8.3 THE TRIGONOMETRIC FUNCTIONS. skipped 8.4 THE ALGEBRAIC COMPLETENESS OF THE COMPLEX FIELD. skipped 8.5 FOURIER SERIES


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1 8.5 FOURIER SERIES THE TRIGONOMETRIC FUNCTIONS skipped 8.4 THE ALGEBRAIC COMPLETENESS OF THE COMPLEX FIELD skipped 8.5 FOURIER SERIES 8.9 Orthogonl Functions, Orthonorl: Let { n }, n, 2, 3,...,besequence of coplex functions on [, b], such tht n(x) (x)dx 0, n 6. Then { n } is sid to be n orthogonl syste of functions on [, b]. In ddition, if for ll n, { n } is sid to be orthonorl. Exple n(x) 2 dx () (2) /2 e inx, n 0, ±, ±2,..., is n orthonorl syste on [ (b), cos x, sin x, cos 2x, sin 2x,... for n orthogonl syste on [ (c) The sequence p, cos p x, sin x cos 2x sin 2x p, p, p,..., 2, ]., ]. for n orthonorl syste on [, ]. 8.0 Fourier Series: Let { n } be orthonorl on [, b]. Put c n f(t) n (t)dt, n, 2, 3,... We cll c n the nth Fourier coe cient of f reltive to { n }.Wewrite f(x) X n c n n (x) nd cll this series the Fourier series of f (reltive to { n }).
2 02 8 SOME SPECIAL FUNCTIONS 8. Theore Let { n } be orthonorl on [, b]. Let s n c (x) be the nth prtil su of the Fourier series of f, nd suppose t n (x) (x). Then nd equlity holds if nd only if f s n 2 dx pple f t n 2 dx, c,,...,n. Proof Since { n } is orthonorl, eleentry clcultions give! b ft n dx f dx nd f dx c, t n 2 dx t n t n dx!! dx 2. Hence f t n 2 dx (f t n ) (f t n )dx f 2 dx f 2 dx f 2 dx ft n dx ft n dx + X n c c + c 2 + t n 2 dx 2 c 2.
3 8.5 FOURIER SERIES 03 In prticulr, if c,wehve f s n 2 dx f 2 dx c 2. These give f t n 2 dx f s n 2 dx + c 2. The results follow this inequlity. 8.2 Theore (Bessel Inequlity) If { n } is orthonorl on [, b], nd if f(x) X n c n n (x), then the Bessel inequlity holds: X c n 2 pple n In prticulr, li n! c n 0. Proof In the proof of Theore 8., we hve 2 dx. 0 pple f s n 2 dx f 2 dx c 2. Let n!, we hve the Bessel inequlity. 8.3 Trigonoetric Series: We shll liit our ttention to the orthonorl syste {(2) /2 e inx } on [, ]. Consider functions f tht hve period 2 nd tht re Riennintegrble on [, ]. The Fourier series of f is given by where Let f(x) X n c n e inx, c n f(x)e inx dx. 2 s N (x) s N (f; x) be the Nth prtil su of the Fourier series of f. c n e inx
4 04 8 SOME SPECIAL FUNCTIONS For the convenience of rguent, we introduce the Dirichlet kernel D N (x) By the definition, s N (f; x) e inx e inx [ + e ix + + e 2iNx ] e inx e2inx+ix e ix sin(n + 2 )x sin( x 2 ). 2 2 f(t)e int dte inx 2 f(t) f(t)d N (x e in(x For f being 2periodic, the lst integrl is the se s s N (f; x) 2 f(x t)dt. t) dt t)d N (t)dt. 8.4 Theore Suppose f is Riennintegrble on [, ] nd 2periodic. If, for soe x, there re constnts >0 nd M<such tht f(x + t) f(x) pplem t for ll t 2 ( Proof Since we hve If we put, ), then li s N (f; x) f(x). N! D N (x)dx 2 2 s N (f; x) f(x) 2 2 f(x t)d N (t)dt e inx dx, [f(x t) f(x)] sin(n + 2 )t sin(t/2) 8 >< f(x t) f(x), 0 < t pple, g(t) sin(t/2) >: 0, t 0, f(x)d N (t)dt 2 dt.
5 8.5 FOURIER SERIES 05 by the hypothesis, it is cler tht g is continuous t t 0, so tht g is Riennintegrble. Fro s N (f; x) f(x) [g(t) cos(t/2)] sin Ntdt + [g(t) sin(t/2)] cos Ntdt, 2 2 by Theore 8.2, we know tht both integrls tends to 0 s N!. 8.4 We cn siilrly prove weker result: Theore Suppose f is Riennintegrble on [, ] nd 2periodic. If, for soe x, both f(x ) nd f(x+) exist, nd there exists constnts >0 nd M<such tht f(x + t) f(x+) pplemt, f(x t) f(x ) pplemt, for ll 0 <t<,then li s N (f; x) N! f(x+) + f(x ). 2 Corollry: Iff(x) 0 for ll x in soe segent J, thenlis N (f; x) 0 for every x 2 J. This result show tht the behvior of {s N (f; x)} depends only on the vlues of f in the neighborhood of x. This is very di erent fro power series. 8.5 Theore Suppose f is continuous nd 2periodic. For ny >0, there is trigonoetric polynoil P such tht P (x) f(x) < for ll rel x, where trigonoetric polynoil is finite su of the for Proof Consider where n 0 + u n (x) n ( n cos nx + b n sin nx). n n + cos x, n, 2, 3,..., 2 n + cos x dx. Then u n (x) 0, u n (x +2) u n (x), nd 2 u n (x)dx. We cli tht for ny fixed 2 (0,), u n (x)dx! 0 pple x pple s n!. In fct, since + cos x is decresing on [0,], for ny x 2 [, ] nd y 2 [0, /2], we hve + cos x + cos pple + cos y + cos( /2),
6 06 8 SOME SPECIAL FUNCTIONS so tht + cos x<r( + cos y), where r + cos + cos( /2) <. Tking powers nd dividing by n,wehveu n (x) pple r n u n (y). This iplies 2 u n(x) /2 0 /2 u n (x)dy pple r n u n (y)dy pple r n! 0, s n!. Thus u n (x)! 0 uniforly on [, ] s n!. By Theore 7.6, u n (x)dx! 0. pple x pple By Euler s forul e ix cos x + i sin x, we see tht u n (x) e ix for soe constnts. For f continuous nd 2periodic, if we put P n (x) 0 u n (y)f(x y)dy, then P n is trigonoetric polynoil for ech n, since P n (x) x u n (x z)f(z)( ) dz x+ x+ e i(x x n e i(x n n z) f(z)dz z) f(z)dz e ix f(z)e inz dz. n Let >0 be given. We shll prove tht there is N such tht n N iplies for ll x 2 [ P n (x) f(x) <,, ]. Then the ssertion of the theore follows. Since f is uniforly continuous on [ 2, 2], there exists >0 such tht for ll x 2 [, ] nd y with y <, f(x y) f(x) < /2. Put M sup <y< f(y) <. Then, fro the discussion bove, there is N such tht n N iplies u n (y)dy< /(4M). pple y pple
7 8.5 FOURIER SERIES 07 Thus, for n N, P n (x) f(x) pple y < < 2 u n (y)f(x y)dy u n (y)f(x)dy u n (y) f(x y) f(x) dy u n (y) f(x y) f(x) dy + y < u n (y)dy +2M < 2 +2M 4M. u n (y)dy pple y pple u n (y) f(x y) f(x) dy pple y pple piecewise continuous, piecewise di erentible, continuously di erentible: A function f is sid piecewise continuous on [, b], if there is prtition P {x 0,...,x n } of [, b] such tht f is continuous on ech (x i,x i ). Siilrly, function f is sid piecewise di erentible on [, b] iff 0 exists on ech (x i,x i ). A function f is sid continuously di erentible on [, b] iff 0 is continuous on [, b]. 8.5U Theore Suppose f is continuous, 2periodic, nd piecewise continuously di erentible. Then s N converges to f uniforly. Proof Without loss of generlity, we ssue tht f 0 hs only one discontinuity t 2 [, ]. In the cse of ore discontinuities, the proof is siilr. We expnd f s Fourier series: with s N (f; x) f(x) The proof is divided into two steps: X n c n e inx, c n e inx nd c n f(x)e inx dx. 2 first we prove tht the series X n converges uniforly to function, sy S(x); next we prove f(x) S(x) for ll x. c n e inx To prove the Fourier series converges uniforly, by Theore 7.0, we only need to prove tht P c n <. Sincef is continuously di erentible on (, ) nd (, ), we integrte by prts on ech of these intervls: 2c n " f(x)e inx dx + f(x) e inx in + pplef(x) e inx in x x f(x)e inx dx # f 0 (x) e inx in dx f 0 (x) e inx in dx.
8 08 8 SOME SPECIAL FUNCTIONS Put Since f( n f 0 (x)e inx dx. 2 ) f(), the bove clcultion gives c n By Bessel s inequlity for f 0,wehve By the Schwrz inequlity (Theore.35), X pple n pplen c n X pple n pplen n, n ±, ±2, ±3,... in n 2 pple f 0 (x) 2 dx. 2 0 n n 0 X A n 2 pple n pplen X A n 2 pple n pplen /2 0 X pple n pplen n 2 A /2 f 0 (x) 2 dx, 2 which iplies P c n converges. Thus, P c n e inx converges uniforly to function, sy S(x). X Next we prove tht f(x) S(x). In fct, since c n e inx converges to S(x) uniforly, by Theore 7.2, S is continuous. Notice tht, by Theore 8., the vlue of the integrl decreses s N increses. Hence li N! s N (f; x) 2 dx s N (f; x) 2 dx exists. By the first step, we know tht s N converges uniforly to S, thus, for ny N, S(x) 2 dx li s N (f; x) 2 dx pple s N (f; x) 2 dx. N! This iplies tht, by Theore 8., for ny trigonoetric polynoil P (x), S(x) 2 dx pple P (x) 2 dx.
9 8.5 FOURIER SERIES 09 Let >0 be given. By Theore 8.5, there is trigonoetric polynoil P such tht P (x) f(x) < for ll x. Thus, S(x) 2 dx<2 2, which iplies the vlue of the integrl is zero. We conclude tht f(x) S(x) for ll x, since f(x) nd S(x) re continuous. 8.6 Theore (Prsevl s Theore) Suppose f nd g re 2periodic Riennintegrble function, nd X f(x) c n e inx, n g(x) X n ne inx. Then li N! Moreover, the following equlities hold: s N (f; x) 2 dx 0. 2 f(x)g(x)dx 2 X n c n n, nd 2 dx 2 X n c n 2. Proof Let >0begiven. We show tht for 2periodic Riennintegrble function f, there is function h tht is continuous, 2periodic nd piecewise di erentible such tht h(x) 2 dx<. 2 In fct, we only prove this ssertion for rel f. Iff is coplex, we just need to pply the conclusion for rel nd iginry prts seprtely. Since f is Riennintegrble, by Theore 6.6, there is prtition P {x 0,...,x n } of [, ], such tht U(P, f) L(P, f) (M i i ) X i < M, n where M i sup f(x), i inf f(x), nd M sup x i pplexpplex i x i pplexpplex i define piecewise liner (continuous) function on [x i,x i ]by h(x) x i x x i x i f(x i )+ x x i x i x i f(x i ). pplexpple. Now we
10 0 8 SOME SPECIAL FUNCTIONS The fcts tht h(x i )f(x i ), h(x i )f(x i ), nd f is 2periodic iply tht h cn be continuously extended to the whole rel line. We still denote the extended function s h. Hence we hve continuous, 2periodic, nd piecewise di erentible function h. To see the estite, we notice tht on [x i,x i ], i pple h(x) pple M i, pple i pple n. Hence, h(x) 2 dx pple X (M i i ) 2 x i pple 2M X (M i i ) x i < 2. To prove the liit in the theore, for the function h, we know tht, by Theore 8.5U, there exists N such tht n N iplies for ll x 2 [ 2 h(x) s N (h; x) < p, ]. By the Bessel inequlity, we know tht s N (f; x) s N (h; x) 2 dx 2 Thus, if n N, 2 2 pple 3 2 pple < 3( + + ) 9. Tht is, s N (f; x) 2 dx s N (f h; x) 2 dx pple 2 h(x)+h(x) s N (h; x)+s N (h; x) s N (f; x) 2 dx h(x) 2 dx + li N! Rewriting the integrl in the liit gives s N (f; x) 2 dx h(x) 2 dx<. h(x) s N (h; x) 2 dx + s N (h; x) s N (f; x) 2 dx s N (f; x) 2 dx 0. 2 [f(x) s N (f; x)] [f(x) s N (f; x)] dx 2 dx 2 dx 2 f(x) 2 c n e inx! f(x)dx + 2 c n 2, c n e inx! dx c n e inx! c n e inx! dx
11 8.5 FOURIER SERIES so tht 2 dx 2 X n c n 2. In generl, we cn siilrly prove li N! 2 which leds to the generl Prsevl s identity. [f(x) s N (f; x)]g(x)dx 0,
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