1.3 Convergence Theorems of Fourier Series. k k k k. N N k 1. With this in mind, we state (without proof) the convergence of Fourier series.

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1 .3 Covergece Theorems of Fourier Series I this sectio, we preset the covergece of Fourier series. A ifiite sum is, by defiitio, a limit of partial sums, that is, a cos( kx) b si( kx) lim a cos( kx) b si( kx) k k k k k k provided that the limit exists. Therefore, we say that the Fourier series of f coverges to f at the poit x if f ( x) lim S ( x) a lim a cos( kx) b si( kx) 0 k k k where S ( x) a a cos( kx) b si( kx) 0 k k k With this i mid, we state (without proof) the covergece of Fourier series. Theorem. Suppose f is a cotiuous ad periodic fuctio. The for each poit x, where the derivative of f is defied, the Fourier series of f at x coverges to f( x), or S ( x) f ( x) 0 as. ow, we preset some variatios of the above Theorem. ote that the hypothesis of this theorem requires the fuctio f to be cotiuous ad periodic. However, there are may fuctios of iterest that are either cotiuous or periodic. Before we state the theorem o covergece ear a discotiuity, we eed the followig defiitio. Defiitio : The left ad right limits of f at a poit x id defied as follows. eft limit: Right limit: f ( x 0) lim f ( x h) h0 f ( x 0) lim f ( x h) h0 The fuctio f is said to be left differetiable at x if the followig limit exists: f '( x0) lim h0 f ( x h) f ( x) h The fuctio f is said to be right differetiable at x if the followig limit exists: f '( x0) lim h0 f ( x h) f ( x) h

2 Ituitively, f '( x 0) represets the slope of the taget lie to f at x cosiderig oly the part of the graph of y f () t that lies to the left of t x. The value of f '( x 0) is the slope of the taget lie to f at x cosiderig oly the part of the graph of y f () t that lies to the right of t x. Example : et f( x ) be the periodic extesio of y x, x. The f( x ) is discotiuous at x,,, The left, right limits, left ad right derivatives of f at x are f( 0) f( 0) f '( 0) f '( 0) Example. et x, x[0, / ] f( x) x, x [ /, ] The graph of f is the sawtooth wave. This fuctio is cotiuous, but ot differetiable at x /. The left ad right derivatives at x / are f '( / 0) ad f '( / 0) ow, we are ready to state the covergece theorem for Fourier series at a poit where f is ot ecessarily cotiuous. Theorem. Suppose f( x ) is periodic ad piecewise cotiuous. Suppose x is a poit where f is left ad right differetiable (but ot ecessarily cotiuous). The the Fourier series of f at x coverges to

3 f ( x 0) f ( x 0) Remark: This theorem stated that at a poit of discotiuity of f, the Fourier series of f coverges to the average of the left ad right limits of f. At a poit of cotiuity, the left ad right limits are the same, ad so i this case, Theorem reduces to Theorem. Defiitio : A fuctio is said to be piecewise smooth if it is cotiuous ad its derivative is defied everywhere except possibly for a discrete set of poits. For example, the sawtooth fuctio is piecewise smooth sice the derivative of f exists at all poits except at multiples of / (which is a discrete set of poits). We say that the Fourier series of f( x ) coverges to f( x) o [ aa, ] uiformly if the sequece of partial sums S ( x) a a cos( kx) b si( kx) 0 k k k coverges to f( x ) uiformly as, or max f ( x) S ( x) 0 as. axa Theorem 3: The Fourier series of a piecewise smooth, periodic fuctio f( x ) coverges uiformly to f( x ) o [, ]. Theorem 4: Suppose f is a elemet of ([, ]). et where S ( x) a a cos( kx) b si( kx) 0 k k k a ad b are the Fourier coefficiets of f. The S coverges to f i f ( x) S ( x) 0 as. We also call this mea covergece. Theorem 4 also holds for complex form of Fourier series. ([, ]) ; that is, Theorem 5. Suppose f is a elemet of ([, ]) with (complex) Fourier coefficiets give by k f () t e dt for. The the partial sum S() t ke k coverges to f i ([, ]) orm as. I other words, lim f S 0.

4 Eergy iterpretatio. Aother way of lookig at the theorem is i terms of eergy. I sigal processig the itegral f ( x) dx f is iterpreted as the eergy of the sigal f. The theorem the states that the Fourier series for ay fiite eergy fuctio f coverges i the mea to f. The coverse is true as well. If the Fourier series coverges i the mea to a fuctio, the that fuctio has to have fiite eergy., i.e., it has to be i ([, ]). Aother physical term that is used i coectio with Fourier series is frequecy mode. I the complex case, this is just oe of the terms e k. (The real case is similar.) The eergy of a sigle mode (term) is ke dt k There is a beautiful coectio betwee the eergy i a sigal ad the eergy i its modes. Parseval s Theorem. Suppose f is a elemet of ([, ]) ad its Fourier series is f () t ke k The, f f ( t) dt k k Parseval s Theorem amouts to sayig that the eergy i a sigal f is the sum of the eergies i its modes. The real versio of the equatio i Parseval s Theorem is 0 k k k f a a b Remark: Parseval s theorem ca be used i a umber of ways. Oe of them is to obtai sums of series. Example. Prove. 6 Solutio. From example before, we have the Fourier series for the fuctio f ( x) x o [, ], x si( x) ( ), x [, ]. (ote that b ( ) ) By Parseval s theorem, x ( ) 4 where x x dx. 3 3

5 Thus, we have 6 Questio: Prove ( ) 96 Hit: Please cosider: The Fourier series for f o the iterval 0 x give by x if 0 x / f() t x if / x ad exteds to the iterval x 0 as a eve fuctio.

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 +

62. Power series Definition 16. (Power series) Given a sequence {c n }, the series. c n x n = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + 62. Power series Defiitio 16. (Power series) Give a sequece {c }, the series c x = c 0 + c 1 x + c 2 x 2 + c 3 x 3 + is called a power series i the variable x. The umbers c are called the coefficiets of