Based On Multiresolution Analysis On Sierpinski Fractals

Wavelet is a relatively new development in applied mathematics.It synthesizes ideas developed in many fields and has the origin across multiple disciplines,which is favored by scientists from different backgrounds.In recent years,the fractal theory has made a lot of new achievements in the mathematics,physics,engineering and scientific experiment,and other fields.The emergence of fractal is a breakthrough in the history of human epistemology.It provides a dialectical way of thinking for partial and the whole recognition system and it provides a powerful concise geometrical language to describe the complex phenomena of nature and society.In recent years,the combination of wavelet and fractal has become a hot issue in scientific research.D.Dutkay and P.Jorgensen introduced the notion of multiresolution analysis bases on ?-finite measure spaces built from dilations and translations on a fractal arising from an iterated affine function system.Although their construction works in a very general setting,the details were mainly worked out in the onedimensional setting,in particular for the ordinary Cantor set and its variants.In the case of ordinary Cantor fractal,they used Hutchinson measure H on the inflated fractal measure space R and considered a multiresolution L2(R,H)constructed from dilation by 3 and integer translation.The self-similarity of the Cantor set under dilation by 3 gave a polynomial variant of a low-pass filter,and using "gap-filling" and "detail"high-pass filters allowed them to construct the wavelet.In further work on the Cantor fractal case,D.Dutkay used the polynomial low-pass filter to construct a probability leasure v ol the solenoid ?3 and a mock Fourier transform F:L2(R,H)?L2(?3,v),such that Fourier-transformed version of the dilation operator corresponded to the shift automorphism on ?3,and the translation operator on L2(R,H)corresponded to multiplication operators on L2(?3,v).This paper describes the development of wavelet analysis and scholars' research on wavelet theory on fractal sets.According to the classic theory,the notion of multiresolution analysis based on ?-finite measure spaces built from dilations and translations on a fractal arising from an iterated affine function system,a multiresolution analysis based on Sierpinski gasket is discussed in this paper.By defining unitary dilation operator D and unitary operator T,using the Hausdorff measure H and combining the measure knowledge in fractal geometry,the multiresolution analysis on the inflated fractal set Rs based on Sierpinski gasket is proved.Then,based on the correspondence between the orthonormal basis and the multi-resolution analysis on L2(Rs,H),a framework is presented by using Hilbert spatial isomorphism J and projection K,and finally the wavelet on L2(Rs,H)is constructed.