Properties And Construction Of Hermite Interpolating Refinable Function Vector

In this paper, we discuss a novel type of multivariate refinable functions vectors with interpolation property and investigate their properties. The corresponding construction method is also presented.The main works of this dissertation lie in the following three parts:.1. Based on the present results of interpolating refinable functions and function vectors, by introducing Hermite interpolating, we provide a generalized notion of interpolating refinable function vectors, that is, interpolating refinable function vectors of type Hermite. With the theories of refinable function vectors and vector cascade algorithms, a necessary and sufficient condition for the existence of generalized interpolating refinable function vectors is given strictly.2. We characterize the generalized interpolation property, that is the symmetry property ofΦin terms of their matrix masks.3. We characterize the generalized interpolation property,that is the shifts ofΦis linear independent in terms of their matrix masks. The sum rule structure of the interpolatory masks of type Hermite is also studied. As a complementory, some special characters of the interpolatory masks are also studied. Some substantial examples is presented later.4. In the last part of this paper, We shall present several examples of Hermite-type interpolanting scaling vectors.More precisely,for each given support, we shall provide Hermite-type interpolanting scaling vectors,which possess high approximation orders and symmetry property respectively.This paper is organized as follows. In the introduction, we have briefly introduced the history of wavelet analysis and its applications, also including the background of this dissertation.In chapter 1, we have reviewed some necessary definitions, glossaries and some important theories. In addition, some necessary notations and assumptions were also introduced.In chapter 2, we generalized the notion of the existing interpolating refinable function vectors to compactly supported Hermiteinterpolating refinable function vectors, that is, interpolating refinable function vectors of type Hermite, which is characterized in terms of their masks. Considering some important and necessary properties of wavelets analysis, we also dicussed symmetry property of the interpolating refinable function vectors of type Hermite.In chapter 3, we studied the sum rule structure of the interpolatory masks of type Hermite, which plays a central role in our construction of interpolatory masks of type Hermite with increasing orders of sum rules. Additionally, some special properties of the interpolatory masks were investigated in this section. As an important complementary, some examples of symmetric interpolating refinable function vectors were presented in this section.In chapter 4, We present several examples of Hermite-type interpolanting scaling vectors.More precisely,for each given support, we shall provide Hermite-type interpolating scaling vectors,which possess high approximation orders and symmetry property respectively.Finally, we have briefly summarized the research work in the dissertation and pointed out further problems to resolve.Our main results covers chapter 2 to chapter 4. The main results from the dissertation can be listed as follow.1. IntroductionThroughout this dissertation, R, Z, N₀ denote the real plane, the set of all integers and the set of all non-negative integers, respectively.For 1≤p≤∞, L_p(R) denotes the set of all Lebesgue measurable functions f such that || f ||_LP(R) := (?). And (?) denotes the linear space of all finitely supported sequences of r×s matrices on Z, and (?) for 1≤p≤∞denotes the linear space of all sequences v of r×s matrices on Z suchthat (?) for 1≤p≤∞and (?), where ||·|| denotes any matrix norm on r×s matrices.Definition 1 An MRA of L₂(R) consists of a set of closed subspaces (?) that satisfies following conditions:There exists Φ(x)∈V₀ such that {Φ(x- k)}_k∈z form the Riesz bases ofV₀From the definition of MRA , the following equation holds.Definition 2 We calledΦ= (Φ₁,... ,Φ_r)^T : (?)is a refinable function vector ifΦsatisfies the following vector refinement equation(?), (1)(?) is a finitely supported sequence of r×r matrices on Z, called the matrix mask for the refinable function vectorΦ.In the frequency domain, the vector refinement equation in (1) can be rewritten as(?), (2)where(?), (3)Definition 3 The following iteration scheme is called a (vector) cascade algorithm associated with mask a and dilation factor d = 2 (?), (4)where Q_α,2 is the cascade operator on (?) For discussion conveniently, let us introduce the following assumptions into this dissertation. Assumption 4 Let d be dilation factor with r be multiplicity . ·LetΓ_d := {Ï„₀,…,Ï„_d-1} and (?) be the complete sets of representatives of different cosets of d^-1Z\Z and Z\dZ respectively.Without loss of generality, we set (?). Definition 5 For an r×1 compactly supported function vector f on R, we say that the shifts of f are stable if span(?) for allζ∈R, and the shifts of are linear independent if span(?) for allζ∈C^r×1. Definition 6 For 0≤ν≤1 and 1≤p≤∞, we say that f∈Lip(ν, L_p,(R)) if there is a positive constant C such thatThe L_p smoothness of a function f∈L_p(R) is measured by its L_p critical exponentν_p(f) defined by(?). (5)Particularly with multiplicity r, for a function vector f= [f₁,…,f_r]^T, Definition 7 For a matrix mask (?), we say that a satisfies the sum rules of order k + 1 with respect to 2Z if there exists a sequence (?) such that (?)(?) (6) Let us recall an important quantityν_p(α;d) from [43, 47], which characterizes convergence of a vector cascade algorithm in Sobolev space and L_p smoothness exponent of a refinable function vector. For (?) and a positive integer k, as in [47], we define the space(?). (7) By convention,ν_0,y :=(?).Let us define(?), (8) where (?) andν_k,y defined in (7). Define(?): (6) holds for some k∈N₀and (?) with (?). (9) As in [47], we define the following quantity(?). (10)Particularly, up to a scalar multiplicative constant, the vectors (?),μ∈N₀ are quite often uniquely determined.The quantityν_p(α;d) will play an important role in our investigation of multivariate interpolating refinable function vectors. It is showed in [47, Theorem 4.3] and [43, Theorem 3.1] that for a nonnegative integer k, the vector cascade algorithm converges in the Sobolev Space W_p^k(R) := {f∈(?) if and only ifν_p(a;d) > k. In general,ν_p(α; d)≤ν_p(Φ) always holds. In addition by recalling [43, Theorem 4.1], if the shifts of the refinable function vectorΦassociated with mask a and dilation factor d are stable in L_p(R), thenν_p(a;d) characterizes the L_p smoothness exponent of a refinable function vectorΦ?with mask a and dilation factor d, that is,ν_p(α; d) =ν_p(Φ) in this case.2. Characterization of Interpolating Refinable Function VectorsIn this section, we concentrated on interpolating refinable function vectors of type Hermite on R and investigate their properties.Definition 8 A refinable function vectorΦ= (Φ₁,...,Φ₄)^T∈L₂(R) is Hermite interpolating ifΦis continuously differentiate and for p∈Z,j,ρ∈{0,1},λ∈{1,2},Φsatisfies the following interpolation condition(?), (11)where (?) = 1,..., r , p = 0,..., r - 1. Here, 8 is a Dirac sequence such thatδ₀ = 1 andδ_k = 0 for all k≠0, for a smooth function f,f^(j)(cdot) denotes the j - th derivative of f(cdot). It is easy to verify that the shifts ofΦin (11) are linearly independent.The following theorem characterizes the compactly supported interpolating refinable function vector of type Hermite in R.Theorem 9 Letα: (?) be a finitely supported sequence of 4×4 matrices on Z. LetΦ= [Φ₁,…,Φ₄]^T be a compactly supported scaling vector such that (?). ThenΦis Hermite-type Interpolating, that is ,Φis continuously differentiable and (11) holds if and only if the following statements holds:(iii)αis a Hermite interpolantory mask,that is,αsatisfies the sum rule oforder 2 with respect to a nonzero sequence (?) which satisfies(?)(12)here i² = - 1,and(?) (13)here z =(?).Theorem 10 LetΦ=[Φ₁...,Φ₄]^T is a Hermite interpolating refinable function vector with mask (?).If for all x∈R,ρ∈{0,1} ,λ∈{1,2}, (?), that isΦ₁(-x)=Φ₁(x),Φ₂(-x)= -Φ₂(x),Φ₃(1 - x) =Φ₃(x),Φ₄(1 - x) =-Φ₄(x) Then we have(?). (14)where (?).3. Construction of Interpolatory Masks of Type (Hermite) In this section, we studied the structure of vector y of the family of interpolatory masks of type Hermite, and provided a family of interpolatory masks of type Hermite with increasing orders of sum rules.Lemma 11 Let d = 2 be a dilation factor . Let a be a finitely supported sequence of 4 x 4 matrices on Z. Suppose that a is an interpolatory mask of type Hermite, that is,αsatisfies the sum rules of order k + 1 with a sequence (?),and (2.31) holds,then(?). (15)For discussion conveniently, we denote [A]_k,j as the (k,j)-entry of the matrix A.Theorem 12 Letμ, k be positive integers.Letα∈(?) be a Hermite interpolatory mask, then a satisfies the sum rule of order k + 1 with (k≥1) if and only if(16)holds for 0≤μ≤k , l , p = 0,1 and k∈Z,where (?) is a piecewise function such that (?)! forμ≥k,and (?) = 0 forμ< k.4. The examples of Interpolating Refinable Function VectorsIn this section,we shall present several examples of Hermite-type interpolanting scaling vectors. More precisely,for each given support, we shall provide Hermite-type interpolanting scaling vectors.which possess high approximation orders and symmetry property respectively, we concentrated on interpolating refinable function vectors of type Hermite on R and investigate their properties.Algrthm 13(1) Given the support of the matrix maskα' and give the basic form of maskαby the Theorem 9; (2) Following the Theorem 10 and the Theorem 12,add the necessary properties to the matrix mask a we can solve theα.(3) Compute theν(α; 2) ,ifν(α; 2) > 1 then the a is Hermite- interpolating matrixmask(4) Following the vector Cascade Algrthm,we have the Hermite-type interpolanting scaling vectors...