Multiresolution of Lp Spaces

Multiresolution analysis plays a major role in wavelet theory. In this paper, multiresolution of Lp spaces is studied. Let S be a shift-invariant subspace of Lp(Rs) (1≤p≤∞) generated by a finite number of functions with compact support, and let Sk be the 2k-dilate of S for each integer k∈Z. It is shown that the intersection of Sk (k∈Z) is always trivial. It is more difficult to deal with the problem whether the union of Sk (k∈Z) is dense in Lp(Rs). The case p=1 or 2 can be solved by Wiener′s density theorem. Under the assumption that S is refinable, it is proved in this paper that the union of Sk (k∈Z) is dense in Lp(Rs), provided s=1, 2, or 2≤p<∞. The same is true for p=∞, if L∞(Rs) is replaced by C0(Rs). Counterexamples are given to demonstrate that for s≥3 and 1