ISSN:
0945-3245
Schlagwort(e):
Mathematics Subject Classification (1991):65D15
Quelle:
Springer Online Journal Archives 1860-2000
Thema:
Mathematik
Notizen:
Summary. The functions $f_1(x), \dots, f_r(x)$ are refinable if they are combinations of the rescaled and translated functions $f_i(2x-k)$ . 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 $1, x, \dots, x^{p-1}$ 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, $r \times r$ 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 conditions on the Fourier transforms $\hat f_i(\omega)$ , but for refinable functions they can be explicitly verified from the $c_k$ .
Materialart:
Digitale Medien
URL:
http://dx.doi.org/10.1007/s002110050185
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