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

80 Pages·2014·1.33 MB·English

by  Shlomo Sternberg

Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform
Math212a1406
The Fourier Transform
The Laplace transform
The spectral theorem for bounded self-adjoint
operators, functional calculus form
The Mellin Transform
Shlomo Sternberg
September 18, 2014
Shlomo Sternberg
Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform
Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform
1 Conventions, especially about 2π.
2 Basic facts about the Fourier transform acting on S.
3 The Fourier transform on L2.
4 Sampling.
5 The Heisenberg Uncertainty Principle.
6 Tempered distributions.
′
Examples of Fourier transforms of elements of S .
7 The Laplace transform.
8 The spectral theorem for bounded self-adjoint operators,
functional calculus form.
9 The Mellin trransform
Dirichlet series and their special values
Shlomo Sternberg
Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform
Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform
The space S.
The space S consists of all functions on R which are infinitely
differentiable and vanish at infinity rapidly with all their derivatives
in the sense that
m (n)
‖f ‖m,n := sup{|x f (x)|} < ∞.
x∈R
The ‖ · ‖m,n give a family of semi-norms on S making S into a
Frechet space - that is, a vector space space whose topology is
determined by a countable family of semi-norms.
Shlomo Sternberg
Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform
Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform
The measure on R.
We use the measure
1
√ dx
2π
on R and so define the Fourier transform of an element of S by
∫
1
fˆ(ξ) := √ f (x)e−ixξdx
2π R
and the convolution of two elements of S by
∫
1
(f ⋆ g)(x) := √ f (x − t)g(t)dt.
2π R
Shlomo Sternberg
Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform
d ˆ
So the Fourier transform of (−ix)f (x) is f (ξ).
dξ
Integration by parts (with vanishing values at the end points) gives
∫ ∫
1 1
′ −ixξ −ixξ
√ f (x)e dx = (iξ)√ f (x)e dx.
2π R 2π R
′ ˆ
So the Fourier transform of f is (iξ)f (ξ).
Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform
∫
1 −ixξ
We are allowed to differentiate √ f (x)e dx with respect to
2π R
ξ under the integral sign since f (x) vanishes so rapidly at ∞. We
get
( ∫ ) ∫
d 1 1
−ixξ −ixξ
√ f (x)e dx = √ (−ix)f (x)e dx.
dξ 2π
R 2π R
Shlomo Sternberg
Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform
Integration by parts (with vanishing values at the end points) gives
∫ ∫
1 1
′ −ixξ −ixξ
√ f (x)e dx = (iξ)√ f (x)e dx.
2π R 2π R
′ ˆ
So the Fourier transform of f is (iξ)f (ξ).
Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform
∫
1 −ixξ
We are allowed to differentiate √ f (x)e dx with respect to
2π R
ξ under the integral sign since f (x) vanishes so rapidly at ∞. We
get
( ∫ ) ∫
d 1 1
−ixξ −ixξ
√ f (x)e dx = √ (−ix)f (x)e dx.
dξ 2π
R 2π R
d ˆ
So the Fourier transform of (−ix)f (x) is f (ξ).
dξ
Shlomo Sternberg
Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform
Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform
∫
1 −ixξ
We are allowed to differentiate √ f (x)e dx with respect to
2π R
ξ under the integral sign since f (x) vanishes so rapidly at ∞. We
get
( ∫ ) ∫
d 1 1
−ixξ −ixξ
√ f (x)e dx = √ (−ix)f (x)e dx.
dξ 2π
R 2π R
d ˆ
So the Fourier transform of (−ix)f (x) is f (ξ).
dξ
Integration by parts (with vanishing values at the end points) gives
∫ ∫
1 1
′ −ixξ −ixξ
√ f (x)e dx = (iξ)√ f (x)e dx.
2π R 2π R
′ ˆ
So the Fourier transform of f is (iξ)f (ξ).
Shlomo Sternberg
Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform
Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform
The Fourier transform maps S to S.
Putting these two facts together gives
The Fourier transform is well defined on S and
[( )m ] ( )n
d d
((−ix)nf ) ˆ= (iξ)m fˆ,
dx dξ
as follows by differentiation under the integral sign and by
integration by parts. This shows that the Fourier transform maps
S to S.
Shlomo Sternberg
Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform
Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform
Convolution goes to multiplication.
∫ ∫
1
−ixξ
(f ⋆ g)ˆ(ξ) = f (x − t)g(t)dte dx
2π
∫ ∫
1
−i(u+t)ξ
= f (u)g(t)e dudt
2π
∫ ∫
1 1
−iuξ −itξ
= √ f (u)e du√ g(t)e dt
2π R 2π R
so
(f ⋆ g)ˆ= fˆgˆ.
Shlomo Sternberg
Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform
Outline Conventions, especially about 2π. Basic facts about the Fourier transform acting on S. The Fourier transform on L2. Sampling. The Heisenberg Uncertainty Principle. Tempered distributions. The Laplace transform. The spectral theorem for bounded self-adjoint operators, functional calculus form. The Mellin trransform
Scaling.
For any f ∈ S and a > 0 define Saf by (Sa)f (x) := f (ax). Then
setting u = ax so dx = (1/a)du we have
∫
1
−ixξ
(Saf )ˆ(ξ) = √ f (ax)e dx
2π R
∫
1
−iu(ξ/a)
= √ (1/a)f (u)e du
2π R
so
(Saf )ˆ= (1/a)S1/afˆ.
Shlomo Sternberg
Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform