Transformada de Fourier - Olea, Maria Mercedes - 2012

7/21/2019 Transformada de Fourier - Olea, Maria Mercedes - 2012

1/36

Rn


2/36

R

Rn

L2(Rn)

C()

DK


3/36

R

f : R R

sop(f) [M, M]

L

L/2> M

f

L

f(x) =nN

cn(L)e2i( nL)x

cn(L) = 1

L

L/2L/2

f(t)e2(nL)tdt

x [M, M]

f(x) =nN

1

L

L/2L/2

f(t)e2i(nL)tdte2i(

nL)x

f(x) =nN

1

L

f(t)e2i(nL)tdte2i(

nL)x

f 0

R [M, M]

fnL

:=

f(t)e2inL

tdt

f

L

nN1

Lf(n/L) e2(nL)x

f(t)e2itd

f(x) =

f(t)e2itd


4/36

f() =

f(x)e2itdx

f L1(R)

f() =

f(x)e2ixdx

F :=f

f L1(R)

f || 0

f Ly f

f1

f

R

|f()| = |

Rnf(x)e2ixdx

Rn

|f(x)|dx= f1

g=

kj=1

cjRj Rj =

ni=1

[ajl , bjl ]

() = Rn

Rje2ixdx

=

R

[aj1,bj1](x1)[aj

2,bj2](x2) [ajn,bjn](xn)e

2ix11 e2ixnndx

=n

l=1

R

[ajl,bjl](xl)e

2ixlldxl

[a,b]

[a,b] =R

[a,b](x)e2ixdx=

ba

e2ixdx

e2ix

2i

ba

=e2ib

2i

e2ia

2i

0


5/36

R

[ajl,bjl](xl)e

2ixlldxl

0

nl=1

R

[ajl,bjl](xl)e

2ixlldxl

0

Rj 0 g() 0

f L1(Rn) >0

g

f g1 < f() = f() g() +g()

(f g)() + |g()| f g1+ |g()|

Rn

f(+ h) f() = Rn

f(x)

e2ix(+h) e2ix

dx=Rn

e2ix

e2ixh 1

f(x)dx

f(+ h) f() Rn

e2ix |f(x)| e2ixh 1 dx

|f(x)|

e2ixh 1

2 |f(x)| L1(Rn).

f

f L1(R)

f L1(R)

f() = 2if()

(xf)() = (1/2i)(f)()


6/36

f, g L1(R)

f()g()d=

f()g()d

f()g()d=

f(x)e2ixdx

g()d

e2ixg()df(x)dx=

f(x)

g(x)dx

g L1(R)

g(0) = 1

t > 0

Ag,tf(x) =

f()g(t)e2ixd

Rn

f L1(Rn)

f() = Rn

f(x)e2ixdx

f L1(R)

Ag,tf

Ag,t =

Rn

f()g(t)e2ixd= (f t)(x)

(x) =g(x) t(x) = (1/tn)(x/t)

Rn

L1(Rn)

= 1

t(x) = (1/t

n)(x/t)

f L(Rn)

(f t)(x)

t0 f(x)

f Lp(Rn)

p [1, )

(f t)

Lp f


7/36

= 1

t = 1

(f t)(x) f(x) =

Rn

f(x y)t(y)dy f(x) =

Rn

[f(x y) f(x)] t(y)dy

> 0 >0 |f(x y) f(x)| < |y| <

|f t(x) f(x)|

Rn

|f(x y) f(x)| |t(y)| dy

=

|y|/t

(z)dz

t 0

f t fp

Rn

f( y) f()p |t(y)| dy=

Rn

wp(y) |t(y)| dy

wp(z) := f( y) f()p wp(tz) 0 t 0

wp 2 fp L

1(Rn).

f (Lp(Rn) L1(Rn))

g L1(Rn)

Rng = 1

Ag,tf f

t 0

Lp

1 p <

p=

Ag,tf(x) f(x)

Agtf=f t g(x) =(x)

1 p <

f L1 Lp

f(x) =limt0Rn

f()et||2e2ixd Lp

p=

f(x) =limt0Rn

f()et||2e2ixd


8/36

g(0) =limt0

Rn

g()et||2

d

g L1(R)

g(0) =

Rn

g()d

f L1(Rn)

f L1(Rn)

f(x) =

Rn

f()e2ixd

g(x) =e|x|2

g

g(t) t0 g(0)

f(x) =limt0

Rn

f()et||2e2ixd en L1(Rn)

f() L1(Rn)

f()et||

2

e2ix

f()

L1(Rn)

f(x) =

Rn

f()e2ixd.

f(x) =

Rn

f()e2ixd

L2(Rn)

f (L1(Rn) L2(Rn))

f L2(Rn)

f2

= f2

g(x) =f(x)

f L1(Rn)

f

f(x) = g(x)

g


9/36

g=f=f

h

f g

h= f g=fg=ff= f2 .

f L1(Rn) g L1(Rn)

h L1(Rn)

h= f g

|h(x)| = |(f g)(x)| =

Rn

f(x y)g(y)dy

Rn

|f(x y)g(y)| dy= f g1 f2 g2

h

|h(x0+ k) h(x0)| =Rn

f(x0+ k y)g(y)dy Rn

f(x0 y)g(y)dy

=

Rn

(f(x0+ k y) f(x0 y)) g(y)dy

Rn

|f(x0+ k y) f(x0 y)| |g(y)| dy

f(x0+ k ) f(x0 )2 g2k0 0

h L1

(Rn

)

h

h(x) =limt0

Rn

h()et||2e2ixd

h(0) =limt0

Rn

h()et||2d

h 0

f

0

h Rn

limt0h()et||2d= Rn

h()d

h L1(Rn)

h(x) =

Rn

h()e2ixd


10/36

h(0) =

Rn

h()d= Rn

f()2 d= f22

h(0) = (f g)(0) =Rn

f(y)g(y)dy=Rn

f(y)f(y)dy=Rn

|f(y)|2 dy = f22

f2

= f2 f L2(Rn)

X

K

K = R

C

X

X

K

X X (x, y) x + y X

K X

X X

K X

Tx : XX Tx(y) =x +y

x X

O(x) x + O(0) {x + U=Tx(U) :U O(0)}


11/36

p: X R

x, y X

K

p(x) 0

p(x) = ||p(x)

p(x + y) p(x) +p(y)

X

K

F

X

x=y

p F

p(x y) = 0

dz,p : X R

+

dz,p(x) =p(x z) X

(X, F)

X

dz,p

X

K

(X, F)

F

X

= (xi)iI X

xi x X (X, F) p(xi x) R p F

K

C X

tC+ (1 t)C C

0 t 1

B X

B B

K

|| 1

E X

s > 0

E tV

t > s

X

X

X

X

d

X

F

X

F

X

d

X

X


12/36

{Vn}

Vn+1+ Vn+2+ Vn+3+ Vn+4 Vn n= 1, 2, . . .

D

r

r=

n=1

cn(r)2n

ci(r) r D 0 r 1

A(r) =X

r 1

r D

A(r) =c1(r)V1+ c2(r)V2+

f(x) = nf{r: x A(r)} x X y

d(x, y) =f(x y)

d

A(r) + A(s) A(r+ s) (x X, y X).

A(s)

A(r) A(r) + A(t r) A(t) (si r < t.)

{A(r)}

f(x + y) f(x) + f(y) (x X, y X)

0

r

s

D

f(x)< r, f(y) < s y r+ s < f(x) + f(y) + .

x A(r)

y A(s)

x + y A(r+ s)

f(x + y) r+ s < f(x) + f(y) +

A(r)

Vi f(x) =f(x) x A(r) x A(r)

f(0) = 0

x = 0

x / Vn= A(2

n)

n

f(x) 2n >0

X

B(0) = {x: f(x)< } =r


13/36

< 2n

B(0) Vn {B(0)}

d

X

r+ s 1

A(r+ s) = X

r+ s


14/36

D =

x1

1

xn

n || =1+ + n

|| = 0, Df=f

f

Rn

C()

Df C()

f

{x: f(x) = 0}

K

Rn

DK

f C(Rn)

K

C()

DK

C

()

K

Ki i = 1, 2, 3, . . . Ki Koi+1 =

i=1,2,...

Ki

Ki= Bi(x) x

pN C

() N = 1, 2, 3, . . .

pN(f) = max {|Df(x)| :x KN, || N} .

f = g

f g = 0

D(f g) = 0

N = ||

pN(f g) = 0

C()

x f f(x) fi C()

f

pN(fi f) 0 N = 1, 2, . . .

(fi f)(x) 0 fi(x) f(x) DK

x

K

DK C()

VN = {f C() :pN(f)< 1/N} N= 1, 2, . . .

{fi} C() N fi fj VN i, j

|Dfi D

fj| < 1/N KN || N Dfi

g

fi(x) g0(x)

g0 C() g = Dg0 fi g0 C()

C()

DK

DK

Rn

K

DK DK

D()


15/36

Rn

K K DK pN

W D()

DK W

K

K

+ W

D()

W

D()

D()

D()

V

K DK D() DK

D()

E DK K

MN < E

N MN para N= 0, 1, 2,

{i} D() {i} DK K

limi,j

i

j

N

= 0 para N= 0, 1, 2,

i 0 D() K

i Di

i 0

D()

D()

i 0 D() i 0

DK D()

D

D() D()

D()

D()


16/36

D()

D()

K

C <

|| CN

DK

u

Rn

x Rn

(xu)(y) =u(y x) y u(y) =u(y) para y Rn

(xu)(y) = u(y x) =u(x y).

u

v

Rn

u

v

(u v)(x) =

Rn

u(y)(xv)(y)dy

(u )(x) =u(x

) con u D(Rn), D(Rn), x Rn

(xu).v=

u.(xv)

u

v

xu u D

(xu)() =u(x) para D, x Rn

u D

D

D

x(u ) = (xu) = u (x) x Rn

u C

D(u ) = (Du) = u (D)

u ( ) = (u )


17/36

y Rn

(x(u ))(y) = (u )(y x) =u(yx)

((xu) )(y) = (xu)(y) =u(yx)

(u (x))(y) =u(y (x)) =u(yx)

yx= yx

(x) =x

u

x (D) = (1)||D(x)

(u (D))(x) = ((Du) )(x).

e

Rn

r =r1(0 re) (r >0).

r(u ) =u (r)

r 0

r De n De e

x( (r)) x (De) en n

x Rn

limr0(u (r))(x) = (u (De))(x)

De(u ) =u (De)


18/36


19/36

xf exf

exf xf

(f g) =

f

g

>0

h(x) =f(x/)

h(t) =nf(t)

x et

xf(t) = (xf) et= f xex= f et(x)et= ex(t)f(t)

exf(t) = exf et = f e(tx)= (xf)(t)

f

f C(Rn)

sup||NsupxRn(1 + |x|2)N|(Df)(x)| <

N= 0, 1, 2,

n

P Df Rn

P

n

n

g n

f P f

f gf

f Df

n n

f n


20/36

P(D)f=PfP f=P(D)f

n n

f, g C()

D(f g) =

c

Df

Dg

n

n {fi} n

xDfi(x)

g i

g(x) = x

Dg00(x) fi g00 = = 0 xDfi = x

0D0fi = fi

n

f n Df n

D(P f) =

c

DP

Df

D(gf) =

c

Dg

Df

P f

gf n

f n P f n

(P(D)f) et= f P(D)et = f P(t)et= P(t) [f et]

Rn

(P(D)f)(t) =P(t)f(t)

t= (t1, , tn) t = (t1+ , t2, , tn) = 0

f(t) f(t)i

=

Rn

x1f(x)eix1 1

ix1 eixtdmn(t)

x1f L

1


21/36

1

i

t1f=

Rn

x1f(x)eixtdmn(t).

P(x) =x1

f n g(x) = (1)

||xf(x)

g n

g = Df P Df =P g = (P(D)g) P(D)g L1(Rn)

f n

fi f n fi f L1(Rn) fi(t) f(t) t Rn.

ff n n

n R

n

n(x) =e 1

2|x|2

n n n= n

n(0) =

Rn

ndmn

n n 1

y + xy= 0

1

1/1

1(0) = 1

1(0) = R

1dm1 = (2)1/2

e(1/2)x2

dx= 1

1= 1

n(x) =1(x1) 1(xn) (x Rn)

n(t) =1(t1) 1(tn) (t Rn)

n = n n n(0) = ndmn


22/36

g n

g(x) =

Rn

gexdmn (x Rn)

n n

f L1(Rn),f L1(Rn)

f0(x) =

Rn

f exdmn

f(x) =f0(x)

f, g L1(Rn)

Rn

Rn

f(x)g(y)eixydmn(x)dmn(y)

Rn

Rn

fgdmn= Rn

Rn

fgdmn

g n n f(x) =(x/) >0

Rn

g(t)n(t)dmn(t) = Rn

y

g(y)dmn(y)

Rn

g

t

(t)dmn(t) = Rn

t

g(t)dmn(y)

g

t

g(0)

y

(0)

g(0)Rn

dmn= (0) Rn

gdmn (g, n)

n x = 0

g(x) = (xg)) (0) =

Rn

xgdmn=

Rn

gexdmn.


23/36

2.

g=g

n g = 0 g= 0

2g= g

g(x) =g(x)

4g= g

:n n

1

1 = 3

g n

Rn

f0gdmn= Rn

fgdmn (g n)

g

n D(R) n Rn

(f0 f)dmn= 0

D(Rn)

f0 f = 0 ctp

f n

g n

f g n

f g=fg

f g=fg

(f g) = f g

f g f g (fg) = 2f 2g= fg= (f g) = 2(f g)

1

2

f g n fg n

n n


24/36

: L2(Rn) L2(Rn)

f=f para todaf n

f =f

n L1 L2

n

L2

L1

L2

f

f L1

f =f

L1 L2

L2

f

f

f L2(Rn)

f

g n

Rn

f g= Rn

g(x)dmn(x) Rnf(t)eixtdmn(t) =

Rnf(t)dmn(t)

Rng(x)e

ixt

dmn(x)

g(t)

Rn

fgdmn=

Rn

fgdmn (f, g n)

g= f

f2=f2

(f n)

n

L2

(Rn

)

n

L1

(Rn

)

f f

n L

2(Rn) n

f f

: L2(Rn) L2(Rn)

L2(Rn)

f, g L2(Rn)

L2

n D(Rn))

D(Rn)

n

D(Rn)

n

D(Rn)

n


25/36

f n, D(R

n)

= 1

Rn

fr(x) =f(x)(rx) (x Rn, r >0)

fr D(R

n)

P

P(x)D(f fr)(x) =P(x)

c(Df)(x)r||D[1 ](rx)

D[1](rx) = 0

|x| 1/r

f n P.D

f C0(Rn)

Rn

r 0

fr f n

K

Rn

DK n

i : D(Rn) n

n

uL = L i

uL D(Rn)

L1 L2 n D(Rn) n

n n

u D(Rn)

n

uL

Rn

n

Rn

Rn

(1 + |x|2)kd(x)<

k


26/36

g n

Du

(Du)(f) = (1)||u(Df)

(P u)(f) =u(P f)

(gu)(f) =u(gf)

u

n

u() =u() ( n)

n n n u n

u

u

n n

u n

(P(D)u) =PuP u= P(D)u

W

n 1, , k n

u n: |u(i)|


27/36

u() =u() ( n, u n)

u V

u W

u=u

n

n

4u= u

u n 1 = 3

(P(D) u) () = (P(D) u) =u P(D) =u P =u (P ) = (Pu) ()

(P(D)u) () =u (P(D) ) =u (P(D) ) =u P = (P u) = P u ()

n

w= (1, 0, , 0) Rn, n

(x) = (x + w) (x)

(x Rn), >0,

0 x1 n

0 x1 n

x1

n

0 0 en n

(y) =eiy1 1

iy1 (y R

n, > 0)

P.D() =

cP.(D).(D)

D(y)

y21 si || = 0

|y1| si || = 1

||1 si || >1

Rn

0

n


28/36

u n n

(u ) (x) =u

x

(x Rn)

x n x R

n

n

u C(Rn)

D (u ) = (Du) = u (D)

u

u =u

(u ) = u ( )

n

u =u

D(u ) =u (D)

w 0

(u ) =u

w 0

.

D(u ) =u (D) = (1, 0, , 0).

pN(f) f n

1 + |x + y|2 2(1 + |x|2)(1 + |y|2) para x, y Rn

pN(xf) 2N(1 + |x|2)NpN(f) para x R

n, f n.

u

n pN n

N

C <

|u(f)| CpN(f) para f n.

|(u )(x)| =u(x) 2N.CpN()(1 + |x|2)N,

u

n D(R

n)

K


29/36

(u )() = (u )() = Rn

(u )(x)(x)dmn(x)

= K

u (x)x dmn(x) =u K

(x)xdmn(x)

u

=u =u

u

() = u ()

n

(u )() =u

((u ) ) (0) = (u ( ))(0).

x

u

= u= u.

u=u

L2

L2

C

Cn

f

f

(a1, , an)

gi() =f(a1, , ai1, ai+ , ai+1, , an)

g1, , gn C

Cn


30/36

Cn

z = (z1, , zn) zk C zk = xk +iyk

x= (x1, , xn) y= (y1, , yn) z = x + iy

x= Re(z)

y= I m(z)

z

Rn z Cn Im(z) = 0

|z| = (|z1|2 + + |zn|

2)1/2

|Im(z)| = (y21+ + y2n)

1/2

z =z11 znn

z t= z1t1+ + zntn

ez(t) =eizt

t Rn

Cn

Rn

n

n= 1

Pk f z C

n

k

f(z) = 0 Pn P0 Pi 1 i n

a1, , ai gi

C

Pi1

rB

{x Rn : |x| r}

D(Rn)

rB

f(z) = Rn

(t)eiztdmn

(t)

N <

|f(z)| N(1 + |z|)Ner|Im(z)| (z Cn, N= 0, 1, 2, )

D(Rn)

rB


31/36

t rB

eiz.t =ey.t e|y||t| er|Im(z)|.

(28) L1(Rn) z Cn f Cn f f

zf(z) =

Rn

(D)(t)eiz.tdmn(t).

|z| |f(z)| D1 er|Im(z)|

f

(t) =

Rn

f(x)eit.xdmn(x) con t Rn.

(1 + |x|)Nf(x) L1(Rn)

N

C(Rn)

f(+ i,z2, , zn)ei[t1(+i)+t2z2++tnzn]d

t1, , tn z1, , zn

(+ i)

= 1

f

= 0

= 1

(t) =

Rn

f(x + iy)eit.(x+iy)dmn(x)


32/36

y Rn

t Rn

t = 0

y= t|t| >0

t.y= |t| , |y| =

f(x + iy)eit.(x+iy) N(1 + |x|)Ne(r|t|)

|(t)| Ne(r|t|)

Rn

(1 + |x|)Ndmn(x)

N

|t| > r

(t) = 0

rB

u D(Rn)

f(z) =u(ez) (z Cn)

f

Rn

<

|f(z)| (1 + |z|)Ner|Im(z)| (z Cn)

Cn

u D(Rn)

u

Cn

u(z) = u(ez) z Cn

u

Rn

Rn

L2

K

|f|2 dmn<

K


33/36

u D()

L2

L2

u() =

gdmn

D()

Df

L2

gdmn= (1)||

f Ddmn

D()

C(p)()

Df || p

Dfi (/xi)

k

n > 0, p 0

r > p+ (n/2)

Rn Dkif L

2

1 i n, 0 k r

f0 C(p)() f0(x) = f(x) x

:

gik L

2

gikdmn= (1)k

f Dkidmn con D(),

1 i n, 0 k r.

K

D()

= 1

K

F

Rn

F(x) = (x)f(x) si x

0 si x /

F (L2 L1)(Rn)

D(f g) =

c(Df)(Dg)


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f

g C()

Dri =

r

s=0

rs(Drsi )(Dsi f) =r

s=0

rs(Drsi )gis

0 , Dri F = 0 0

Dri F Rn

L2(Rn)

1 i n

(Drsi )gis L

2()

F, Dr1F, , D

rnF

Rn

F2 dmn<

Rn

y2ri

F(y)2 dmn(y)< (1 i n).

(1 + |y|)2r


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F = F R

n

yF(y) L1

|| p

F C(p)(Rn).

f

F

f=F

F C

(p)(Rn)

f

F =F

f0

f0(x) =F(x) si x .


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Transformada de Fourier - Olea, Maria Mercedes - 2012

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Transcript of Transformada de Fourier - Olea, Maria Mercedes - 2012