| Wavelet analysis is a prosperous research field in contemporary mathematics. The research of its theory and application is very important, it is outstanding not only in signal and image processing, in numerical analysis, approximation theory of generalized functions, but also in numerical solution of differential equations. So the construction of wavelet bases is becoming more and more popular. The theory of multi-resolution analysis (MRA) created by Mallat and Meyer, unionized the construction of basis for orthogonal wavelet function and set a unified framework for wavelet conformation.The development of refinable shift invariant spline function spaces promotes the development of the MRA. In this article, we first study the nature and characteristics of the FSI, and then receive the following conclusions :(1)φis linearly independent if and only if for each z∈d\{0}, there are b1 , b2 , , br∈d, for which the matrix A := [φ? j ( z + 2πb k )]rj , k=1 is non-singular.(2) ifφ= (φ1 ,φ2, ,φr) is a stable, local generator of S, thenψ= (ψ1 ,ψ2, ,ψs) is also a stable, local generator of S if and only if s = r andψ? (u ) =φ? (u ) A( e ?iu ),u∈d where A is an r×r matrix of Laurent polynomials which is unimodular, i.e., detA(z) is a non-trivial monomial.(3) Suppose that V is a symmetric, local FSI space on R with multiplicity r, which is refinable, i.e., for an integer m≥2, f∈V ? f ( m ?1?)∈V Let V1 = { f ( m? ) : f∈V}, Then V ? V1. If V1 = V + W, where V∩W ={0} and W is a symmetric local FSI space, then ( ) 0( ) ( ), ( ), 2 2 ,1, .d We W lr d Vlr e V mm l l ll== ??? + ? = = +∈∈This holds,(4)φ= (φ1 ,φ2, ,φr) is orthogonal if and only if its Grammian matrix is the identity.(5) For f∈S, f has a discontinuity in a derivative atαif and only ifα∈S k ? 2α(mod )∈Sk, k = 0, , n.(6) Let S be a refinable spline space as above with a total of r knots (mod Z), i.e., 0| |nkkS r=∑=. Then S has multiplicity r.(7) If S has multiplicity 2, then S is of the form S for one of the following. (a) For some n≥0, Sn= {0, 12}. (b) For some 0≤k < n, S k = Sn= {0}. (c) For some0≤k < n, S k = Sn= {0}.(8) If S comprises spline functions of degree n≥0 with knots having odd denominators, then S is of the form S for some choice of S 0, , S n. In the fourth chapter, we focusing on the construction processing of wavelet bring forwarded by Daubechies, namely : adding a series of restrictive conditions, putting a cap on the scope of the solutions, and finally, we can get the scale function and the wavelet function. But for the wavelet get by this method, it often mutually exclusive in symmetry nature, compactly supported nature, and so the orthogonal nature. This leads us to the concept of multi-wavelet and its theory is introduced subsequently.In chapter five, in order to construct a new MRA from a known one, we introduce the concept of intertwining multi-resolution analysis, and the following result was obtained: If (V p) is a multi-resolution analysis generated by compactly supported scaling functions then there is some pair of integers (q, n) and some orthogonal multiresolution analysis ( V p) such that Vq ? V0 ? Vq +nThis theorem guarantees the existence of intertwining multi-resolution analysis.At the end of this paper, we give an example of structural intertwining MRA, its graphics of scaling function and wavelet function as the figure below:...
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