Re: [Axiom-mail] Defining piece-wise functions and drawing, integrating, ...

[Ralf Hemmecke]

> Roughly generalized functionals can be taken as the limit of the
> sequence of functions that are constrained.  For instance a sequence of
> Gaussian curves that shrink in width a have an area of one.  It turns
> out the constraints, smaller and smaller footprints, define the limit
> more than the particular functional sequence.  This is similar to
> considering the real numbers being the limit of sequences of rational
> numbers.

Clear to me, but try to explain that to someone who has not studied 
mathematics and who first thinks in terms of piecewise functions.

> You could consider a particular real number as the limit of a
> lot rational series; for instance subsequent approximations in different
> base systems.

Well, of course, a real number is an eqivalence class of all rational 
sequences leading to the same "value".

>  Generalized functions have properties that are  useful in
> a lot of ways.  For instance a piecewise polynomial can be represented
> like I showed.  In the case of Electrical Engineering the representation
> can be directly Fourier transformed and written down by inspection.
> > g(x)==if x>0 then x else -x
> >
> > he/she should rather write
> >
> > g(x)=  -x +2*DD^2,
> Because it's a "true" function that you can integrate and differentiate
> by standard rules; independent of the number of terms.

I have not really something against generalised functions. Maybe it is 
the same problem with the introduction of the imaginary i. One 
temporarily works in a much bigger space and comes back to the reals or 
ordinary functions.

> The only problem is in multiplication, D*D, and such.  But there is a
> mathematical interpretation that apparently also allows this, but so far
> I would say "don't try this at home".

See, this is what I meant. It might be a bit scary for some people.

> I hope to understand this soon.
> Why use it?  Well in EE land it's hard to start and stop things
> symbolically without step functions.  Differentiating and integrating
> step functions drag in the rest of the standard generalized functions. 
> As is typical there are two methods of definition.  Extrinsic and
> intrinsic definitions.   That is to say direct construction or by
> properties.  I like and I think symbolic algebra programs are more
> comfortable with intrinsic definitions.  That is to lay down a set of
> operational rules and the using them like a machine.   As I said, I and
> not 100% clear on how to do this, but am sure it can be done.

Well, if there are rules how to operate with generalised functions, then 
there should be no problem at all of implementing them in Axiom. One 
really has to think about a new domain (of generalised functions) in 
Axiom and what features (signature of the domain) it should export.

Ralf