A question about continuous derivatives

in article 1130018797.195034@ostenberg.wh.uni-dortmund.de, Martin Eisenberg
at martin.eisenberg@udo.edu wrote on 10/22/2005 18:06:

> Seek simplicity and mistrust it.
> --Alfred Whitehead


in article juednc3tHLeRQMfeRVn-tg@comcast.com, Robert E. Beaudoin at
rbeaudoin@comcast.net wrote on 10/22/2005 20:44:

> You're welcome!  I like the Whitehead quote you used as a sig.

i like it, too.  it's kinda a counterweight to Occam's Razor.  the Einstein
quote that Bob Cain puts in his sig, is probably the optimal position
between the two extremes.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

robert bristow-johnson wrote:
> in article 1129968325.560607.57340@z14g2000cwz.googlegroups.com,
> abariska@student.ethz.ch at abariska@student.ethz.ch wrote on 10/22/2005
> 04:05:
>
> > robert bristow-johnson wrote:
...
> >>> What about windowed sinc?
> >>
> >> all derivatives are continuous until you get to the place where the window
> >> is spliced to silence.  then at least *some* derivative at some order has to
> >> have a jump discontinuity.
> >
> > This is not necessarily the case.
>
> it *is* necessarily the case.
>
> > Some windows are infinitely often differentiable.
>
> not at where they get spliced to silence.

I don't know if you have read Robert's post, but it describes the
general idea of the construction of a window that is non-zero only on a
closed interval and continuously differentiable on the whole real
numbers. The existence of such windows ("smooth functions with compact
support") is quite essential for distribution theory.

mike@unknown.no.spam.com wrote:

> am I right to say that the higher the order of a B-spline, the more
> continuous derivatives it has?

> And that sinc interpolation has infinite continuous derivatives?

My first thought when I saw continuous and derivative in the same
sentence was not that the derivative itself was continuous, but
that the derivative order was.

Consider the derivatives of x**y.

First derivative y*x**(y-1)
Second derivative y*(y-1)*x**(y-2).

Nth derivative (y!/(y-N)!)*x**(y-n).
Use the Gamma function instead of factorial, and it
can be defined for non-integer N.

After a few seconds I realized that this was not the original
question, but that was my first thought when I saw it.


-- glen