Lecture 8. Dirac and Weierstrass

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1 Leture 8. Dira ad Weierstrass Audrey Terras May 5, 9 A New Kid of Produt of Futios You are familiar with the poitwise produt of futios de ed by f g(x) f(x) g(x): You just tae the produt of the real umbers f(x) ad g(x). Thus if we de e f(x) for all x, we get f g(x) g(x): So f(x) is the idetity for poitwise produt. Now we wat to de e a ew id of produt, well, ot that ew if you got to ovolutio i the Laplae trasform setio of D. Ad ot so ew if you are taig probability or statistis ourses where give idepedet radom variables with desities f ad g, the desity of the sum of the radom variables is f g. For this produt, f 6 f: Our aim is to use ovolutio i order to uiformly approximate otiuous futios f o [a; b] by polyomials P N a x (or futios of period by trigoometri polyomials P a e ix for Fourier series). We will assume that our futios are pieewise otiuous ad that at least oe of the futios i f g vaishes o a bouded iterval so that we ow the itegrals exist. De itio The ovolutio of f ad g is de ed for all x by f g(x) f(t)g(x Oe a read f g this as f splat g sie it does splat the properties of the futios together, preservig the best properties. If oe futio is disotiuous, but the other is a polyomial, the result is a polyomial. Example. Let g(x) x ; for all x. De e 8 ; if < x < >< ; if < x < f(x) ; if x >: ; if x or x : The, sie f has di erig formulas o di eret itervals (ad merifully g does ot), we get (f g)(x) g(x t)dt + g(x t)dt: (x t) dt + (x t) dt (x t) x + (x + ) (x t) dt (x t) + x + x x + x t)dt (x t) dt (x ) x

2 y x Figure : g(x) x Figure : The futio f(x)

3 y 6 4 x 4 6 Figure : f g So we see that eve though f is disotiuous, whe ovolved with a polyomial, we get a polyomial. Properties of Covolutio. Assume f; g; h are pieewise otiuous ad at least oe vaishes o a bouded iterval whe eessary to mae a itegral exist. Let R: ) f g g f ) f (g + h) f g + f h ) (f) g (f g) 4) f (g h) (f g) h 5) Suppose that f(x) if jxj > ad g(x) if jxj > d: The (f g)(x) if jxj > + d: 6) If g(x) is a polyomial the (f g)(x) is a polyomial. 7) (f g) f (g ) assumig g di eretiable. Proof. ) Mae the hage of variables u x t. The t x u ad du dt ad the order of itegratio hages so we get: f g(x) )-4) are Exerises i homewor 7. 5) f(t)g(x t)dt f(t)g(x f(t)g(x t)dt t)dt: f(x u)g(u)du g(u)f(x u)du g f(x): Now if x > + d, ad t ; we see that x t > + d d ad g(x t) : So x > + d implies (f g). If x < d ad t ; we see that x t < d + d ad g(x t) : Thus x < d implies (f g). 6) It su es usig ) ad ) to osider the ase that g(x) x : The supposig f(x) if jxj > ; we have the followig, usig the biomial theorem f g(x) f(t)g(x t)dt f(t)(x t) dt f(t) X x ( t) dt X x f(t)( t) dt:

4 Let The f g(x) f(t)( t) dt : X x ; whih is a polyomial. 7) We will ot eed this so we leave it as a extra redit exerise. This oludes our disussio of ovolutio. It is importat i the theory of probability ad Fourier aalysis, Laplae trasforms. I fat the Fourier ad Laplae trasforms hage ovolutio ito ordiary poitwise produt of the trasformed futios. Covolutio is also used to smooth data thas to property 7). We wat to use it to approximate otiuous futios by polyomials. A Futio Whih is Not a Ordiary Futio We wat to thi about somethig alled the Dira delta "futio" deoted (x): It is used i physis to represet a impulse. It is ofte said to be a futio that is for x 6 ad at x. Of ourse, o mathematiia would all that legal. The graph usually assoiated with shows a uit arrow or spie at the origi - ot a poit at i ite height. Figure 4: Dira delta Aother "de itio" of is that for ay otiuous futio f(x) it is supposed to give the formula f(t)(t)dt f(): We show i the homewor that this formula fores (x) for x 6 ad thus fores the itegral to be. Yet aother de itio of delta is that it is the idetity for ovolutio. That is just as bad. 4

5 This disturbed mathematiias mightily i the 9 s ad 94 s. Egieers ad physiists just happily used delta ad eve its derivative or i ite sums of deltas. Fially Lauret Shwartz ad others legalized delta ad its relatives by allig it a distributio or geeralized futio ad developig the alulus of distributios. See my boo Harmoi Aalysis o Symmetri Spaes ad Appliatios, I, for a itrodutio. Other referees are Korevaar, Mathematial Methods ad Shwartz, Math. for the Physial Siees. The subjet of aalysis o distributios does ot seem to appear i udergrad math. ourses. So what to do? I this setio we osider sequees of futios that have the behavior of delta i the limit as! : These are alled Dira sequees or approximate idetities (for ovolutio). De itio A Dira sequee K (x) is a sequee of futios K : R! R suh that Dir. K (x) ; for all x ad. Dir. K (x)dx ; for all. Dir. 8 " > ad 8 > ; 9 N + s.t. N implies K (x)dx + K (x)dx < ": The properties say that the area uder the urve y K (x) beomes more ad more oetrated at the origi as! : You might thi it hard to d suh a sequee but we have the followig example from the rst problem i Lag. Example. Let K(x) be suh that K(x) for all x ad K(x)dx : The K (x) K(x) is a Dira sequee. Example. The Ladau Kerel. This erel is amed for a mathematiia (umber theorist) who lived at the same time as Dira. De e the Ladau erel to be ( ( t ) L (x) ; t where t dt: otherwise: It is possible to use itegratio by parts to d that (!) + ()! ( + ) : See Courat ad Hilbert, Methods of Mathematial Physis, Vol. I, p. 84. Uless you ow Stirlig s formula, this is ot too helpful i provig the Ladau erel gives a Dira sequee. Lag gives a simple iequality whih is all we eed. I Figure 5, we plot L (t) t ()!(+) ; for t [ ; ]; whe i blue; 5 i gree ; i turquoise; ad (!) + i purple. Figure 5 shows the area uder the urve (whih is ) begis to oetrate at the origi as ireases. Lemma. + : Proof. t dt ( t) ( + t) dt ( t) dt + : It is lear that the Ladau erel has the rst properties of a Dira sequee. To prove the rd property, we argue as i Lag. Give " > ad < < ; we eed to d N so that N maes the followig itegral < " : t + dt t + dt dt + ( ) : We a mae the stu o the right < " for large, sie we a show that beause < < ; lim Use l Hôpital s rule.! + : 5

6 y x Figure 5: L (t); ;(blue); 5(gree); (turquoise); (purple). A Dira Sequee Approahes The Dira Delta We wat to prove that ay Dira sequee K behaves lie a idetity for ovolutio i the limit as! : Some people all Dira sequees "approximate idetities" for this reaso. Theorem Suppose f is a bouded pieewise otiuous futio o R. Let I be ay ite iterval o whih f is otiuous. De e g max jg(x)j. The lim f K f XI! : This says that the sequee K f overges uiformly to f o the iterval I. Proof. Usig the st ad d properties of Dira sequees plus the fat that itegrals preserve, we have: j(k f) (x) f(x)j K (t)f(x t)dt f(x) K (t)dt K (t) (f(x t) f(x)) dt K (t) jf(x t) f(x)j dt: Sie f is uiformly otiuous o I, give " there is a suh that jf(x t) f(x)j < " whe jtj < : Sie f is bouded, we there is a boud M so that jf(x)j M for all x. Now we a brea up the last itegral K (t) jf(x t) f(x)j dt K (t) jf(x t) f(x)j dt + K (t) jf(x t) f(x)j dt + K (t) jf(x t) f(x)j dt: For the rst itegrals, use the boud o f ad the rd property of Dira sequees to see that, for large eough, they are less tha M" or eve "; if you prefer. 6

7 For the last itegral, use the uiform otiuity of f to see that the itegral is " K (t)dt " K (t)dt ": " This ompletes the proof that j(k f) (x) f(x)j (M + ) ". You a replae " by M+ if you are paraoid. Corollary. Weierstrass Theorem. Ay otiuous futio f o a ite losed iterval [a; b] a be uiformly approximated by polyomials. Proof. We a use the preedig theorem for the iterval [; ] alog with the Ladau erel. See Lag, p. 88 for a geeral iterval. We still eed to see that L f is a polyomial. Suppose x [; ]: The L (x t)f(t)dt L (x Also we see that for x; t [; t]; we have x t ad so L (x t) t)f(t)dt: (x t) ad the itegral is (x t) f(t)dt: This is a polyomial usig the same reasoig as i our proof i the st setio that ovolutio of f with ay polyomial is a polyomial. This ishes our story for the momet. 7