19. Fourier Transform - Probability Tutorials

1 Tutorial 19: Fourier Transform119. Fourier TransformExercise ;b2R,a< :[a;b]!Cbe a map suchthatf0(t) exists for allt2[a;b]. We assume that:Zbajf0(t)jdt <+11. Show thatf0:([a;b];B([a;b]))!(C;B(C)) is Show that:f(b) f(a)=Zbaf0(t)dtExercise de ne the maps :R2!Cand :R!C:8(u;x)2R2; (u;x)4=eiux x2=28u2R; (u)4=Z+1 1 (u;x) 19: Fourier Transform21. Show that for allu2R,themapx! (u;x) is Show that for allu2R,wehave:Z+1 1j (u;x)jdx=p2 <+1and conclude that is well de Letu2 Rand (un)n 1be a sequence inRconverging that (un)! (u) and conclude that is Show that:Z+10xe x2=2dx=15. Show that for allu2R,wehave:Z+1 1 @ @u(u;x) dx=2<+ 19: Fourier Transform36.

2 Leta;b2R;a<b. Show that:eib eia=Zbaieixdx7. Leta;b2R;a<b. Show that:jeib eiaj jb aj8. Leta;b2R;a6=b. Show that for allx2R: (b;x) (a;x)b a jxje x2=29. Letu2 Rand (un)n 1be a sequence inRconverging tou,withun6=ufor alln. Show that:limn!+1 (un) (u)un u=Z+1 1@ 19: Fourier Transform410. Show that is di erentiable with:8u2R; 0(u)=Z+1 1@ Show that is of Show that for all (u;x)2R2,wehave:@ @u(u;x)= u (u;x) i@ Show that for allu2R:Z+1 1 @ @x(u;x) dx <+114. Leta;b2R;a<b. Show that for allu2R: (u;b) (u;a)=Zba@ 19: Fourier Transform515. Show that for allu2R:Z+1 1@ Show that for allu2R: 0(u)= u (u)Exercise the set of functions de ned by:S4=fh:h2C1(R;R);8u2R;h0(u)= uh(u)g1.

3 Let be as in ex. (2). Show thatRe( )andIm( ) lie Givenh2S, we de neg:R!R,by:8u2R;g(u)4=h(u)eu2=2 Show thatgis of classC1withg0= 19: Fourier Transform63. Leta;b2R;a<b. Show the existence ofc2]a;b[, such that:g(b) g(a)=g0(c)(b a)4. Conclude that for allh2S,wehave:8u2R;h(u)=h(0)e u2=25. Prove the following:Theorem 124 For al lu2R, we have:1p2 Z+1 1eiux x2=2dx=e u2= 19: Fourier Transform7De nition 135 Let 1;:::; pbe complex measures onRn,wheren;p 1;:::; p,denoted 1?:::? p,theimage measure of the product measure 1 ::: pby the measurablemapS:(Rn)p!Rnde ned by:S(x1;:::;xp)4=x1+:::+xpIn other words, 1?:::? pis the complex measure onRn, de ned by: 1?

4 :::? p4=S( 1 ::: p)Recall that the product 1 ::: pis de ned in theorem (66).Exercise , be complex measures Show that for allB2B(Rn): ? (B)=ZRn Rn1B(x+y)d (x;y) 19: Fourier Transform82. Show that for allB2B(Rn): ? (B)=ZRn ZRn1B(x+y)d (x) d (y)3. Show that for allB2B(Rn): ? (B)=ZRn ZRn1B(x+y)d (x) d (y)4. Show that ? = ? .5. Letf:Rn!Cbe bounded and measurable. Show that:ZRnfd ? =ZRn Rnf(x+y)d (x;y) 19: Fourier Transform9 Exercise ; be complex measures Rnandx2Rn, we de neB x=fy2Rn;y+ Show that for allB2B(Rn)andx2Rn,B x2B(Rn).2. Showx! (B x) is measurable and bounded, forB2B(Rn).3. Show that for allB2B(Rn): ?

5 (B)=ZRn (B x)d (x)4. Show that for allB2B(Rn): ? (B)=ZRn (B x)d (x) 19: Fourier Transform10 Exercise 1; 2; 3be complex measures Show that for allB2B(Rn): 1?( 2? 3)(B)=ZRn Rn1B(x+y)d 1 ( 2? 3)(x;y)2. Show that for allB2B(Rn)andx2Rn:ZRn1B(x+y)d 2? 3(y)=ZRn Rn1B(x+y+z)d 2 3(y;z)3. Show that for allB2B(Rn): 1?( 2? 3)(B)=ZRn Rn Rn1B(x+y+z)d 1 2 3(x;y;z)4. Show that 1?( 2? 3)= 1? 2? 3=( 1? 2)? 19: Fourier Transform11De nition 136 Letn 1anda2Rn. We de ne a:B(Rn)!R+:8B2B(Rn); a(B)4=1B(a) ais cal led theDirac Probability measureonRn, Show that ais indeed a Probability measure Show for allf:Rn![0;+1] non-negative and measurable:ZRnfd a=f(a)3.

6 Show iff:Rn!Cis measurable,f2L1C(Rn;B(Rn); a) and:ZRnfd a=f(a) 19: Fourier Transform124. Show that for any complex measure onRn: ? 0= 0? = 5. Let a(x)=a+xde ne the translation of vectorainRn. Showthat for any complex measure onRn: ? a= a? = a( )Exercise ;g:( ;F)!(C;B(C)) be two measurable maps,where ( ;F) is a measurable space. Letu=Re(f),v=Im(f),u0=Re(g)andv0=Im(g).1 . Show thatu;v;u0;v0:( ;F)!(R;B(R)) are all Show thatu+u0,v+v0,uu0 vv0anduv0+u0vare Show thatf+g;fg:( ;F)!(C;B(C)) are Show thatC=R2has a countable 19: Fourier Transform135. Show thatB(C C)=B(C) B(C).6. Show that (z;z0)!z+z0and (z;z0)!

7 Zz0are Show that!!(f(!);g(!)) is measurable toB(C) B(C).8. Conclude once more thatf+gandfgare 1and ; be complex measures that <<dx, that is absolutely continuous with respectto the Lebesgue measure Show there isf2L1C(Rn;B(Rn);dx), such that = Show that for allB2B(Rn), we have: ? (B)=ZRn (B x)d (x) 19: Fourier Transform143. Show that for allB2B(Rn)andx2Rn: (B x)=ZRn1B(y)f(y x)dy4. Show that for allB2B(Rn) the map:(x;y)!1B(y)f(y x)lies inL1C(Rn Rn;B(Rn) B(Rn);j j dy).5. Leth2L1C(Rn;B(Rn);j j)withjhj=1, =Rhdj j. Show:(x;y)!1B(y)f(y x)h(x)also lies inL1C(Rn Rn;B(Rn) B(Rn);j j dy).6. Show that for allB2B(Rn), we have: ?

8 (B)=ZB ZRnf(y x)d (x) 19: Fourier Transform157. Letgbe the map de ned byg(y)4=RRnf(y x)d (x). Recallwhygisdy-almost surely well-de ned, anddy-almost surelyequal to an element ofL1C(Rn;B(Rn);dy).8. Show that ? =Rgdxand ? << 125 Let ; be two complex measures onRn,n <<dx, is absolutely continuous with respect to the Lebesguemeasure onRn,withdensityf2L1C(Rn;B(Rn);dx), then the convo-lution ? = ? is itself absolutely continuous with respect to theLebesgue measure onRn,withdensity:g(y)=ZRnf(y x)d (x);dy a:s:In other words, ? = ? = 19: Fourier Transform16 Exercise ( ;F; )where( ;F; ) is a measure be the complex measure on ( ;F) de ned by =Rfd.

9 Letg:( ;F)!(C;B(C)) be a measurable Show thatg2L1C( ;F; ),gf2L1C( ;F; ).2. Show that for allg2L1C( ;F; ):Zgd =Zgfd Exercise to theorem (125), show that if =Rhdxforsomeh2L1C(Rn;B(Rn);dx), then:g(y)=ZRnf(y x)h(x)dx ; dy a: 19: Fourier Transform17De nition 137 Let be a complex measure on(Rn;B(Rn)),n callFourier transformof ,themapF :Rn!Cde ned by:8u2Rn;F (u)4=ZRneihu;xid (x)whereh ; iis the usual inner-product to de nition (137):1. Show thatF is well-de Show thatF 2 CbC(Rn), is continuous and Show that for alla;u2Rn,wehaveF a(u)=eihu; Let be the Probability measure on (R;B(R)) de ned by:8B2B(R); (B)4=1p2 ZBe x2=2dxShow thatF (u)=e u2=2, for 19: Fourier Transform18 Exercise 1;:::; pbe complex measures onRn,p Show that for allu2Rn,wehave:F( 1?)

10 :::? p)(u)=Z(Rn)peihu;x1+:::+xpid 1 ::: p(x)2. Show that ifp 3then 1?:::? p=( 1?:::? p 1)? Show thatF( 1?:::? p)= pj=1F 1, >0andg :Rn!R+de ned by:8x2Rn;g (x)4=1(2 )n2 ne kxk2=2 21. Show thatRRng (x)dx= Show that for allu2Rn,wehave:ZRng (x)eihu;xidx=e 2kuk2= 19: Fourier Transform193. Show thatP =Rg dxis a Probability onRn, and:8u2Rn;FP (u)=e 2kuk2=24. Show that for allx2Rn,wehave:g (x)=1(2 )nZRneihx;ui 2kuk2=2duExercise r th e r to e x . (14), let be a complex measure Show that ?P =R dxwhere: (x)=ZRng (x y)d (y);dx a:s:2. Show that we also have: (x)=ZRng (y x)d (y);dx a: 19: Fourier Transform203.