[PPT] - Lebesgue integration of oscillating and subanalytic functions Tamara PowerPoint Presentation

SLIDE 1

Lebesgue integration of oscillating and subanalytic functions Tamara Servi (University of Pisa)

(joint work with R. Cluckers, G. Comte, D. Miller, J.-P. Rolin) 19th July 2015

SLIDE 2

Motivation and background

SLIDE 3

Motivation and background

Oscillatory integrals. λ ∈ R, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiλϕ(x)ψ (x) dx, where:

the phase ϕ is analytic, 0 ∈ Rn is an isolated singular point of ϕ;
the amplitude ψ is C∞ with support a compact nbd of 0.

SLIDE 4

Motivation and background

Oscillatory integrals. λ ∈ R, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiλϕ(x)ψ (x) dx, where:

the phase ϕ is analytic, 0 ∈ Rn is an isolated singular point of ϕ;
the amplitude ψ is C∞ with support a compact nbd of 0.

These objects are classically studied in optical physics (Fresnel, Airy,...).

SLIDE 5

Motivation and background

Oscillatory integrals. λ ∈ R, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiλϕ(x)ψ (x) dx, where:

the phase ϕ is analytic, 0 ∈ Rn is an isolated singular point of ϕ;
the amplitude ψ is C∞ with support a compact nbd of 0.

These objects are classically studied in optical physics (Fresnel, Airy,...).

Aim. To study the behaviour of I (λ) when λ → ∞.

SLIDE 6

Motivation and background

Oscillatory integrals. λ ∈ R, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiλϕ(x)ψ (x) dx, where:

the phase ϕ is analytic, 0 ∈ Rn is an isolated singular point of ϕ;
the amplitude ψ is C∞ with support a compact nbd of 0.

These objects are classically studied in optical physics (Fresnel, Airy,...).

Aim. To study the behaviour of I (λ) when λ → ∞.

n = 1 I (λ) ∼ eiλϕ(0)

j∈N

aj (ψ) λ

−

j N(ϕ)

aj (ψ) ∈ R, N (ϕ) ∈ N fixed.

SLIDE 7

Motivation and background

Oscillatory integrals. λ ∈ R, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiλϕ(x)ψ (x) dx, where:

the phase ϕ is analytic, 0 ∈ Rn is an isolated singular point of ϕ;
the amplitude ψ is C∞ with support a compact nbd of 0.

These objects are classically studied in optical physics (Fresnel, Airy,...).

Aim. To study the behaviour of I (λ) when λ → ∞.

n = 1 I (λ) ∼ eiλϕ(0)

j∈N

aj (ψ) λ

−

j N(ϕ)

aj (ψ) ∈ R, N (ϕ) ∈ N fixed. n > 1 reduce to the case n = 1 by monomializing the phase (res. of sing.).

SLIDE 8

Motivation and background

Oscillatory integrals. λ ∈ R, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiλϕ(x)ψ (x) dx, where:

the phase ϕ is analytic, 0 ∈ Rn is an isolated singular point of ϕ;
the amplitude ψ is C∞ with support a compact nbd of 0.

These objects are classically studied in optical physics (Fresnel, Airy,...).

Aim. To study the behaviour of I (λ) when λ → ∞.

n = 1 I (λ) ∼ eiλϕ(0)

j∈N

aj (ψ) λ

−

j N(ϕ)

aj (ψ) ∈ R, N (ϕ) ∈ N fixed. n > 1 reduce to the case n = 1 by monomializing the phase (res. of sing.). Suitable blow-ups act as changes of variables in Rn, outside a set of measure 0.

SLIDE 9

Motivation and background

Oscillatory integrals. λ ∈ R, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiλϕ(x)ψ (x) dx, where:

the phase ϕ is analytic, 0 ∈ Rn is an isolated singular point of ϕ;
the amplitude ψ is C∞ with support a compact nbd of 0.

These objects are classically studied in optical physics (Fresnel, Airy,...).

Aim. To study the behaviour of I (λ) when λ → ∞.

n = 1 I (λ) ∼ eiλϕ(0)

j∈N

aj (ψ) λ

−

j N(ϕ)

aj (ψ) ∈ R, N (ϕ) ∈ N fixed. n > 1 reduce to the case n = 1 by monomializing the phase (res. of sing.). Suitable blow-ups act as changes of variables in Rn, outside a set of measure 0. Using Fubini and the case n = 1, one proves: I (λ) ∼ eiλϕ(0)

q∈Q n−1

k=0

aq,k (ψ) λq (log λ)k .

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Oscillatory integrals in several variables

A more general situation. λ = (λ1, . . . , λm) ∈ Rm, x = (x1, . . . , xn) ∈ Rn

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Oscillatory integrals in several variables

A more general situation. λ = (λ1, . . . , λm) ∈ Rm, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiϕ(λ,x)ψ (x) dx

(the parameters λ and the variables x are “intertwined” in the expression for ϕ).

SLIDE 12

Oscillatory integrals in several variables

A more general situation. λ = (λ1, . . . , λm) ∈ Rm, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiϕ(λ,x)ψ (x) dx

(the parameters λ and the variables x are “intertwined” in the expression for ϕ).

Example. Fourier transforms ˆ

ψ (λ) = ˆ

Rne−2πiλ·xψ (x) dx.

SLIDE 13

Oscillatory integrals in several variables

A more general situation. λ = (λ1, . . . , λm) ∈ Rm, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiϕ(λ,x)ψ (x) dx

(the parameters λ and the variables x are “intertwined” in the expression for ϕ).

Example. Fourier transforms ˆ

ψ (λ) = ˆ

Rne−2πiλ·xψ (x) dx.

Aim. Understand the nature of I (λ) (depending on the nature of ϕ and ψ).

SLIDE 14

Oscillatory integrals in several variables

A more general situation. λ = (λ1, . . . , λm) ∈ Rm, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiϕ(λ,x)ψ (x) dx

(the parameters λ and the variables x are “intertwined” in the expression for ϕ).

Example. Fourier transforms ˆ

ψ (λ) = ˆ

Rne−2πiλ·xψ (x) dx.

Aim. Understand the nature of I (λ) (depending on the nature of ϕ and ψ).

Tool needed. Monomialize the phase while keeping track of the different nature of the variables λ and x.

SLIDE 15

Oscillatory integrals in several variables

A more general situation. λ = (λ1, . . . , λm) ∈ Rm, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiϕ(λ,x)ψ (x) dx

(the parameters λ and the variables x are “intertwined” in the expression for ϕ).

Example. Fourier transforms ˆ

ψ (λ) = ˆ

Rne−2πiλ·xψ (x) dx.

Aim. Understand the nature of I (λ) (depending on the nature of ϕ and ψ).

Tool needed. Monomialize the phase while keeping track of the different nature of the variables λ and x. Natural framework and natural tool:

SLIDE 16

Oscillatory integrals in several variables

A more general situation. λ = (λ1, . . . , λm) ∈ Rm, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiϕ(λ,x)ψ (x) dx

(the parameters λ and the variables x are “intertwined” in the expression for ϕ).

Example. Fourier transforms ˆ

ψ (λ) = ˆ

Rne−2πiλ·xψ (x) dx.

Aim. Understand the nature of I (λ) (depending on the nature of ϕ and ψ).

Tool needed. Monomialize the phase while keeping track of the different nature of the variables λ and x. Natural framework and natural tool: Framework: ϕ, ψ globally subanalytic (i.e. definable in Ran).

SLIDE 17

Oscillatory integrals in several variables

A more general situation. λ = (λ1, . . . , λm) ∈ Rm, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiϕ(λ,x)ψ (x) dx

(the parameters λ and the variables x are “intertwined” in the expression for ϕ).

Example. Fourier transforms ˆ

ψ (λ) = ˆ

Rne−2πiλ·xψ (x) dx.

Aim. Understand the nature of I (λ) (depending on the nature of ϕ and ψ).

Tool needed. Monomialize the phase while keeping track of the different nature of the variables λ and x. Natural framework and natural tool: Framework: ϕ, ψ globally subanalytic (i.e. definable in Ran). Tool: the Lion-Rolin Preparation Theorem.

SLIDE 18

Oscillatory integrals in several variables

A more general situation. λ = (λ1, . . . , λm) ∈ Rm, x = (x1, . . . , xn) ∈ Rn I (λ) = ˆ

Rn eiϕ(λ,x)ψ (x) dx

(the parameters λ and the variables x are “intertwined” in the expression for ϕ).

Example. Fourier transforms ˆ

ψ (λ) = ˆ

Rne−2πiλ·xψ (x) dx.

Aim. Understand the nature of I (λ) (depending on the nature of ϕ and ψ).

Tool needed. Monomialize the phase while keeping track of the different nature of the variables λ and x. Natural framework and natural tool: Framework: ϕ, ψ globally subanalytic (i.e. definable in Ran). Tool: the Lion-Rolin Preparation Theorem.

Proviso. For the rest of the talk, subanalytic means “globally subanalytic”.

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Our framework: parametric integrals and subanalytic functions

Def. For X ⊆ Rm and f : X × Rn → R, define, ∀x ∈ X s.t. f (x, ·) ∈ L1 (Rn),

the parametric integral If (x) = ´

Rn f (x, y) dy.

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Our framework: parametric integrals and subanalytic functions

Def. For X ⊆ Rm and f : X × Rn → R, define, ∀x ∈ X s.t. f (x, ·) ∈ L1 (Rn),

the parametric integral If (x) = ´

Rn f (x, y) dy.

Def. For X ⊆ Rm subanalytic, let

S (X) := {f : X → R subanalytic} and S =

X sub.

S (X)

SLIDE 21

Our framework: parametric integrals and subanalytic functions

Def. For X ⊆ Rm and f : X × Rn → R, define, ∀x ∈ X s.t. f (x, ·) ∈ L1 (Rn),

the parametric integral If (x) = ´

Rn f (x, y) dy.

Def. For X ⊆ Rm subanalytic, let

S (X) := {f : X → R subanalytic} and S =

X sub.

S (X) (Comte - Lion - Rolin). f ∈ S (X × Rn) ⇒ If ∈ C (X),

SLIDE 22

Our framework: parametric integrals and subanalytic functions

Def. For X ⊆ Rm and f : X × Rn → R, define, ∀x ∈ X s.t. f (x, ·) ∈ L1 (Rn),

the parametric integral If (x) = ´

Rn f (x, y) dy.

Def. For X ⊆ Rm subanalytic, let

S (X) := {f : X → R subanalytic} and S =

X sub.

S (X) (Comte - Lion - Rolin). f ∈ S (X × Rn) ⇒ If ∈ C (X), where C (X) := R-algebra generated by {g, log h : g, h ∈ S (X) , h > 0}

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Our framework: parametric integrals and subanalytic functions

Def. For X ⊆ Rm and f : X × Rn → R, define, ∀x ∈ X s.t. f (x, ·) ∈ L1 (Rn),

the parametric integral If (x) = ´

Rn f (x, y) dy.

Def. For X ⊆ Rm subanalytic, let

S (X) := {f : X → R subanalytic} and S =

X sub.

S (X) (Comte - Lion - Rolin). f ∈ S (X × Rn) ⇒ If ∈ C (X), where C (X) := R-algebra generated by {g, log h : g, h ∈ S (X) , h > 0} (“constructible” or “log-subanalytic” functions:

j≤J

gj

k≤K

log hj,k).

SLIDE 24

Our framework: parametric integrals and subanalytic functions

Def. For X ⊆ Rm and f : X × Rn → R, define, ∀x ∈ X s.t. f (x, ·) ∈ L1 (Rn),

the parametric integral If (x) = ´

Rn f (x, y) dy.

Def. For X ⊆ Rm subanalytic, let

S (X) := {f : X → R subanalytic} and S =

X sub.

S (X) (Comte - Lion - Rolin). f ∈ S (X × Rn) ⇒ If ∈ C (X), where C (X) := R-algebra generated by {g, log h : g, h ∈ S (X) , h > 0} (“constructible” or “log-subanalytic” functions:

j≤J

gj

k≤K

log hj,k). (Cluckers - Miller). f ∈ C (X × Rn) ⇒ If ∈ C (X) .

SLIDE 25

Our framework: parametric integrals and subanalytic functions

Def. For X ⊆ Rm and f : X × Rn → R, define, ∀x ∈ X s.t. f (x, ·) ∈ L1 (Rn),

the parametric integral If (x) = ´

Rn f (x, y) dy.

Def. For X ⊆ Rm subanalytic, let

S (X) := {f : X → R subanalytic} and S =

X sub.

S (X) (Comte - Lion - Rolin). f ∈ S (X × Rn) ⇒ If ∈ C (X), where C (X) := R-algebra generated by {g, log h : g, h ∈ S (X) , h > 0} (“constructible” or “log-subanalytic” functions:

j≤J

gj

k≤K

log hj,k). (Cluckers - Miller). f ∈ C (X × Rn) ⇒ If ∈ C (X) .

Aim. Study oscillatory integrals I (λ) =

´

Rn eiλϕ(x)ψ (x) dx, with ϕ, ψ ∈ S (Rn)

and Fourier transforms ˆ f (ξ) = ´

Rn f (x) e−2πiξ·xdx with f ∈ S (Rn).

SLIDE 26

Our framework: parametric integrals and subanalytic functions

Def. For X ⊆ Rm and f : X × Rn → R, define, ∀x ∈ X s.t. f (x, ·) ∈ L1 (Rn),

the parametric integral If (x) = ´

Rn f (x, y) dy.

Def. For X ⊆ Rm subanalytic, let

S (X) := {f : X → R subanalytic} and S =

X sub.

S (X) (Comte - Lion - Rolin). f ∈ S (X × Rn) ⇒ If ∈ C (X), where C (X) := R-algebra generated by {g, log h : g, h ∈ S (X) , h > 0} (“constructible” or “log-subanalytic” functions:

j≤J

gj

k≤K

log hj,k). (Cluckers - Miller). f ∈ C (X × Rn) ⇒ If ∈ C (X) .

Aim. Study oscillatory integrals I (λ) =

´

Rn eiλϕ(x)ψ (x) dx, with ϕ, ψ ∈ S (Rn)

and Fourier transforms ˆ f (ξ) = ´

Rn f (x) e−2πiξ·xdx with f ∈ S (Rn).

Question. D (X) := C- algebra generated by C (X) and
eiϕ(x) : ϕ ∈ S (X)
.

SLIDE 27

Our framework: parametric integrals and subanalytic functions

Def. For X ⊆ Rm and f : X × Rn → R, define, ∀x ∈ X s.t. f (x, ·) ∈ L1 (Rn),

the parametric integral If (x) = ´

Rn f (x, y) dy.

Def. For X ⊆ Rm subanalytic, let

S (X) := {f : X → R subanalytic} and S =

X sub.

S (X) (Comte - Lion - Rolin). f ∈ S (X × Rn) ⇒ If ∈ C (X), where C (X) := R-algebra generated by {g, log h : g, h ∈ S (X) , h > 0} (“constructible” or “log-subanalytic” functions:

j≤J

gj

k≤K

log hj,k). (Cluckers - Miller). f ∈ C (X × Rn) ⇒ If ∈ C (X) .

Aim. Study oscillatory integrals I (λ) =

´

Rn eiλϕ(x)ψ (x) dx, with ϕ, ψ ∈ S (Rn)

and Fourier transforms ˆ f (ξ) = ´

Rn f (x) e−2πiξ·xdx with f ∈ S (Rn).

Question. D (X) := C- algebra generated by C (X) and
eiϕ(x) : ϕ ∈ S (X)
.

f ∈ D (X × Rn)

?

⇒ If ∈ D (X)

SLIDE 28

Oscillating and subanalytic functions

The answer is NO: the C- algebra D (X) generated by

g (x) , log h (x) , eiϕ(x) : g, h, ϕ ∈ S (X)
is not stable under parametric integration.

SLIDE 29

Oscillating and subanalytic functions

The answer is NO: the C- algebra D (X) generated by

g (x) , log h (x) , eiϕ(x) : g, h, ϕ ∈ S (X)
is not stable under parametric integration.
Example. Si (x) =

ˆ x sin t t dt

SLIDE 30

Oscillating and subanalytic functions

The answer is NO: the C- algebra D (X) generated by

g (x) , log h (x) , eiϕ(x) : g, h, ϕ ∈ S (X)
is not stable under parametric integration.
Example. Si (x) =

ˆ x sin t t dt = ˆ

R

χ[0,x] (t) 2it

eit − e−it

dt

SLIDE 31

Oscillating and subanalytic functions

The answer is NO: the C- algebra D (X) generated by

g (x) , log h (x) , eiϕ(x) : g, h, ϕ ∈ S (X)
is not stable under parametric integration.
Example. Si (x) =

ˆ x sin t t dt = ˆ

R

χ[0,x] (t) 2it

eit − e−it

dt / ∈ D

R+

.

SLIDE 32

Oscillating and subanalytic functions

The answer is NO: the C- algebra D (X) generated by

g (x) , log h (x) , eiϕ(x) : g, h, ϕ ∈ S (X)
is not stable under parametric integration.
Example. Si (x) =

ˆ x sin t t dt = ˆ

R

χ[0,x] (t) 2it

eit − e−it

dt / ∈ D

R+

. Why?

SLIDE 33

Oscillating and subanalytic functions

The answer is NO: the C- algebra D (X) generated by

g (x) , log h (x) , eiϕ(x) : g, h, ϕ ∈ S (X)
is not stable under parametric integration.
Example. Si (x) =

ˆ x sin t t dt = ˆ

R

χ[0,x] (t) 2it

eit − e−it

dt / ∈ D

R+

. Why?

It is well-known that Si (x) ∼

x→+∞

π 2 − cos x x

k≥0

(−1)k (2k)! x2k − sin x x

k≥0

(−1)k (2k + 1)! x2k+1 ,

SLIDE 34

Oscillating and subanalytic functions

The answer is NO: the C- algebra D (X) generated by

g (x) , log h (x) , eiϕ(x) : g, h, ϕ ∈ S (X)
is not stable under parametric integration.
Example. Si (x) =

ˆ x sin t t dt = ˆ

R

χ[0,x] (t) 2it

eit − e−it

dt / ∈ D

R+

. Why?

It is well-known that Si (x) ∼

x→+∞

π 2 − cos x x

k≥0

(−1)k (2k)! x2k − sin x x

k≥0

(−1)k (2k + 1)! x2k+1 , i.e. Si ∼ to a polynomial in {cos x, sin x} with coefficients divergent series ∈ R 1

x

.

SLIDE 35

Oscillating and subanalytic functions

The answer is NO: the C- algebra D (X) generated by

g (x) , log h (x) , eiϕ(x) : g, h, ϕ ∈ S (X)
is not stable under parametric integration.
Example. Si (x) =

ˆ x sin t t dt = ˆ

R

χ[0,x] (t) 2it

eit − e−it

dt / ∈ D

R+

. Why?

It is well-known that Si (x) ∼

x→+∞

π 2 − cos x x

k≥0

(−1)k (2k)! x2k − sin x x

k≥0

(−1)k (2k + 1)! x2k+1 , i.e. Si ∼ to a polynomial in {cos x, sin x} with coefficients divergent series ∈ R 1

x

.

However, if f ∈ D

R+

, then f is asymptotic to a polynomial in {log x} ∪ {cos (cjxrj ) , sin (cjxrj )}N

j=1 with convergent coefficients ∈ R

x− 1

d

,

(for some N, d ∈ N, cj ∈ R, rj ∈ Q+).

SLIDE 36

Oscillating and subanalytic functions

The answer is NO: the C- algebra D (X) generated by

g (x) , log h (x) , eiϕ(x) : g, h, ϕ ∈ S (X)
is not stable under parametric integration.
Example. Si (x) =

ˆ x sin t t dt = ˆ

R

χ[0,x] (t) 2it

eit − e−it

dt / ∈ D

R+

. Why?

It is well-known that Si (x) ∼

x→+∞

π 2 − cos x x

k≥0

(−1)k (2k)! x2k − sin x x

k≥0

(−1)k (2k + 1)! x2k+1 , i.e. Si ∼ to a polynomial in {cos x, sin x} with coefficients divergent series ∈ R 1

x

.

However, if f ∈ D

R+

, then f is asymptotic to a polynomial in {log x} ∪ {cos (cjxrj ) , sin (cjxrj )}N

j=1 with convergent coefficients ∈ R

x− 1

d

,

(for some N, d ∈ N, cj ∈ R, rj ∈ Q+).

To see this:

For g ∈ S
R+

, ∃c ∈ R, r ∈ Q, d ∈ N, ∃H ∈ R {Y }∗ s.t. g (x) = cxrH

x− 1

d

.

SLIDE 37

Oscillating and subanalytic functions

The answer is NO: the C- algebra D (X) generated by

g (x) , log h (x) , eiϕ(x) : g, h, ϕ ∈ S (X)
is not stable under parametric integration.
Example. Si (x) =

ˆ x sin t t dt = ˆ

R

χ[0,x] (t) 2it

eit − e−it

dt / ∈ D

R+

. Why?

It is well-known that Si (x) ∼

x→+∞

π 2 − cos x x

k≥0

(−1)k (2k)! x2k − sin x x

k≥0

(−1)k (2k + 1)! x2k+1 , i.e. Si ∼ to a polynomial in {cos x, sin x} with coefficients divergent series ∈ R 1

x

.

However, if f ∈ D

R+

, then f is asymptotic to a polynomial in {log x} ∪ {cos (cjxrj ) , sin (cjxrj )}N

j=1 with convergent coefficients ∈ R

x− 1

d

,

(for some N, d ∈ N, cj ∈ R, rj ∈ Q+).

To see this:

For g ∈ S
R+

, ∃c ∈ R, r ∈ Q, d ∈ N, ∃H ∈ R {Y }∗ s.t. g (x) = cxrH

x− 1

d

.
log (g (x)) = rlogx + log
cH
x− 1

d

, eig(x) = ei

j≤rd cj xr−j/d

· ei

j>rd cj xr−j/d

.

SLIDE 38

Oscillating and subanalytic functions

The answer is NO: the C- algebra D (X) generated by

g (x) , log h (x) , eiϕ(x) : g, h, ϕ ∈ S (X)
is not stable under parametric integration.
Example. Si (x) =

ˆ x sin t t dt = ˆ

R

χ[0,x] (t) 2it

eit − e−it

dt / ∈ D

R+

. Why?

It is well-known that Si (x) ∼

x→+∞

π 2 − cos x x

k≥0

(−1)k (2k)! x2k − sin x x

k≥0

(−1)k (2k + 1)! x2k+1 , i.e. Si ∼ to a polynomial in {cos x, sin x} with coefficients divergent series ∈ R 1

x

.

However, if f ∈ D

R+

, then f is asymptotic to a polynomial in {log x} ∪ {cos (cjxrj ) , sin (cjxrj )}N

j=1 with convergent coefficients ∈ R

x− 1

d

,

(for some N, d ∈ N, cj ∈ R, rj ∈ Q+).

To see this:

For g ∈ S
R+

, ∃c ∈ R, r ∈ Q, d ∈ N, ∃H ∈ R {Y }∗ s.t. g (x) = cxrH

x− 1

d

.
log (g (x)) = rlogx + log
cH
x− 1

d

, eig(x) = ei

j≤rd cj xr−j/d

· ei

j>rd cj xr−j/d

.

if g is bounded, then log (g (x)) , cos (g (x)) , sin (g (x)) ∈ S
R+

.

SLIDE 39

One-dimensional transcendentals

X ⊆ Rm subanalytic. What do we need to add to D (X) to make it stable under parametric integration?

SLIDE 40

One-dimensional transcendentals

X ⊆ Rm subanalytic. What do we need to add to D (X) to make it stable under parametric integration? γh,ℓ(x) = ´

R h(x, t)(log |t|)ℓeitdt,

ℓ ∈ N, h ∈ S (X × R) , h (x, ·) ∈ L1 (R)

SLIDE 41

One-dimensional transcendentals

X ⊆ Rm subanalytic. What do we need to add to D (X) to make it stable under parametric integration? γh,ℓ(x) = ´

R h(x, t)(log |t|)ℓeitdt,

ℓ ∈ N, h ∈ S (X × R) , h (x, ·) ∈ L1 (R)
Def. E (X) := the D (X)-module generated by {γh,ℓ}h,ℓ

SLIDE 42

One-dimensional transcendentals

X ⊆ Rm subanalytic. What do we need to add to D (X) to make it stable under parametric integration? γh,ℓ(x) = ´

R h(x, t)(log |t|)ℓeitdt,

ℓ ∈ N, h ∈ S (X × R) , h (x, ·) ∈ L1 (R)
Def. E (X) := the D (X)-module generated by {γh,ℓ}h,ℓ

Main Theorem. f ∈ E (X × Rn) ⇒ If ∈ E (X) .

SLIDE 43

One-dimensional transcendentals

X ⊆ Rm subanalytic. What do we need to add to D (X) to make it stable under parametric integration? γh,ℓ(x) = ´

R h(x, t)(log |t|)ℓeitdt,

ℓ ∈ N, h ∈ S (X × R) , h (x, ·) ∈ L1 (R)
Def. E (X) := the D (X)-module generated by {γh,ℓ}h,ℓ

Main Theorem. f ∈ E (X × Rn) ⇒ If ∈ E (X) . More precisely, let Int (f , X) :=

x ∈ X : f (x, ·) ∈ L1 (Rn)
(integrability locus).

SLIDE 44

One-dimensional transcendentals

X ⊆ Rm subanalytic. What do we need to add to D (X) to make it stable under parametric integration? γh,ℓ(x) = ´

R h(x, t)(log |t|)ℓeitdt,

ℓ ∈ N, h ∈ S (X × R) , h (x, ·) ∈ L1 (R)
Def. E (X) := the D (X)-module generated by {γh,ℓ}h,ℓ

Main Theorem. f ∈ E (X × Rn) ⇒ If ∈ E (X) . More precisely, let Int (f , X) :=

x ∈ X : f (x, ·) ∈ L1 (Rn)
(integrability locus).

Then there exists F ∈ E (X) s.t. F (x) = ˆ

Rnf (x, y) dy

∀x ∈ Int (f , X).

SLIDE 45

One-dimensional transcendentals

X ⊆ Rm subanalytic. What do we need to add to D (X) to make it stable under parametric integration? γh,ℓ(x) = ´

R h(x, t)(log |t|)ℓeitdt,

ℓ ∈ N, h ∈ S (X × R) , h (x, ·) ∈ L1 (R)
Def. E (X) := the D (X)-module generated by {γh,ℓ}h,ℓ

Main Theorem. f ∈ E (X × Rn) ⇒ If ∈ E (X) . More precisely, let Int (f , X) :=

x ∈ X : f (x, ·) ∈ L1 (Rn)
(integrability locus).

Then there exists F ∈ E (X) s.t. F (x) = ˆ

Rnf (x, y) dy

∀x ∈ Int (f , X).

Corollary. E (X) is a C-algebra.

SLIDE 46

One-dimensional transcendentals

X ⊆ Rm subanalytic. What do we need to add to D (X) to make it stable under parametric integration? γh,ℓ(x) = ´

R h(x, t)(log |t|)ℓeitdt,

ℓ ∈ N, h ∈ S (X × R) , h (x, ·) ∈ L1 (R)
Def. E (X) := the D (X)-module generated by {γh,ℓ}h,ℓ

Main Theorem. f ∈ E (X × Rn) ⇒ If ∈ E (X) . More precisely, let Int (f , X) :=

x ∈ X : f (x, ·) ∈ L1 (Rn)
(integrability locus).

Then there exists F ∈ E (X) s.t. F (x) = ˆ

Rnf (x, y) dy

∀x ∈ Int (f , X).

Corollary. E (X) is a C-algebra.
Proof. By Fubini,

γh,ℓ (x) · γh′,ℓ′ (x) = ˜

R2 h (x, t) · h′ (x, t′) · (log |t|)ℓ · (log |t′|)ℓ′

ei(t+t′)dtdt′

SLIDE 47

One-dimensional transcendentals

X ⊆ Rm subanalytic. What do we need to add to D (X) to make it stable under parametric integration? γh,ℓ(x) = ´

R h(x, t)(log |t|)ℓeitdt,

ℓ ∈ N, h ∈ S (X × R) , h (x, ·) ∈ L1 (R)
Def. E (X) := the D (X)-module generated by {γh,ℓ}h,ℓ

Main Theorem. f ∈ E (X × Rn) ⇒ If ∈ E (X) . More precisely, let Int (f , X) :=

x ∈ X : f (x, ·) ∈ L1 (Rn)
(integrability locus).

Then there exists F ∈ E (X) s.t. F (x) = ˆ

Rnf (x, y) dy

∀x ∈ Int (f , X).

Corollary. E (X) is a C-algebra.
Proof. By Fubini,

γh,ℓ (x) · γh′,ℓ′ (x) = ˜

R2 h (x, t) · h′ (x, t′) · (log |t|)ℓ · (log |t′|)ℓ′

ei(t+t′)dtdt′, which is the parametric integral of a function in D (X), and hence, by the Main Theorem, belongs to E (X).

SLIDE 48

One-dimensional transcendentals

X ⊆ Rm subanalytic. What do we need to add to D (X) to make it stable under parametric integration? γh,ℓ(x) = ´

R h(x, t)(log |t|)ℓeitdt,

ℓ ∈ N, h ∈ S (X × R) , h (x, ·) ∈ L1 (R)
Def. E (X) := the D (X)-module generated by {γh,ℓ}h,ℓ

Main Theorem. f ∈ E (X × Rn) ⇒ If ∈ E (X) . More precisely, let Int (f , X) :=

x ∈ X : f (x, ·) ∈ L1 (Rn)
(integrability locus).

Then there exists F ∈ E (X) s.t. F (x) = ˆ

Rnf (x, y) dy

∀x ∈ Int (f , X).

Corollary. E (X) is a C-algebra.
Proof. By Fubini,

γh,ℓ (x) · γh′,ℓ′ (x) = ˜

R2 h (x, t) · h′ (x, t′) · (log |t|)ℓ · (log |t′|)ℓ′

ei(t+t′)dtdt′, which is the parametric integral of a function in D (X), and hence, by the Main Theorem, belongs to E (X).

Corollary. E = E (X) is the smallest collection of C-algebras containing

S ∪

eiϕ : ϕ ∈ S
and stable under parametric integration.

Moreover, E is closed under taking Fourier transforms.

SLIDE 49

Generators

Rem. An element of E (X × Rn) can be written as a finite sum of generators:

SLIDE 50

Generators

Rem. An element of E (X × Rn) can be written as a finite sum of generators:

T (x, y) = f (x, y) · eiϕ(x,y) · γ (x, y) , where f ∈ C (X × Rn) , ϕ ∈ S (X × Rn) and γ (x, y) = ˆ

R

h(x, y, t)(log |t|)ℓeitdt

SLIDE 51

Generators

Rem. An element of E (X × Rn) can be written as a finite sum of generators:

T (x, y) = f (x, y) · eiϕ(x,y) · γ (x, y) , where f ∈ C (X × Rn) , ϕ ∈ S (X × Rn) and γ (x, y) = ˆ

R

h(x, y, t)(log |t|)ℓeitdt

Def. A generator T (x, y) ∈ E (X × Rn) is strongly integrable if

y − → |f (x, y)| ˆ

R

h (x, y, t) (log |t|)ℓ
dt ∈ L1 (Rn).

SLIDE 52

Generators

Rem. An element of E (X × Rn) can be written as a finite sum of generators:

T (x, y) = f (x, y) · eiϕ(x,y) · γ (x, y) , where f ∈ C (X × Rn) , ϕ ∈ S (X × Rn) and γ (x, y) = ˆ

R

h(x, y, t)(log |t|)ℓeitdt

Def. A generator T (x, y) ∈ E (X × Rn) is strongly integrable if

y − → |f (x, y)| ˆ

R

h (x, y, t) (log |t|)ℓ
dt ∈ L1 (Rn).
Proposition. If T is strongly integrable, then IT ∈ E (X).

SLIDE 53

Generators

Rem. An element of E (X × Rn) can be written as a finite sum of generators:

T (x, y) = f (x, y) · eiϕ(x,y) · γ (x, y) , where f ∈ C (X × Rn) , ϕ ∈ S (X × Rn) and γ (x, y) = ˆ

R

h(x, y, t)(log |t|)ℓeitdt

Def. A generator T (x, y) ∈ E (X × Rn) is strongly integrable if

y − → |f (x, y)| ˆ

R

h (x, y, t) (log |t|)ℓ
dt ∈ L1 (Rn).
Proposition. If T is strongly integrable, then IT ∈ E (X).
Proof. By Fubini-Tonelli,

ˆ

RnT (x, y) dy =

¨

Rn+1f (x, y) h (x, y, t) (log |t|)ℓ ei(t+ϕ(x,y))dydt, so we may

suppose T = f (x, y) eiϕ(x,y) ∈ D (X × Rn).

SLIDE 54

Generators

Rem. An element of E (X × Rn) can be written as a finite sum of generators:

T (x, y) = f (x, y) · eiϕ(x,y) · γ (x, y) , where f ∈ C (X × Rn) , ϕ ∈ S (X × Rn) and γ (x, y) = ˆ

R

h(x, y, t)(log |t|)ℓeitdt

Def. A generator T (x, y) ∈ E (X × Rn) is strongly integrable if

y − → |f (x, y)| ˆ

R

h (x, y, t) (log |t|)ℓ
dt ∈ L1 (Rn).
Proposition. If T is strongly integrable, then IT ∈ E (X).
Proof. By Fubini-Tonelli,

ˆ

RnT (x, y) dy =

¨

Rn+1f (x, y) h (x, y, t) (log |t|)ℓ ei(t+ϕ(x,y))dydt, so we may

suppose T = f (x, y) eiϕ(x,y) ∈ D (X × Rn). O-minimality does the rest.

SLIDE 55

Generators

Rem. An element of E (X × Rn) can be written as a finite sum of generators:

T (x, y) = f (x, y) · eiϕ(x,y) · γ (x, y) , where f ∈ C (X × Rn) , ϕ ∈ S (X × Rn) and γ (x, y) = ˆ

R

h(x, y, t)(log |t|)ℓeitdt

Def. A generator T (x, y) ∈ E (X × Rn) is strongly integrable if

y − → |f (x, y)| ˆ

R

h (x, y, t) (log |t|)ℓ
dt ∈ L1 (Rn).
Proposition. If T is strongly integrable, then IT ∈ E (X).
Proof. By Fubini-Tonelli,

ˆ

RnT (x, y) dy =

¨

Rn+1f (x, y) h (x, y, t) (log |t|)ℓ ei(t+ϕ(x,y))dydt, so we may

suppose T = f (x, y) eiϕ(x,y) ∈ D (X × Rn). O-minimality does the rest.

Def. A generator T (x, y) ∈ E (X × Rn) is naive in y if γ does not depend on y.

SLIDE 56

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t.

SLIDE 57

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj,

SLIDE 58

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj, where the Tj are strongly integrable, the Sj are naive in y and ∀x, x ∈ Int (f , X) ⇒ ∀j ∈ JNaive, x / ∈ Int (Sj, X).

SLIDE 59

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj, where the Tj are strongly integrable, the Sj are naive in y and ∀x, x ∈ Int (f , X) ⇒ ∀j ∈ JNaive, x / ∈ Int (Sj, X). Ingredients of the proof.

cell decomposition, definable choice

SLIDE 60

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj, where the Tj are strongly integrable, the Sj are naive in y and ∀x, x ∈ Int (f , X) ⇒ ∀j ∈ JNaive, x / ∈ Int (Sj, X). Ingredients of the proof.

cell decomposition, definable choice
“nested” subanalytic preparation (after Lion-Rolin):

h (x, y, t) = h0 (x, y) |t − θ (x, y)|r U (x, y, t)

SLIDE 61

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj, where the Tj are strongly integrable, the Sj are naive in y and ∀x, x ∈ Int (f , X) ⇒ ∀j ∈ JNaive, x / ∈ Int (Sj, X). Ingredients of the proof.

cell decomposition, definable choice
“nested” subanalytic preparation (after Lion-Rolin):

h (x, y, t) = h0 (x, y) |t − θ (x, y)|r U (x, y, t)

integration by parts creates a naive term and an integrable term.

SLIDE 62

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj, where the Tj are strongly integrable, the Sj are naive in y and ∀x, x ∈ Int (f , X) ⇒ ∀j ∈ JNaive, x / ∈ Int (Sj, X). Ingredients of the proof.

cell decomposition, definable choice
“nested” subanalytic preparation (after Lion-Rolin):

h (x, y, t) = h0 (x, y) |t − θ (x, y)|r U (x, y, t)

integration by parts creates a naive term and an integrable term.

Proof of the Main Theorem. Let x ∈ Int (f , X) and F (x) =

j∈JInt

ˆ

R

Tjdy.

SLIDE 63

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj, where the Tj are strongly integrable, the Sj are naive in y and ∀x, x ∈ Int (f , X) ⇒ ∀j ∈ JNaive, x / ∈ Int (Sj, X). Ingredients of the proof.

cell decomposition, definable choice
“nested” subanalytic preparation (after Lion-Rolin):

h (x, y, t) = h0 (x, y) |t − θ (x, y)|r U (x, y, t)

integration by parts creates a naive term and an integrable term.

Proof of the Main Theorem. Let x ∈ Int (f , X) and F (x) =

j∈JInt

ˆ

R

Tjdy.

Claim. x /

∈ Int

j∈JNaiveSj, X
.

SLIDE 64

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj, where the Tj are strongly integrable, the Sj are naive in y and ∀x, x ∈ Int (f , X) ⇒ ∀j ∈ JNaive, x / ∈ Int (Sj, X). Ingredients of the proof.

cell decomposition, definable choice
“nested” subanalytic preparation (after Lion-Rolin):

h (x, y, t) = h0 (x, y) |t − θ (x, y)|r U (x, y, t)

integration by parts creates a naive term and an integrable term.

Proof of the Main Theorem. Let x ∈ Int (f , X) and F (x) =

j∈JInt

ˆ

R

Tjdy.

Claim. x /

∈ Int

j∈JNaiveSj, X
. Then
j∈JNaiveSj (x, ·) ≡ 0

SLIDE 65

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj, where the Tj are strongly integrable, the Sj are naive in y and ∀x, x ∈ Int (f , X) ⇒ ∀j ∈ JNaive, x / ∈ Int (Sj, X). Ingredients of the proof.

cell decomposition, definable choice
“nested” subanalytic preparation (after Lion-Rolin):

h (x, y, t) = h0 (x, y) |t − θ (x, y)|r U (x, y, t)

integration by parts creates a naive term and an integrable term.

Proof of the Main Theorem. Let x ∈ Int (f , X) and F (x) =

j∈JInt

ˆ

R

Tjdy.

Claim. x /

∈ Int

j∈JNaiveSj, X
. Then
j∈JNaiveSj (x, ·) ≡ 0 and

´

R f (x, y) dy = F (x)

SLIDE 66

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj, where the Tj are strongly integrable, the Sj are naive in y and ∀x, x ∈ Int (f , X) ⇒ ∀j ∈ JNaive, x / ∈ Int (Sj, X). Ingredients of the proof.

cell decomposition, definable choice
“nested” subanalytic preparation (after Lion-Rolin):

h (x, y, t) = h0 (x, y) |t − θ (x, y)|r U (x, y, t)

integration by parts creates a naive term and an integrable term.

Proof of the Main Theorem. Let x ∈ Int (f , X) and F (x) =

j∈JInt

ˆ

R

Tjdy.

Claim. x /

∈ Int

j∈JNaiveSj, X
. Then
j∈JNaiveSj (x, ·) ≡ 0 and

´

R f (x, y) dy = F (x)∈ E (X).

SLIDE 67

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj, where the Tj are strongly integrable, the Sj are naive in y and ∀x, x ∈ Int (f , X) ⇒ ∀j ∈ JNaive, x / ∈ Int (Sj, X). Ingredients of the proof.

cell decomposition, definable choice
“nested” subanalytic preparation (after Lion-Rolin):

h (x, y, t) = h0 (x, y) |t − θ (x, y)|r U (x, y, t)

integration by parts creates a naive term and an integrable term.

Proof of the Main Theorem. Let x ∈ Int (f , X) and F (x) =

j∈JInt

ˆ

R

Tjdy.

Claim. x /

∈ Int

j∈JNaiveSj, X
. Then
j∈JNaiveSj (x, ·) ≡ 0 and

´

R f (x, y) dy = F (x)∈ E (X). This proves the case n = 1.

SLIDE 68

Key step: preparation of functions in E (X×R)

Preparation Theorem. Given f ∈ E (X×R), up to cell decomposition of X × R, there are finite index sets JInt, JNaive ⊆ N and generators Tj, Sj s.t. f =

j∈JInt

Tj +

j∈JNaive

Sj, where the Tj are strongly integrable, the Sj are naive in y and ∀x, x ∈ Int (f , X) ⇒ ∀j ∈ JNaive, x / ∈ Int (Sj, X). Ingredients of the proof.

cell decomposition, definable choice
“nested” subanalytic preparation (after Lion-Rolin):

h (x, y, t) = h0 (x, y) |t − θ (x, y)|r U (x, y, t)

integration by parts creates a naive term and an integrable term.

Proof of the Main Theorem. Let x ∈ Int (f , X) and F (x) =

j∈JInt

ˆ

R

Tjdy.

Claim. x /

∈ Int

j∈JNaiveSj, X
. Then
j∈JNaiveSj (x, ·) ≡ 0 and

´

R f (x, y) dy = F (x)∈ E (X). This proves the case n = 1.

The case n > 1 follows by Fubini and induction on n.

SLIDE 69

Finite sums of exponentials of polynomials

SLIDE 70

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.

SLIDE 71

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.
Proof. Fix such an x.

SLIDE 72

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.
Proof. Fix such an x. We may assume that Sj (y) = fjy rj (log y)sj eipj (y), with

fj = 0 and pj distinct polynomials in y 1/d and pj (0) = 0.

SLIDE 73

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.
Proof. Fix such an x. We may assume that Sj (y) = fjy rj (log y)sj eipj (y), with

fj = 0 and pj distinct polynomials in y 1/d and pj (0) = 0. Let G (y) =

j∈J fjeipj (y).

SLIDE 74

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.
Proof. Fix such an x. We may assume that Sj (y) = fjy rj (log y)sj eipj (y), with

fj = 0 and pj distinct polynomials in y 1/d and pj (0) = 0. Let G (y) =

j∈J fjeipj (y). Notice that y rj (log y)sj > y −1 for y >

> 0.

SLIDE 75

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.
Proof. Fix such an x. We may assume that Sj (y) = fjy rj (log y)sj eipj (y), with

fj = 0 and pj distinct polynomials in y 1/d and pj (0) = 0. Let G (y) =

j∈J fjeipj (y). Notice that y rj (log y)sj > y −1 for y >

> 0. Then, ˆ

R+

j∈J Sj (y)
dy ≥

ˆ

R+ 1 y |G (y)| dy.

SLIDE 76

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.
Proof. Fix such an x. We may assume that Sj (y) = fjy rj (log y)sj eipj (y), with

fj = 0 and pj distinct polynomials in y 1/d and pj (0) = 0. Let G (y) =

j∈J fjeipj (y). Notice that y rj (log y)sj > y −1 for y >

> 0. Then, ˆ

R+

j∈J Sj (y)
dy ≥

ˆ

R+ 1 y |G (y)| dy.

Since G ≡ 0, by continuity ∃ε, δ > 0 s.t. |G (y)| > ε on some interval I of length ≥ δ.

SLIDE 77

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.
Proof. Fix such an x. We may assume that Sj (y) = fjy rj (log y)sj eipj (y), with

fj = 0 and pj distinct polynomials in y 1/d and pj (0) = 0. Let G (y) =

j∈J fjeipj (y). Notice that y rj (log y)sj > y −1 for y >

> 0. Then, ˆ

R+

j∈J Sj (y)
dy ≥

ˆ

R+ 1 y |G (y)| dy.

Since G ≡ 0, by continuity ∃ε, δ > 0 s.t. |G (y)| > ε on some interval I of length ≥ δ. Idea: If G were periodic, of period ν, then |G| ≥ ε on Vε :=

k∈N (I + kν).

SLIDE 78

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.
Proof. Fix such an x. We may assume that Sj (y) = fjy rj (log y)sj eipj (y), with

fj = 0 and pj distinct polynomials in y 1/d and pj (0) = 0. Let G (y) =

j∈J fjeipj (y). Notice that y rj (log y)sj > y −1 for y >

> 0. Then, ˆ

R+

j∈J Sj (y)
dy ≥

ˆ

R+ 1 y |G (y)| dy.

Since G ≡ 0, by continuity ∃ε, δ > 0 s.t. |G (y)| > ε on some interval I of length ≥ δ. Idea: If G were periodic, of period ν, then |G| ≥ ε on Vε :=

k∈N (I + kν).

Then, ˆ

R+ 1 y |G (y)| dy ≥ ε

ˆ

R+∩Vε 1 y dy ∼ ∞

k=1

δ kν = ∞.

SLIDE 79

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.
Proof. Fix such an x. We may assume that Sj (y) = fjy rj (log y)sj eipj (y), with

fj = 0 and pj distinct polynomials in y 1/d and pj (0) = 0. Let G (y) =

j∈J fjeipj (y). Notice that y rj (log y)sj > y −1 for y >

> 0. Then, ˆ

R+

j∈J Sj (y)
dy ≥

ˆ

R+ 1 y |G (y)| dy.

Since G ≡ 0, by continuity ∃ε, δ > 0 s.t. |G (y)| > ε on some interval I of length ≥ δ. Idea: If G were periodic, of period ν, then |G| ≥ ε on Vε :=

k∈N (I + kν).

Then, ˆ

R+ 1 y |G (y)| dy ≥ ε

ˆ

R+∩Vε 1 y dy ∼ ∞

k=1

δ kν = ∞.

Now, G is not periodic.

SLIDE 80

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.
Proof. Fix such an x. We may assume that Sj (y) = fjy rj (log y)sj eipj (y), with

fj = 0 and pj distinct polynomials in y 1/d and pj (0) = 0. Let G (y) =

j∈J fjeipj (y). Notice that y rj (log y)sj > y −1 for y >

> 0. Then, ˆ

R+

j∈J Sj (y)
dy ≥

ˆ

R+ 1 y |G (y)| dy.

Since G ≡ 0, by continuity ∃ε, δ > 0 s.t. |G (y)| > ε on some interval I of length ≥ δ. Idea: If G were periodic, of period ν, then |G| ≥ ε on Vε :=

k∈N (I + kν).

Then, ˆ

R+ 1 y |G (y)| dy ≥ ε

ˆ

R+∩Vε 1 y dy ∼ ∞

k=1

δ kν = ∞.

Now, G is not periodic. But, using the theory of almost periodic functions (H. Bohr), we show that the set Vε := {y : |G (y)| ≥ ε} is relatively dense in R, i.e. it intersects every interval of size ν (for some ν > 0), and such an intersection has measure ≥ δ (for some δ > 0).

SLIDE 81

Finite sums of exponentials of polynomials

Claim. Let x /

∈ Int (Sj, X) ∀j ∈ J. Then x / ∈ Int

j∈J Sj, X
.
Proof. Fix such an x. We may assume that Sj (y) = fjy rj (log y)sj eipj (y), with

fj = 0 and pj distinct polynomials in y 1/d and pj (0) = 0. Let G (y) =

j∈J fjeipj (y). Notice that y rj (log y)sj > y −1 for y >

> 0. Then, ˆ

R+

j∈J Sj (y)
dy ≥

ˆ

R+ 1 y |G (y)| dy.

Since G ≡ 0, by continuity ∃ε, δ > 0 s.t. |G (y)| > ε on some interval I of length ≥ δ. Idea: If G were periodic, of period ν, then |G| ≥ ε on Vε :=

k∈N (I + kν).

Then, ˆ

R+ 1 y |G (y)| dy ≥ ε

ˆ

R+∩Vε 1 y dy ∼ ∞

k=1

δ kν = ∞.

Now, G is not periodic. But, using the theory of almost periodic functions (H. Bohr), we show that the set Vε := {y : |G (y)| ≥ ε} is relatively dense in R, i.e. it intersects every interval of size ν (for some ν > 0), and such an intersection has measure ≥ δ (for some δ > 0).

SLIDE 82

Almost periodic functions

Example. f (x) = sin (2πx) + sin
2

√ 2πx

is not periodic.

SLIDE 83

Almost periodic functions

Example. f (x) = sin (2πx) + sin
2

√ 2πx

is not periodic. However,

∀ε > 0 ∃ ∞ many τ s.t. x ∈ R |f (x + τ) − f (x)| < ε.

SLIDE 84

Almost periodic functions

Example. f (x) = sin (2πx) + sin
2

√ 2πx

is not periodic. However,

∀ε > 0 ∃ ∞ many τ s.t. x ∈ R |f (x + τ) − f (x)| < ε. Given f , an ε-period is a number τ such that x ∈ R |f (x + τ) − f (x)| < ε.

SLIDE 85

Almost periodic functions

Example. f (x) = sin (2πx) + sin
2

√ 2πx

is not periodic. However,

∀ε > 0 ∃ ∞ many τ s.t. x ∈ R |f (x + τ) − f (x)| < ε. Given f , an ε-period is a number τ such that x ∈ R |f (x + τ) − f (x)| < ε. Tf ,ε := {τ : τ is an ε − period}.

SLIDE 86

Almost periodic functions

Example. f (x) = sin (2πx) + sin
2

√ 2πx

is not periodic. However,

∀ε > 0 ∃ ∞ many τ s.t. x ∈ R |f (x + τ) − f (x)| < ε. Given f , an ε-period is a number τ such that x ∈ R |f (x + τ) − f (x)| < ε. Tf ,ε := {τ : τ is an ε − period}.

Def. A continuous function f is almost periodic if for every ε > 0, the set Tf ,ε

is relatively dense, i.e. it intersects every interval of size ν (for some ν > 0).

SLIDE 87

Almost periodic functions

Example. f (x) = sin (2πx) + sin
2

√ 2πx

is not periodic. However,

∀ε > 0 ∃ ∞ many τ s.t. x ∈ R |f (x + τ) − f (x)| < ε. Given f , an ε-period is a number τ such that x ∈ R |f (x + τ) − f (x)| < ε. Tf ,ε := {τ : τ is an ε − period}.

Def. A continuous function f is almost periodic if for every ε > 0, the set Tf ,ε

is relatively dense, i.e. it intersects every interval of size ν (for some ν > 0). This definition extends to F : Rn → R.

SLIDE 88

Almost periodic functions

Example. f (x) = sin (2πx) + sin
2

√ 2πx

is not periodic. However,

∀ε > 0 ∃ ∞ many τ s.t. x ∈ R |f (x + τ) − f (x)| < ε. Given f , an ε-period is a number τ such that x ∈ R |f (x + τ) − f (x)| < ε. Tf ,ε := {τ : τ is an ε − period}.

Def. A continuous function f is almost periodic if for every ε > 0, the set Tf ,ε

is relatively dense, i.e. it intersects every interval of size ν (for some ν > 0). This definition extends to F : Rn → R.

Lemma. If F : Rn → R is almost periodic and G (y) = F
y, y 2, . . . , y n

, then ∃ε > 0 s.t. the set Vε := {y : |G (y)| ≥ ε} intersects every interval of size ν (for some ν > 0), and such an intersection has measure ≥ δ (for some δ > 0).

SLIDE 89

Almost periodic functions

Example. f (x) = sin (2πx) + sin
2

√ 2πx

is not periodic. However,

∀ε > 0 ∃ ∞ many τ s.t. x ∈ R |f (x + τ) − f (x)| < ε. Given f , an ε-period is a number τ such that x ∈ R |f (x + τ) − f (x)| < ε. Tf ,ε := {τ : τ is an ε − period}.

Def. A continuous function f is almost periodic if for every ε > 0, the set Tf ,ε

is relatively dense, i.e. it intersects every interval of size ν (for some ν > 0). This definition extends to F : Rn → R.

Lemma. If F : Rn → R is almost periodic and G (y) = F
y, y 2, . . . , y n

, then ∃ε > 0 s.t. the set Vε := {y : |G (y)| ≥ ε} intersects every interval of size ν (for some ν > 0), and such an intersection has measure ≥ δ (for some δ > 0). Recall: we have G (y) =

j∈J fjeipj (y), which is not almost periodic, and we

want to prove that ˆ

Vε 1 y dy = ∞.

SLIDE 90

Almost periodic functions

Example. f (x) = sin (2πx) + sin
2

√ 2πx

is not periodic. However,

∀ε > 0 ∃ ∞ many τ s.t. x ∈ R |f (x + τ) − f (x)| < ε. Given f , an ε-period is a number τ such that x ∈ R |f (x + τ) − f (x)| < ε. Tf ,ε := {τ : τ is an ε − period}.

Def. A continuous function f is almost periodic if for every ε > 0, the set Tf ,ε

is relatively dense, i.e. it intersects every interval of size ν (for some ν > 0). This definition extends to F : Rn → R.

Lemma. If F : Rn → R is almost periodic and G (y) = F
y, y 2, . . . , y n

, then ∃ε > 0 s.t. the set Vε := {y : |G (y)| ≥ ε} intersects every interval of size ν (for some ν > 0), and such an intersection has measure ≥ δ (for some δ > 0). Recall: we have G (y) =