An improvement of Papadakis' theorem*
Zhihua Zhang, Mu Lehua, Peixuan Zhang · Progress in Natural Science Materials International · 2004
There exist many orthonormal wavelets which cannot be derived by multiresolution analysis (MRA) with a single scaling function. In 2000, Papadakis announced that any orthonormal wavelet is derived by a generalized MRA with countable scaling functions at most. We improve Papadakis' theorem and find that for any othonormal wavelet, the least number of the corresponding scaling functions is just the essential supremum of the dimension function of the orthonormal wavelet. Moreover, we construct directly the fewest scaling functions.