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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.