Roman Haefeli reduzierer at yahoo.de
Thu Jul 3 20:15:33 CEST 2008

```On Thu, 2008-07-03 at 04:01 -0400, Matt Barber wrote:
> >> On the other hand, doesn't [tabread4~]'s Lagrange interpolator have a
> >> continuous 2nd derivative while the [tabread4c~] Hermite one does not?
> >
> > No. A Lagrange interpolator on N points is a polynomial of degree N-1, and
> > so its Nth derivative is a flat zero function without holes, and so it is
> > infinitely differentiable. However, those are pieced together as a disparate
> > mosaïc in a way that is not even C1 (continuous 1st derivative), which is
> > what prompted Cyrille to work on a replacement in the first place. Note that
> > a discontinuous 1st derivative implies that all other orders of derivatives
> > are discontinuous.
> >
>
> I'm with you on the general piecewise Lagrange not being C1, but I
> don't think it follows that all other orders are discontinuous --
> can't they alternate?  At any rate, check out the 2nd derivatives of
> the piecewise cubic Lagrange.  I believe that at x=0 it will be y[-1]
> - 2*y[0] + y[1], while at x=1 it will be y[0] - 2*y[1] + y[2].
> Therefore, since the terms match at the points on adjacent pieces, the
> 2nd derivative is continuous even though the first isn't.  I'd imagine
> you could run into this kind of phenomenon especially with piecewise
> functions.  Not sure what it means for the spectral response of the
> interpolator, though.

yo, i am not too much a math guy, so correct me, if i am talking
non-sense, but doesn't the a derivative describe the slope of of the
original function at any point? if so, a function with one ore more
discontinuities cannot have continuous derivative, because a jump at a
certain point would result in a infinitely high value at this point of
the derivative. one could argue, that in an analogue continuous world -
if the jump is short enough - the peak would be too short to be noticed,
but this certainly wouldn't be true in a digital, time discrete domain.

after all, i still don't get, how it could be figured out in the digital
domain, whether a curve is continuous or not.

roman

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