# Download Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded PDF

Title Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded Shlomo Sternberg English 1.3 MB 80
```                            Conventions, especially about 2.
Basic facts about the Fourier transform acting on S.
The Fourier transform on L2.
Sampling.
The Heisenberg Uncertainty Principle.
Tempered distributions.
Examples of Fourier transforms of elements of S'.
The Laplace transform.
The spectral theorem for bounded self-adjoint operators, functional calculus form.
The Mellin trransform
Dirichlet series and their special values
```
##### Document Text Contents
Page 1

Outline
ˇ
.
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.
TheMellintrransform
Math212a1406
TheFourierTransform
TheLaplacetransform
operators,functionalcalculusform
TheMellinTransform
ShlomoSternberg
September18,2014
ShlomoSternberg

Page 2

Outline
ˇ
.
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.
TheMellintrransform
1
ˇ
.
2
S
.
3
TheFouriertransformon
L
2
.
4
Sampling.
5
TheHeisenbergUncertaintyPrinciple.
6
Tempereddistributions.
ExamplesofFouriertransformsofelementsof
S
0
.
7
TheLaplacetransform.
8
functionalcalculusform.
9
TheMellintrransform
Dirichletseriesandtheirspecialvalues
ShlomoSternberg

Page 40

Outline
ˇ
.
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.
TheMellintrransform
ExamplesofFouriertransformsofelementsof
S
0
.
TheFouriertransformofthe

function.
If

denotestheDirac

-function,then
(
F
(

)(

)=

(
F

1
(

))=

(
F

1
(

)

(0)=
1
p
2
ˇ
Z
R

(
x
)
dx
:
SotheFouriertransformoftheDirac

functionisthefunction
whichisidenticallyone.
Infact,thislastexamplefollowsfromthe
precedingone:If
m
=
F
(
`
)then
(
F
(
m
)(
˚
)=
m
(
F

1
(
˚
))=
`
(
F

1
(
F

1
(
˚
))
:
But
F

2
(
˚
)(
x
)=
˚
(

x
)
:
Soif
m
=
F
(
`
)then
F
(
m
)=

`
where

`
(
˚
):=
`
(
˚
(

))
:
ShlomoSternberg

Page 41

Outline
ˇ
.
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.
TheMellintrransform
ExamplesofFouriertransformsofelementsof
S
0
.
TheFouriertransformofthe

function.
If

denotestheDirac

-function,then
(
F
(

)(

)=

(
F

1
(

))=

(
F

1
(

)

(0)=
1
p
2
ˇ
Z
R

(
x
)
dx
:
SotheFouriertransformoftheDirac

functionisthefunction
whichisidenticallyone.
Infact,thislastexamplefollowsfromthe
precedingone:If
m
=
F
(
`
)then
(
F
(
m
)(
˚
)=
m
(
F

1
(
˚
))=
`
(
F

1
(
F

1
(
˚
))
:
But
F

2
(
˚
)(
x
)=
˚
(

x
)
:
Soif
m
=
F
(
`
)then
F
(
m
)=

`
where

`
(
˚
):=
`
(
˚
(

))
:
ShlomoSternberg

Page 79

Outline
ˇ
.
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.
TheMellintrransform
Dirichletseriesandtheirspecialvalues
Jacobi'sfunctionalequationofthethetafunction
ShlomoSternberg

Page 80

Outline
ˇ
.
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.