Subdivision algorithms with nonnegative masks generally converge

Abstract

In this paper, a subdivision scheme consists of an operator froml ^∞(ℤ) tol ^∞(ℤ) determined by a doubly infinite sequence, called the mask. This operator convolutes, in a certain sense, sequencesλ ∈l ^∞ (ℤ) with the mask, thus producing a new sequence inl ^∞(ℤ). Moreover, this new sequence is placed on a finer grid. If we iterate this process with a positive mask infinitely many times, it is known that this process will produce a continuous function, which we callf _λ. In this paper, we consider the extent to which non-negative masks yield similar results. An important application of subdivision schemes in computer graphics is the generation of curves and surfaces from an initial sequenceλ.