By FranÃ§ois-Eric Racicot
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The coefficient of x k , in the polynomial p k (x) of degree < k which agrees with f(x) at x0 , . . , xk. This coefficient depends only on the values of f(x) at the points x0, . . , xk; it is called the kth divided difference of f(x) at the points x0, . . 11) The first divided difference, at any rate, is a ratio of differences. 2-1 Prove that (x - xn). 2-l as 22-2 Calculate the limit of the formula for while all other points remain fixed. 2-3 Prove that the polynomial of degree < n which interpolates f(x) at n + 1 distinct points is f(x) itself in case f(x) is a polynomial of degree < n.
Hence, induction on the number k of zeros may now be used to complete the proof. 36 INTERPOLATION BY POLYNOMIALS Corollary If p(x) and q(x) are two polynomials of degree < k which agree at the k + 1 distinct points z0, . . , zk, then p(x) = q(x) identically. 1, be written in the form with r(x) some polynomial. Suppose that Then some coefficients c0, . . , cm with for which is nonsense. Hence, r(x) = 0 identically, and so p(x) = q(x). ” These considerations concerning zeros of polynomials can be refined through the notion of multiplicity of a zero.
3. 3 Let f(x) be a real-valued function defined on [a, b] and n + 1 times differentiable on (a, b). If p n (x) is the polynomial of degree < n which interpolates f(x) at the n + 1 distinct points there exists x0, . . 18) It is important to note that depends on the point at which the error estimate is required. This dependence need not even be continuous. As we have need in Chap. 16). For, as we show in Sec. 7, f[x0, . . , xn, x] is a well-behaved function of x. 18) to obtain a (usually crude) bound on the error of the interpolating polynomial in that interval.