Advanced Calculus fi..

y the Schwarz inequality (7.46). By (7.65),Chapter 7 Fourier Series and Orthogonal Functions 51 1By part (i) of Theorem 17 the Weierstrass M-test is applicable to the Fourier-Legendre series of f (x): for n ? 2,The series C Mn converges by the integral test. Therefore the Fourier-Legendreseries converges uniformly for - 15 x 5 1.aCOROLLARY If f (x) is very smooth for - 1 ( x 5 1, then the Fourier-Legendreseries of f (x) converges uniformly to f (x) for - 1 < x 5 1.This is a consequence of Theorems 1 l,19, and 20.Remark. No condition of periodicity or other condition at x = f 1 is imposedon f (x), as in the analogous theorem (Theorem 5) for trigonometric series. Thisis because of a symmetry of the Legendre polynomials, somewhat like that of thefunctions cos nx (Problem 2 following Section 7.13).THEOREM 21for the interval - 1 < x < 1.The Legendre polynomials form a complete orthogonal systemaThis follows at once from the two preceding theorems, by virtue of the generalTheorem 12 of Section 7.12.Remark.Given a sequence fn(x) of functions continuous for a < x 5 b, no finitenumber of which are linearly dependent, one can construct linear combinations= fl, @2 = al fl + a2 f2, . . . such that the functions {+n(x)) form an orthogonalsystem in the interval a 5 x 5 b. This is carried out by the Gram-Schmidt orthogonalizationprocess, described in Section 1.14. If we choose the sequence fn(x) tobe 1, x, x2, . . . and the interval to be - 1 p x p 1, the functions +,(x) turn out to beconstants times the Legendre polynomials.The case of a function f (x) that is piecewise very smooth is covered as forFourier series by studying a particular jump function s(x). We do not carry throughthe details here but merely state the result: Just as does the Fourier series, theFourier-Legendre series converges uniformly to the function in each closed intervalcontaining no discontinuity and converges at each jump discontinuity to the halfwayvalueIt is to be emphasized that there is no peculiar behavior at the points x = f 1; herethe series converges to the function (provided that the function remains continuous).