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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