An order n Bezier curve (vector function) can represent smooth curves not possible with the order n polynomial. Does it offer any advantages over ordinary polynomial approximation? (The setup [url]http://www.geogebratube.org/material/show/id/145552[/url].)
My answer is, generally, [i]no.[/i] A search for the best possible general approximation leads me to choose, for the 3rd order Bezier curve, the ordinary order 3 polynomial matching the curve, and tangent, at the two given points.
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But perhaps I can do better than this. Firstly, what specific behaviors of f(x) suggest that spline approximation would be superior (or at least that polynomials would be bad)? Can I reduce them to a set of conditions? Second, suppose f(x) is an anonymous function, and I am given a table of values f(x) and some its derivatives, at a set of points. How might I determine if this unknown function meets --or is likely to meet-- the conditions?
