6 Interpolation and Polynomial Approximation
A function is often known only through a table of values: measurements, or values computed at great cost. Interpolation constructs a simple function, usually a polynomial, that passes through the tabulated points, and uses it to estimate values between them, to differentiate, or to integrate. Polynomials are the natural choice: they are easy to evaluate, differentiate and integrate, and they can approximate any continuous function as closely as we please.
Theorem 6.0.1 (Weierstrass Approximation Theorem). If \(f\) is continuous on \([a,b]\), then for every \(\varepsilon >0\) there is a polynomial \(P\) with \(|f(x)-P(x)|<\varepsilon \) for all \(x\in [a,b]\).
(A constructive proof, using Bernstein polynomials, is given in the Real Analysis course on this site.) The theorem says that good polynomial approximations exist; it does not say that the obvious ones are good. Taylor polynomials, for example, are accurate only near the point of expansion: the Taylor polynomials of \(f(x) = \frac 1x\) about \(x_0 = 1\) give, at \(x = 3\), the values \(1, -1, 3, -5, 11, -21,\dots \), getting worse as the degree grows, while \(f(3) = \frac 13\). Interpolating polynomials use information spread over the whole interval.
6.2 Neville’s method
6.3 Divided differences and Newton’s interpolation formulas
6.4 Hermite interpolation
6.5 Practice Problems
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