interpolation

[Duane K. Wise]

What is classically known as "cubic splines" would not be suitable  
for interpolation.  While it is made up of several cubic  
polynomials, the coefficients for the polynomials are functions of  
*all* data points in the interpolation space.  To do it right, you  
would have to use all the samples in your file to calculate each  
polynomial.  One can chop up the file into segments, but there may  
be problems at the meeting of two segments.

Lagrange polynomial interpolation requires knowledge of samples  
only in the immediate neighborhood of the interpolation point, thus  
making it suitable for a time-varying FIR.  Lagrange polynomials are  
OK for low-order interpolation, and the formula can be derived for  
an arbitrary order.  However, Lagrange interpolation does have  
first-derivative discontinuities at the sample points, as several  
have pointed out.  The Lagrange interpolation formula, however, is  
only the first-order example of another family of interpolation  
formulae (called, for some reason, osculating polynomials).  The  
second order case is called Hermite interpolation.  The higher the  
order, the higher the number of derivatives where the interpolation  
will be smooth.  Robert Bristow-Johnson and I stumbled upon this  
earlier this year, and we will likely write a paper next year on the  
case of third-order interpolation.

Robert Bristow-Johnson wrote:

> Duane Wise (he _is_ subscribed to music-dsp) might have
> something intelligent to say about this issue.

Scott Gargash wrote:

> I didn't know Duane was on this list. Hi Duane.

Good evening to you, one and all.

Enjoy, Duane Wise (dwise at wholegrain-ds.com)