Using Sampled Data to Estimate Derivatives, Integrals, and Interpolated Values

In applied mathematics, a function is often only available as a set of sampled points. Even so, much can be inferred about the unknown function. This Demonstration shows how to approximate a linear operation (selected by dropdown menu) on .
Sampled points are represented here by locator positions. The results are calculated from a polynomial of selected order that best fits these points (using least-squared error).


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More interesting than the numerical result is the formula used to calculate it. This tool demonstrates an astonishing fact:
The least squares estimator of any linear operator applied to a continuous real analytic function will always be a linear combination of the sampled data. Moreover, the coefficients will be rational if the values of the samples are rational.
These formulas are often very simple. A look at some of the bookmarked examples shows that many famous rules in numerical analysis are a result of this principle. These rules are perfect for real-time data sampling and analysis.
The ability to generate rules for over-sampled and over-constrained data is important for estimating derivatives. Derivatives are very sensitive to errors or noise in the data.
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