Advanced Calculus fi..

Chapter 7 Fourier Series and Orthogonal Functions 501If the orthogonal system {c$,(x)} is complete, then the square error En convergesto 0. This does not imply convergence of the Fourier series C c,#,(x) to f (x),although the successive partial sums do approach f (x) in the sense of square error.We describe the situation by saying that the series converges in the mean to f (x),and we writewhere "L.i.m." stands for "limit in the mean." In general, if functions f (x) and fn(x)(n = 1,2, . . .) are all piecewise continuous for a 5 x 5 b, we writeL.i.m. fn(x) = f (x)n-rm(7.58)if the sequence 11 fn(x) - f (x)ll converges to 0, that is, ifRemark. The following assertions can be proved: (a) if fn(x) converges uniformlyto f (x) for a 5 x 5 b, then fn(x) also converges in the mean to f (x), provided thatall functions are piecewise continuous; (b) if f,(x) converges in the mean to f (x),then f,(x) need not converge uniformly to f (x); in fact, the sequence f,(x) neednot converge for a 5 x 5 b; (c) if fn(x) converges to f (x) for a 5 x 5 b, but notuniformly, then f,(x) need not converge in the mean to f (x). For proofs, refer toProblem 9 following Section 7.13.DEFINITION An orthogonal system {c$,(x)} for the interval a 5 x 5 b has theuniqueness property if every piecewise continuous function f (x) for a 5 x 5 b isuniquely determined by its Fourier coefficients with respect to {@n(x)}; that is, iff (x) and g(x) are piecewise continuous for a 5 x 5 b and ( f, 4,,) = (g, 4,) for alln, then f (x)- g(x) = O*. This is equivalent to the statement that h(x) = 0* is theonly piecewise continuous function orthogonal to all the functions Gn(x); the systemof orthogonal functions can therefore not be enlarged.THEOREM 10 Let {@,,(x)) be a complete system of orthogonal functions for theinterval a 5 x 5 b. Then {+n(x)) has the uniqueness property.The proof is left as an exercise (Problem 8). One might expect the converse tohold, that is, that the uniqueness property implies completeness. However, examplescan be given to show that this is not the case. Theorem 12 gives more informationon this point.THEOREM 11 Let {4,(x)} be an orthogonal system of continuous functions forthe interval a 5 x 5 b and let {c$,,(x)} have the uniqueness property. Let f (x) becontinuous for a 5 x 5 b and let the Fourier series of f (x) with respect to {@n(x)}converge uniformly for a 5 x 5 b. Then the Fourier series converges to f (x).The proof is left as an exercise (Problem 10 following Section 7.13).