MULTIRESOLUTION OF LP SPACES

Multiresolution analysis plays a major role in wavelet theory. In this paper, multiresolution of L(p) spaces is studied. Let S be a shift-invariant subspace of L(p)(R(s)) (1 less-than-or-equal-to p less-than-or-equal-to infinity) generated by a finite number of functions with compact support, and let S(k) be the 2k-dilate of S for each integer k is-an-element-of Z. It is shown that the intersection of S(k) (k is-an-element-of Z) is always trivial. It is more difficult to deal with the problem whether the union of S(k) (k is-an-element-of Z) is dense in L(p)(R(s)). 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 S(k) (k is-an-element-of Z) is dense in L(p)(R(s)), provided s = 1, 2, or 2 less-than-or-equal-to p < infinity. The same is true for p = infinity, if L(infinity)(R(s)) is replaced by C0(R(s)). Counterexamples are given to demonstrate that for s greater-than-or-equal-to 3 and 1 < p < 2, the aforementioned results on density are no longer true in general. (C) 1994 Academic Press, Inc.