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

TitleMath212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded
Author
LanguageEnglish
File Size1.3 MB
Total Pages80
Table of Contents
                            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
Conventions,especiallyabout2
ˇ
.
BasicfactsabouttheFouriertransformactingon
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.
Thespectraltheoremforboundedself-adjointoperators,functionalcalculusform.
TheMellintrransform
Math212a1406
TheFourierTransform
TheLaplacetransform
Thespectraltheoremforboundedself-adjoint
operators,functionalcalculusform
TheMellinTransform
ShlomoSternberg
September18,2014
ShlomoSternberg
Math212a1406TheFourierTransformTheLaplacetransformThespectraltheoremforboundedself-adjointoperators,functionalcalculusformTheMellinTransform



Page 2




Outline
Conventions,especiallyabout2
ˇ
.
BasicfactsabouttheFouriertransformactingon
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.
Thespectraltheoremforboundedself-adjointoperators,functionalcalculusform.
TheMellintrransform
1
Conventions,especiallyabout2
ˇ
.
2
BasicfactsabouttheFouriertransformactingon
S
.
3
TheFouriertransformon
L
2
.
4
Sampling.
5
TheHeisenbergUncertaintyPrinciple.
6
Tempereddistributions.
ExamplesofFouriertransformsofelementsof
S
0
.
7
TheLaplacetransform.
8
Thespectraltheoremforboundedself-adjointoperators,
functionalcalculusform.
9
TheMellintrransform
Dirichletseriesandtheirspecialvalues
ShlomoSternberg
Math212a1406TheFourierTransformTheLaplacetransformThespectraltheoremforboundedself-adjointoperators,functionalcalculusformTheMellinTransform



Page 40




Outline
Conventions,especiallyabout2
ˇ
.
BasicfactsabouttheFouriertransformactingon
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.
Thespectraltheoremforboundedself-adjointoperators,functionalcalculusform.
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
Math212a1406TheFourierTransformTheLaplacetransformThespectraltheoremforboundedself-adjointoperators,functionalcalculusformTheMellinTransform



Page 41




Outline
Conventions,especiallyabout2
ˇ
.
BasicfactsabouttheFouriertransformactingon
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.
Thespectraltheoremforboundedself-adjointoperators,functionalcalculusform.
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
Math212a1406TheFourierTransformTheLaplacetransformThespectraltheoremforboundedself-adjointoperators,functionalcalculusformTheMellinTransform



Page 79




Outline
Conventions,especiallyabout2
ˇ
.
BasicfactsabouttheFouriertransformactingon
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.
Thespectraltheoremforboundedself-adjointoperators,functionalcalculusform.
TheMellintrransform
Dirichletseriesandtheirspecialvalues
Jacobi'sfunctionalequationofthethetafunction
ShlomoSternberg
Math212a1406TheFourierTransformTheLaplacetransformThespectraltheoremforboundedself-adjointoperators,functionalcalculusformTheMellinTransform



Page 80




Outline
Conventions,especiallyabout2
ˇ
.
BasicfactsabouttheFouriertransformactingon
S
.
TheFouriertransformon
L
2
.
Sampling.
TheHeisenbergUncertaintyPrinciple.
Tempereddistributions.
TheLaplacetransform.
Thespectraltheoremforboundedself-adjointoperators,functionalcalculusform.
TheMellintrransform
Dirichletseriesandtheirspecialvalues
Thefunctionalequationofthezetafunction
ShlomoSternberg
Math212a1406TheFourierTransformTheLaplacetransformThespectraltheoremforboundedself-adjointoperators,functionalcalculusformTheMellinTransform

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