# interpolation

Duane K. Wise dwise at wholegrain-ds.com
Thu Dec 4 22:34:34 EST 1997

```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

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)

```