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

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.

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."

Martin Eisenberg wrote:
> Robert E. Beaudoin wrote:
> 
> 
>>Consider the function f(x) = exp(-1/x^2) for x other than 0 and
>>f(0) = 0.
> 
> 
> Thanks for straightening me out.
> 
> 
> Martin
> 

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

Robert E. Beaudoin

Robert E. Beaudoin wrote:

> Consider the function f(x) = exp(-1/x^2) for x other than 0 and
> f(0) = 0.

Thanks for straightening me out.


Martin

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

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:
>> in article 4358f7d8$0$24646$4fafbaef@reader3.news.tin.it,
>> mike@unknown.no.spam.com at mike@unknown.no.spam.com wrote on 10/21/2005
>> 10:14:
>> 
>>> am I right to say that the higher the order of a B-spline, the more
>>> continuous derivatives it has?
>> 
>> yes.
>> 
>>> And that sinc interpolation has infinite continuous derivatives?
>> 
>> yes.
>> 
>>> 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.  conceptually, you can have
windows like the Gaussian that goes on forever and never gets spliced to
anything, but practically it is kinda difficult to store and operate on an
infinite buffer of data.  in practice, *all* windows are eventually spliced
to silence.

-- 

r b-j                  rbj@audioimagination.com

"Imagination is more important than knowledge."

CH is (still) Switzerland. I'm recuperating from a two-week stroke of
pneumonia. Every now and again, I manage to turn on the home laptop.

Martin Eisenberg wrote:
> abariska@student.ethz.ch wrote:
> 
>>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. Some windows are infinitely
>>often differentiable.
> 
> 
> But we would moreover require that the value and all derivatives be 
> zero in the splice location, only true of the all-zero "window".
> 
> 
> Martin
> 

Consider the function f(x) = exp(-1/x^2) for x other than 0 and
f(0) = 0.  (This choice of value at zero removes the discontinuity
in f that would otherwise exist there.)  Now f and all its derivatives
are easily seen to have value zero at x=0, despite the fact that
f is not identically zero.  So we can form a splice g like so:  let
g(x) = 0 if x <= 0 and g(x) = exp(-1/x^2) if x > 0, and g will also
be infinitely often (continuously) differentiable.  It isn't much
more work to concoct a window function which is smooth (i.e. has
infinitely many continuous derivatives everywhere), takes on only
values between 0 and 1 inclusive, is 0 except on ]-1,1[, is 0
nowhere in ]-1,1[, and is 1 only at 0.  Windowing sincs with such
functions allows one to interpolate regularly-spaced data points
with a smooth interpolant that exactly passes through each data point.
But in practice something like a spline will be cheaper to compute,
and one seldom needs the interpolant to have infinitely many
continuous derivatives.

Robert E. Beaudoin

abariska@student.ethz.ch wrote:

Are you in China now?

Jerry
-- 
Engineering is the art of making what you want from things you can get.
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abariska@student.ethz.ch wrote:
> 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. Some windows are infinitely
> often differentiable.

But we would moreover require that the value and all derivatives be 
zero in the splice location, only true of the all-zero "window".


Martin

-- 
Quidquid latine scriptum sit, altum viditur.