Approximation by translates of refinable functions |
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Authors: | Christopher Heil Gilbert Strang Vasily Strela |
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Institution: | (1) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332-0160, USA (heil@math. gatech.edu) , US;(2) Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (gs@math.mit.edu) , US;(3) Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA (strela@math.mit.edu) , US |
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Abstract: | Summary.
The functions
are
refinable if they are
combinations of the rescaled and translated functions
.
This is very common in scientific computing on a regular mesh.
The space of approximating functions with meshwidth
is a
subspace of with meshwidth
.
These refinable spaces have refinable basis functions.
The accuracy of the computations
depends on , the
order of approximation, which is determined by the degree of
polynomials
that lie in .
Most refinable functions (such as scaling functions in the theory
of wavelets) have no simple formulas.
The functions
are known only through the coefficients
in the refinement equation – scalars in the traditional case,
matrices for multiwavelets.
The scalar "sum rules" that determine
are well known.
We find the conditions on the matrices
that
yield approximation of order
from .
These are equivalent to the Strang–Fix conditions on the Fourier
transforms
, but for refinable
functions they can be explicitly verified from
the .
Received
August 31, 1994 / Revised version received May 2, 1995 |
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Keywords: | Mathematics Subject Classification (1991):65D15 |
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