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 -----
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
>
>
>
>
>
>
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