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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