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What is an interpolation?

What is an interpolation? Here are some definitions.

Noun
  1. (music) An abrupt change in elements, with continuation of the first idea.
  2. (mathematics) The process of estimating the value of a function at a point from its values at nearby points.
  3. (computing) The process of including and processing externally-fetched data in a document or program; see interpolate.
  4. That which is introduced or inserted, especially something foreign or spurious.
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Examples
He would interpolate values between his data points and he did this using a cubic interpolation formula.
During the 18th century, the interpolation of additional songs was often dictated by the presence of good singers.
If the individual who marked the Tavola is the same as the interpolator, perhaps the other two were marked but the interpolation never completed.
All interpolators have a stop method which is called when the interpolation is finished.
The interpolation adds nothing to the dramatic situation, which has been swamped by then anyway.
This work also contains a mean value type formula for inverse interpolation of the sine.

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