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Answer: about 2D-refinement equation (WD 3.2 #12)
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Lars Villemoes (

PostPosted: Mon Dec 02, 2002 12:45 pm    
Subject: Answer: about 2D-refinement equation (WD 3.2 #12)
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Answer: about 2D-refinement equation (WD 3.2 #12)

Reply to question in Wavelet Digest 3.2 about 2D-refinement equation
regularity. (Topic #12):

The Sobolev regularity of solutions to multidimensional refinement equations
with finite masks can be found from the spectral radius of a finite matrix.
This is true when some power of the dilation matrix is proportional to the
identity, as in the quincunx or FCO cases. This kind of result can be
found implicitly in [1] and explicitly for the quincunx case in [4].

Hoelder regularity is harder. The technology for continuity is known
to the CAGD community, [2], [3]. The Hoelder exponent can be related
to the spectral radius of the adjoint of the subdivision operator when
it acts on a certain subspace of l^1, see [5].

Lars Villemoes
Dept. Mathematics, Royal Inst. Technology, S 10044 Stockholm.

1. A. Cohen and I. Daubechies: "Non-separable bidimensional wavelet bases"
Rev. Mat. Iberoamericana 1993 pp. 51-137 vol 9 No. 1 (1993)

2. G. Deslauriers, J. Dubois and S. Dubuc: "Multidimensional iterative
interpolation", Can. J. Math. vol 43 pp. 297-312 (1991)

3. N. Dyn and D. Levin "Interpolating subdivision schemes for the generation
of curves and surfaces", pp. 91-106 in Multivariate interpolation and
approximation ed. W. Haussmann and K. Jetter, Birkauser Verlag Basel (1990)

4. L.F. Villemoes: "Sobolev regularity of wavelets and stability of
iterated filter banks", pp. 243-251 in Progress in wavelet analysis and
applications, proc. Toulouse June 1992, ed. Y. Meyer and S. Roques.
Editions Frontieres, France.

5. L.F. Villemoes: "Continuity of quincunx wavelets", to appear in
Applied and Computational Harmonic Analysis (1994).
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