Curves
APROXIMATION THEORY
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Optimal polynomials
Chebyshev approximation
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Remez’s algorithm
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I. Lagrange polynomials
– published by Joseph-Louis Lagrange 1795 (but also 1779 mentioned by E. Waring and were undirectly specified in 1783 by Leonhard Euler).
This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial L(x) (dashed, black), which is the sum of the scaled basis polynomials y0ℓ0(x), y1ℓ1(x), y2ℓ2(x) and y3ℓ3(x). The interpolation polynomial passes through all four control points, and each scaled basis polynomial passes through its respective control point and is 0 where x corresponds to the other three control points.