#
Analysis of Quasi-Uniform Subdivision

# Abstract

We study the smoothness of quasi-uniform bivariate subdivision.
A quasi-uniform bivariate scheme consists of different
uniform rules on each side of the $y$-axis, far enough from the axis,
some different rules near the $y$-axis,
and is uniform in the $y$ direction.
For schemes that generate polynomials up to degree m,
we derive a sufficient condition for $C^m$ continuity of
the limit function, which is simple enough to be used in practice.
It amounts to showing
that the joint spectral radius of a certain pair of matrices
has to be less than $2^{-m}$. We also relate the H\"older exponent of the
$m$-th order derivatives to that joint spectral radius.
The main tool is an extension of existing analysis techniques
for uniform subdivision schemes, although a different proof is
required for the quasi-uniform case.
The same idea is also applicable to the analysis of quasi-uniform
subdivision processes in higher dimension.
Along with the analysis we present a `tri-quad' scheme,
which is combined of a scheme on a triangular grid on
the half plane $x<0$ and a scheme on a square grid on the
other half plane $x>0$ and special rules near the $y$-axis.
Using the new analysis tools it is shown that
the tri-quad scheme is globally $C^2$.

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### BiBTeX entry

@article(LevinLevin:2003:acha:analyis-quasi-uniform,

author = "A. Levin and D. Levin",

title = "Analysis of quasi uniform subdivision",

journal = "Applied and Computational Harmonic Analysis",

pages = "18-32",

year = 2003,

volume = "15(1)")