Smoothness of Wavelets
The regularity of multivariate wavelet frames with an arbitrary dilation is studied by the matrix approach. The formulas for the Hölder exponents in spaces C and \(L_p\) are obtained in terms of the joint spectral radius of the corresponding transition ma
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Smoothness of Wavelets
Abstract The regularity of multivariate wavelet frames with an arbitrary dilation is studied by the matrix approach. The formulas for the Hölder exponents in spaces C and L p are obtained in terms of the joint spectral radius of the corresponding transition matrices. Some results on higher order regularity, on the local regularity, and on the asymptotics of the moduli of continuity in various spaces are presented.
In this chapter, we address the problem of regularity of compactly supported wavelets. How to decide whether the wavelet function belongs to a given functional space and how to compute its exponent of regularity? Since the wavelet function is compactly supported, it is generated by refinement Eq. (2.16) with finitely many nonzero terms. It will be convenient to change the notation to the form used in most of literature: We denote coefficients h i = ci and change the summation index from k to −k. Thus, we consider refinement equation of the form ϕ(x) =
ck ϕ(M x − k) .
(6.1)
k∈I
where c = {ck }k∈I is a finite set of coefficients, I ⊂ Zd is a finite index set. The corresponding transition operator (2.49) now gets the form: [T ϕ](x) =
ck ϕ(M x − k) .
(6.2)
k∈I
Thus, the solution of refinement Eq. (6.1) is the eigenvector of T with the eigenvalue 1: T ϕ = ϕ. Clearly, the problem of regularity of compactly supported wavelets is reduced to the regularity of solutions of Eq. (6.1). For the univariate refinement equations, there are several efficient methods to compute or to estimate the exponents of regularity of solutions: the brute force method, the method of estimating invariant cycles, the method of computing the Sobolev regularity, and the matrix method. The first and the second ones approximate the exponents of regularity, while the latter two compute the exact values. For the multivariate equations with an arbitrary dilation matrix, only the methods of computing of the Sobolev regularity can be efficiently extended (see, for instance, [1–4], and references therein). The other methods are © Springer Nature Singapore Pte Ltd. 2016 A. Krivoshein et al., Multivariate Wavelet Frames, Industrial and Applied Mathematics, DOI 10.1007/978-981-10-3205-9_6
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6 Smoothness of Wavelets
generalized only in some special cases. For instance, when the dilation matrix M is isotropic, i.e., it is orthogonal in some basis [4–7]. Another approach is to consider regularity in special Besov spaces corresponding to the matrix M [8]. We focus on the matrix method that computes the Hölder regularity in the spaces C and L p by means of the joint spectral radius of transition matrices associated with refinement equations. Let us recall that the Hölder exponent of a function ϕ ∈ C(Rd ) is αϕ = sup α ≥ 0 ϕ(· + h) − ϕ(·) ∞ ≤ C h α . The results of this chapter are based on recent work [9], where it was shown that the matrix method can be generalized to multivariate refinement equations with arbitrary dilation matrices, not necessarily isotropic. First, we are going to establi
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