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The Intelligent Algorithm of the Biorthogonal
Quarternary Wavelet Packs and Applications in Physics
Hailin Gao 1,* and Ruohui Liu 2
1 Department of Fundamentals, Henan Polytechnic Institute, Nanyang, 473009
2 Dept. of Computer Science, Huanghuai University, Zhumadian 463000, China
{zxc123wer,sxxa66xauat}@126.com
Abstract. The rise of wavelet analysis in applied mathematics is due to its
applications and the flexibility. In this article, the notion of orthogonal nonsepa-
rable four-dimensional wavelet packets, which is the generalization of orthogo-
nal univariate wavelet packets, is introduced. An approach for designing a sort
of biorthogonal vector-valued wavelet wraps in three-dimensional space is pre-
sented and their biorthogonality traits are characterized by virtue of iteration
method and time-frequency representation method. The biorthogonality formu-
las concerning the-se wavelet wraps are established. Moreover, it is shown how
to draw new Riesz bases of space
2 ()
R
from these wavelet wraps. The quar-
L
ternary dual frames ia also discussed.
Keywords: B-spline function; quarternary; vector-valued wavelet wraps; Riesz
bases; iteration method; time-frequency analysis representation.
1 Introduction
Although the Fourier transfer has been a major tool in analysis for overa century, it has
a serious lacking for signal analysis in that it hides in its phases information concerning
the moment of emission and duration of a signal. What was needed was a localed time-
frquency representation which has this information encoded in it. Transform and Gabor
Transform were used for harmonic studies of nonstationary power system waveforms
which are basically Fourier Transform-based methods. Wavelet analysis has become a
developing branch of mathematics for over twenty years. The main feature of the wav-
elet transform is to hierarchically decompose general functions, as a signal or a proc-
ess, into a set of approximation functions with different scales. The last two decades or
so have witnessed the development of wavelet theory[1]. Wavelet packs, owing to their
good properties, have attracted considerable attention. They can be widely applied in
science and engineering [4,5]. Coifman R. R. and Meyer Y. firstly introduced the no-
tion for orthogonal wavelet packs which were used to decompose wavelet compo-
nents. Chui C K.and Li Chun L.[6] generalized the concept of orthogonal wavelet
packs to the case of non-orthogonal wavelet packs so that wavelet packets can be em-
ployed in the case of the spline wavelets and so on. The introduction for biorthogonal
* Corresponding author.
 
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