4-dimensional anti-Kahler manifolds and Weyl curvature by Kim J.

By Kim J.

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2 M U LT I S C A L E A P P ROX I M AT I O N A N D M U LT I R E S O L U T I O N When does a mask define a refinable function? 32) [55, 97]. In the following, we describe a corresponding criterion that is due to Cavaretta, Dahmen, and Micchelli. e. 33) for two integers 1 < 2 . Some of the subsequent statements in fact also hold for infinite masks. However, since any computer can only store finite arrays, an application of an infinite mask would require a truncation of this mask. This means that we would introduce an error due to the truncation.

8 from left to right). This was an open problem for quite some time, and was finally solved by Ingrid Daubechies [96, 97]. Nowadays, these functions are widely used in various applications. There exists a whole family of such functions labeled by a parameter N , which roughly means that the functions are smoother (and less local) for increasing values of N . There are no closed formulas for the arising functions N ϕ. The functions are only given in terms of their refinement coefficients. The determination of these coefficients is far from being trivial; it can be found in [96].

We do not give the proof here and refer to the cited literature. 35) converges for all c ∈ ∞ (Z) and for some c∗ ∈ ∞ (Z) we have fc∗ ≡ 0. Then, the finite mask a determines a compactly supported refinable function ϕ. 37) k∈Z the function ϕ is uniquely determined. Moreover, ck ϕ(2x − k), fc = x ∈ R, k∈Z holds for all c ∈ ∞ (Z). In the above proposition, the convergence of the subdivision scheme is assumed. It is of course an interesting question to investigate sufficient conditions on the mask coefficients to ensure convergence of the subdivision scheme.

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