Introduction to the constructive theory of functions. by J. Todd

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By J. Todd

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Suppose given a set (or space) of functions provided with an inner product (,). Two functions f, g of this space are said to be orthogonal if U. g) = 0. S. if, in addition, (n, n) = I. S. 5) Proceeding formally, on this assumption, it is easy to calculate the an. Indeed These an are called the Fourier coefficients off with respect to {n}. 5). g. 3). It is clear that if there were non-trivial functions f such that ' (f, n) = 0, n = 0, I, 2, ... , then different functions could have the same Fourier coefficients.

Repeated application of RoLLE's Theorem shows that p< 2 n+ 2l(z) must vanish at some point Cin this interval. We note that H(f, z) is a polynomial of degree (2 n + 1) at most and so H< 211 + 2> = 0. Since II (z - xc)8 is a polynomial of degree (2 n + 2) with leading coefficient unity, it follows that its (2 n + 2)th derivative is (2 n + 2) ! Hence 0=pc211+2J(C)=1<211+2J(C) - (2n + 2) ! 3). 3) available we can discuss some problems about the optimal choice of the nodes. 1) Suppose we are going to interpolate in [ - 1, 1), using an n-point Lagrangian formula, for a function f(x), whose n-th derivative is bounded in [- l, 1).

2 n) ! 2n. 53 Orthogonal Polynomials Jacobi Polynomials These are the polynomials orthogonal in [ - 1, 1] with respect to the weight function w(x) = (1 - x)"' (1 + x)ll where oc. > - 1, fJ > - 1. The ultraspherical polynomials correspond to the case oc. = fJ. Various normalizations are in use. If the basic interval is taken to be [O, 1] the polynomials orthogonal with respect to xq-1 (1 - x)f>-q are denoted by { Gn(P, q, x)}; these have been tabulated by KARMAZINA (1954). Bibliography There is an enormous literature dealing with orthogonal polynomials.