The Construction Of Some Scaling Functions And Wavelets With Nice Properties

Wavelet has the time-frequency localization character which isn't owned by other analyzing tools and it can deal with odd signals effectively, so it has been the hotspot of theory research since it was presented. Many wavelets and multi-wavelets have been constructed, which have nice properties. These theory fruits have been using in many fields of science and production more and more widely, making good effects in scientific research and economy. So it is greatly significative to study the construction of scaling function and wavelet, which arc the two vital composed elements of wavelet system. The paper explore this field from various angles and construct several kinds of scaling function and a balanced multiwavelet.Firstly, based on the wavelet-sampling theorem, we construct a sort of cardinal orthogonal scaling function (COSF) which has nice decay. It avoid the shortcoming of the sampling function in the classical Shanon sampling theorem, which is only suitable for band-limited signals and decays slowly. The paper give the conditions of their one and two order regularity and prove that Haar function is the only symmetrical one in the kind of COSF. Finally some examples are presented.Secondly, the paper choose a Chui-Lian multiscaling function with multiplicity 3, which is compactly supported, symmetric, orthogonal and has approximation order 3. At first, by applying two-scaling similarity transform(TST) arithmetic to it, we make two new multiscaling functions which have approximation order 4 and 5 respectively. Then we construct a balanced multiwavelet system with multiplicity 3 by using paraunitary two-scaling similarity trans-form(PTST) algorithm to its corresponding CL multiwavelet system.Finally, we draw the figures of these vector functions by the MATLAB tool and compare them. It can be found that each new multiscaling function with higher approximation order is more smooth than the original one. Moreover, because the TST transform matrix is chosen strictly at each transform, new...