# Math212a1406 The Fourier Transform The Laplace transform ... shlomo/212a/06.pdf OutlineConventions,

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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

‖f ‖m,n := sup x∈R {|xmf (n)(x)|}

The measure on R.

We use the measure 1√ 2π

dx

on R and so define the Fourier transform of an element of S by

f̂ (ξ) := 1√ 2π

∫

R f (x)e−ixξdx

and the convolution of two elements of S by

(f ? g)(x) := 1√ 2π

∫

R f (x − t)g(t)dt.

Shlomo Sternberg

Math212a1406 The Fourier Transform The Laplace transform The spectral theorem for bounded self-adjoint operators, functional calculus form The Mellin Transform

We are allowed to differentiate 1√ 2π

∫ R f (x)e

−ixξdx with respect to

ξ under the integral sign since f (x) vanishes so rapidly at ∞. We get

d

dξ

( 1√ 2π

∫

R f (x)e−ixξdx

) =

1√ 2π

∫

R (−ix)f (x)e−ixξdx .

So the Fourier transform of (−ix)f (x) is ddξ f̂ (ξ).

Integration by parts (with vanishing values at the end points) gives

1√ 2π

∫

R f ′(x)e−ixξdx = (iξ)

1√ 2π

∫

R f (x)e−ixξdx .

So the Fourier transform of f ′ is (iξ)f̂ (ξ).

Shlomo Sternberg

We are allowed to differentiate 1√ 2π

∫ R f (x)e

−ixξdx with respect to

ξ under the integral sign since f (x) vanishes so rapidly at ∞. We get

d

dξ

( 1√ 2π

∫

R f (x)e−ixξdx

) =

1√ 2π

∫

R (−ix)f (x)e−ixξdx .

So the Fourier transform of (−ix)f (x) is ddξ f̂ (ξ).

Integration by parts (with vanishing values at the end points) gives

1√ 2π

∫

R f ′(x)e−ixξdx = (iξ)

1√ 2π

∫

R f (x)e−ixξdx .

So the Fourier transform of f ′ is (iξ)f̂ (ξ).

Shlomo Sternberg

We are allowed to differentiate 1√ 2π

∫ R f (x)e

−ixξdx with respect to

ξ under the integral sign since f (x) vanishes so rapidly at ∞. We get

d

dξ

( 1√ 2π

∫

R f (x)e−ixξdx

) =

1√ 2π

∫

R (−ix)f (x)e−ixξdx .

So the Fourier transform of (−ix)f (x) is ddξ f̂ (ξ).

Integration by parts (with vanishing values at the end points) gives

1√ 2π

∫

R f ′(x)e−ixξdx = (iξ)

1√ 2π

∫

R f (x)e−ixξdx .

So the Fourier transform of f ′ is (iξ)f̂ (ξ).

Shlomo Sternberg

The Fourier transform maps S to S.

Putting these two facts together gives

The Fourier transform is well defined on S and [(

d

dx

)m ((−ix)nf )

] ˆ= (iξ)m

( d

dξ

)n f̂ ,

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

Convolution goes to multiplication.

(f ? g )̂(ξ) = 1

2π

∫ ∫ f (x − t)g(t)dte−ixξdx

= 1

2π

∫ ∫ f (u)g(t)e−i(u+t)ξdudt

= 1√ 2π

∫

R f (u)e−iuξdu

1√ 2π

∫

R g(t)e−itξdt

so (f ? g )̂ = f̂ ĝ .

Shlomo Sternberg

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

(Saf )̂(ξ) = 1√ 2π

∫

R f (ax)e−ixξdx

= 1√ 2π

∫

R (1/a)f (u)e−iu(ξ/a)du

so (Saf )̂ = (1/a)S1/a f̂ .

Shlomo Sternberg

Fourier transform of a Gaussian is a Gaussian.

The polar coordina

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