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   -> Volume 6, Issue 4


Preprint: Regularity of refinable functions.
 
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Amos Ron (amos@cs.wisc.edu)
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PostPosted: Mon Mar 31, 1997 9:33 pm    
Subject: Preprint: Regularity of refinable functions.
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#7 Preprint: Regularity of refinable functions.

Regularity of refinable functions.: a complete characterization

Zuowei Shen and I (Amos Ron) had completed a ms on the smoothness of refinable
functions. That ms can be found at
ftp://ftp.cs.wisc.edu/Approx/reg.ps or
ftp://ftp.cs.wisc.edu/Approx/reg.ps.Z.

We establish in the paper a formula that computes, for isotropic
dilation and in any number of variables, the sharp L_2-regularity of
the refinable function $phi$ in terms of the spectral radius of the
restriction of the associated transfer operator to a specific
invariant subspace $H_phi$ of it. For a compactly supported
refinable function $phi$, the relevant invariant space is proved to
be finite dimensional, and is completely characterized in terms of the
dependence relations among the shifts of $phi$ together with the
polynomials that these shifts reproduce. The previously known formula
for this compact support case requires the further assumptions that
the mask is finitely supported, and that the shifts of $phi$ are
stable. Adopting a stability assumption (but without assuming the
finiteness of the mask), we derive that known formula from our general
one. Moreover, we show that in the absence of stability, the lower
bound provided by that previously known formula may be abysmal.

Our characterization is further extended to the FSI (i.e., vector)
case, to the unisotropic dilation matrix case, and to even more
general setups. We also establish corresponding results for refinable
distributions. It is worthwhile to mention that, in the vector case,
we get separately the regularity of each function in the vector (and
not only the common smoothness of all the refinable functions).
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