Pointwise Convergence of Bounded Cascade Sequences
نویسندگان
چکیده
The cascade algorithm plays an important role in computer graphics and wavelet analysis. For an initial function φ0, a cascade sequence (φn)n=0 is constructed by the iteration φn = Caφn−1, n = 1, 2, . . . , where Ca is defined by Cag = ∑ α∈Z a(α)g(2 · −α), g ∈ Lp(R). In this paper, under a condition that the sequence (φn)n=0 is bounded in L∞(R), we prove that the following three statements are equivalent: (i) (φn)n=0 converges a.e. x ∈ R. (ii) For a.e. x ∈ R, there exist a positive constant c and a constant γ ∈ (0, 1) such that |φn+1(x) − φn(x)| ≤ cγn ∀n = 1, 2, . . . . (iii) For some p ∈ [1,∞), (φn)n=0 converges in Lp(R). An example is presented to illustrate our result.
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