MULTIRESOLUTION OF Lp SPACES
نویسنده
چکیده
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(IR) (1 ≤ p ≤ ∞) generated by a finite number of functions with compact support, and let Sk be the 2-dilate of S for each integer k ∈ ZZ. It is shown that the intersection of Sk (k ∈ ZZ) is always trivial. It is more difficult to deal with the problem whether the union of Sk (k ∈ ZZ) is dense in Lp(IR). 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 ∈ ZZ) is dense in Lp(IR), provided s = 1, 2, or 2 ≤ p < ∞. The same is true for p = ∞, if L∞(IR) is replaced by C0(IR). Counterexamples are given to demonstrate that for s ≥ 3 and 1 < p < 2, the aforementioned results on density are no longer true in general. AMS Subject Classification: 41 A 63, 41 A 30, 42 C 99, 46 C 99
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