Wavelet Analysis of Refinement Equations

Villemoes, Lars

doi:10.1137/s0036141092228179

Cited by 133 publications

(105 citation statements)

References 21 publications

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“…We classify the splines for which the corresponding subdivision schemes converge, and for all of them we compute the rate of convergence explicitly. In §4 we prove the following factorization theorem: any smooth refinable function (of class C l+1 , l ≥ 0) is the convolution of a continuous refinable function and a refinable spline of order l. This extends a series of earlier results in this direction, obtained in [1,2,10] and, in the case of one variable, answers a question stated by Caveretta, Dahmen, and Micchelli in [6]. §2.…”

supporting

confidence: 62%

“…As to general refinement equations, for l = 0 an analog of Theorem 4 was established in [2]. In [10] Villemoes proved that any refinable function is a linear combination of integral shifts of a stable refinable function having the same smoothness. In [12] it was shown that any smooth refinable function is the convolution of a continuous (and even stable) refinable function and a spline.…”

Section: Now It Remains To Apply Item C) Of Proposition 3 and Formulamentioning

confidence: 97%

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Piecewise-smooth refinable functions

Protasov

2005

St. Petersburg Math. J.

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Abstract. Univariate piecewise-smooth refinable functions (i.e., compactly supported solutions of the equation ϕ( x 2 ) = N k=0 c k ϕ(x−k)) are classified completely. Characterization of the structure of refinable splines leads to a simple convergence criterion for the subdivision schemes corresponding to such splines, and to explicit computation of the rate of convergence. This makes it possible to prove a factorization theorem about decomposition of any smooth refinable function (not necessarily stable or corresponding to a convergent subdivision scheme) into a convolution of a continuous refinable function and a refinable spline of the corresponding order. These results are applied to a problem of combinatorial number theory (the asymptotics of Euler's partition function). The results of the paper generalize several previously known statements about refinement equations and help to solve two open problems. §1. Introduction and the statement of the problem Refinement equations of the type(univariate two-scale difference equations with compactly supported mask) have found many applications in wavelets and subdivision algorithms in approximation theory, as well as in the design of curves and surfaces, in probability theory, combinatorial number theory, mathematical physics, and so on (see

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supporting

confidence: 62%

Section: Now It Remains To Apply Item C) Of Proposition 3 and Formulamentioning

confidence: 97%

Piecewise-smooth refinable functions

Protasov

2005

St. Petersburg Math. J.

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“…a is, in a sense, responsible for the regularity of wavelets and refinable functions [6,8,15]. This observation led to intensive studies of the spectral radius of A (q) a .…”

Section: It Is Well Known That the Operators D (Q)mentioning

confidence: 99%

Spectral radii of refinement and subdivision operators

Didenko

2005

Proc. Amer. Math. Soc.

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Abstract. The spectral radii of refinement and subdivision operators considered on the space L 2 can be estimated by using norms of their symbols. In several cases, including those arising in wavelet analysis, the exact value of the spectral radius is found. For example, if T is the unit circle and if the symbol a of a refinement operator satisfies the conditions |a(z)| 2 + |a(−z)| 2 = 4, z ∈ T, and a(1) = 2, then the spectral radius of this operator is equal to √ 2.

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“…The class of slant Toeplitz operators having the property that the matrices with respect to the standard orthonormal basis could be obtained by eliminating every alternate row of the matrices of the corresponding Toeplitz operators introduced by Ho [4] in 1995.These operators are very useful in many applications like prediction theory [2], solution of differential equation [6] and wavelet analysis [3]. However, these studies were made in context of the usual Hardy Spaces and Lorentz spaces.…”

Section: Introductionmentioning

confidence: 99%