of formulas can become new and practical programs.

In one word, this is a new branch of morden harmonic analysis. No
mater whether you are a professor in mathematics or physics, or an
engineer in electronics or signal processor, only if you are
interested in Easily- generated Function Analysis, I would like to
coorperate with you in all possible forms including: meeting,
workshop, report, talk, training, course, publishing book, sircuit
designing, software developement and so on.

Yours sincerely,
Dr. Yuchuan Wei
Easily-generated Function Analysis Group (EGFAG),
Applied Physics Department,
Beijing University of Science and Technology,
Beijing 100083, China.
weiyuch@bltda.com.bta.net.cn
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