RE: Stupid question

From: I.Kalaykov (igkal@computer.org)
Date: Sat Dec 08 2001 - 17:06:53 MET

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    Some extension and generalization of this issue.

    If the functions form an orthogonal family, then it is
    possible to apply a nonsingular transformation of the basis.
    In other words to transform the Haar basis to binary step
    function basis or any other basis. As the only requirement
    is nonsigularity, then you can find that basis giving smallest
    number of terms of the function approximation. Or, to get
    the most rational structure of the pice of hardware that can
    implement the respective processing (if your problem is to
    design a hardware).

    The transformations between some popular orthogonal function
    families can be found in the literature for digital signal
    processing from 1970s-1980s. You may derive yourself applying
    basic calculus knowledge.

    Ivan Kalaykov
    Orebro University, Sweden

    -----Original Message-----
    From: fuzzy-mail@dbai.tuwien.ac.at
    [mailto:fuzzy-mail@dbai.tuwien.ac.at]On Behalf Of P. Sarma
    Sent: Friday, December 07, 2001 12:19 AM
    To: Multiple recipients of list
    Subject: Re: Stupid question

    Thank you for pointing out the clear and explicit relation between the
    "binary step function" and the Haar functions.

    The Haar functions are a focused set, generating a collection of such
    step-functions, and leading to an extended orthogonal basis. These are
    indeed very similar to the binary-step-functions (bsf). One difference is
    that the bsf has less structure - there is no claim to orthogonality, and in
    that limited sense are more general. Nevertheless, via Cybenko's Theorem,
    the bsf do appear to form a basis set for smooth function approximations.
    The fact that the bsf arises with natural relevance as a simple continuous
    extension of B{0,1}, the numbers of binary logic, was the key to trying them
    out.

    Pramit

    ----- Original Message -----
    From: "Tadeusz Dobrowiecki" <tade@mit.bme.hu>
    To: "Multiple recipients of list" <fuzzy-mail@dbai.tuwien.ac.at>
    Sent: Thursday, December 06, 2001 6:01 AM
    Subject: Re: Stupid question

    >
    > As far as I recall it is a fact (Haar Theorem) that a continuous (i.e.
    > smooth) function can be approximated (pointwise) by step-like functions
    > (the Haar orthogonal function system had been designed to this
    > purpose).
    >
    > Greetings
    >
    > Tadeusz
    >
    >
    >
    >
    >
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