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Approximation By Translates Of Refinable Functions

By Christopher Heil, Gilbert Strang and Vasily Strela


. The functions f 1 (x); : : : ; fr (x) are refinable if they are combinations of the rescaled and translated functions f i (2x \Gamma k). This is very common in scientific computing on a regular mesh. The space V 0 of approximating functions with meshwidth h = 1 is a subspace of V 1 with meshwidth h = 1=2. These refinable spaces have refinable basis functions. The accuracy of the computations depends on p, the order of approximation, which is determined by the degree of polynomials 1; x; : : : ; x p\Gamma1 that lie in V 0 . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions f i (x) are known only through the coefficients c k in the refinement equation---scalars in the traditional case, r \Theta r matrices for multiwavelets. The scalar "sum rules" that determine p are well known. We find the conditions on the matrices c k that yield approximation of order p from V 0 . These are equivalent to the Strang--Fix condition..

Year: 1996
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