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Math Help - A question about multiresolution analysis (from a topological point of view)

  1. #1
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    A question about multiresolution analysis (from a topological point of view)

    Hi,

    I have a problem understanding something

    This is a snapshot of a book I am reading



    Point no. 2 concerns me, because it looks to me like it contradicts itself, with "this or this"

    The first part says

    \sum_{j}V_j = {L^2(R)} which, to me, looks completely equivavalent to
    \lim_{j \rightarrow \infty}V_j = {L^2(R)}
    given the nested nature of these subspaces.

    However, the paper says


    so what troubles me is this: is this countable union \sum_{j}V_j equal to {L^2(R)} or is it only dense in {L^2(R)}?

    I personally think it's the former, and I don't understand this "dense" part. Could someone perhaps clarify this for me?

    Much obliged!
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  2. #2
    Super Member Rebesques's Avatar
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    Re: A question about multiresolution analysis (from a topological point of view)

    It is equal.
    What the author wants to state by saying dense, is that for every element u of L^2, there exists a series (u_n)\subset \cup_j V_j with u=\sum_j u_j.


    Enter the wavelets!
    That is, a Schauder basis for L^2 with exactly one element in each V_j.
    Last edited by Rebesques; January 8th 2015 at 05:27 AM.
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