Theorem: W21.integrable_fourier (PrimeNumberTheoremAnd/Wiener.lean)

State before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
⊢ Integrable (fun u => 𝓕 ψ.toFun (u / c)) volume
Tactic:
have l1 (C) : Integrable (fun u ↦ C / (1 + (u / c) ^ 2)) volume := by
  simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C
State after:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
l1 : ∀ (C : ℝ), Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
⊢ Integrable (fun u => 𝓕 ψ.toFun (u / c)) volume

State before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
l1 : ∀ (C : ℝ), Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
⊢ Integrable (fun u => 𝓕 ψ.toFun (u / c)) volume
Tactic:
have l2 : AEStronglyMeasurable (fun u ↦ 𝓕 ψ (u / c)) volume := by
  apply Continuous.aestronglyMeasurable ; continuity
State after:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
l1 : ∀ (C : ℝ), Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
l2 : AEStronglyMeasurable (fun u => 𝓕 ψ.toFun (u / c)) volume
⊢ Integrable (fun u => 𝓕 ψ.toFun (u / c)) volume

State before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
l1 : ∀ (C : ℝ), Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
l2 : AEStronglyMeasurable (fun u => 𝓕 ψ.toFun (u / c)) volume
⊢ Integrable (fun u => 𝓕 ψ.toFun (u / c)) volume
Tactic:
obtain ⟨C, h⟩ := decay_bounds_cor ψ
State after:
case intro
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
l1 : ∀ (C : ℝ), Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
l2 : AEStronglyMeasurable (fun u => 𝓕 ψ.toFun (u / c)) volume
C : ℝ
h : ∀ (u : ℝ), ‖𝓕 ψ.toFun u‖ ≤ C / (1 + u ^ 2)
⊢ Integrable (fun u => 𝓕 ψ.toFun (u / c)) volume

State before:
case intro
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
l1 : ∀ (C : ℝ), Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
l2 : AEStronglyMeasurable (fun u => 𝓕 ψ.toFun (u / c)) volume
C : ℝ
h : ∀ (u : ℝ), ‖𝓕 ψ.toFun u‖ ≤ C / (1 + u ^ 2)
⊢ Integrable (fun u => 𝓕 ψ.toFun (u / c)) volume
Tactic:
apply @Integrable.mono' ℝ ℂ _ volume _ _ (fun u => C / (1 + (u / c) ^ 2)) (l1 C) l2 ?_
State after:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
l1 : ∀ (C : ℝ), Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
l2 : AEStronglyMeasurable (fun u => 𝓕 ψ.toFun (u / c)) volume
C : ℝ
h : ∀ (u : ℝ), ‖𝓕 ψ.toFun u‖ ≤ C / (1 + u ^ 2)
⊢ ∀ᵐ (a : ℝ), ‖𝓕 ψ.toFun (a / c)‖ ≤ (fun u => C / (1 + (u / c) ^ 2)) a

State before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
l1 : ∀ (C : ℝ), Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
l2 : AEStronglyMeasurable (fun u => 𝓕 ψ.toFun (u / c)) volume
C : ℝ
h : ∀ (u : ℝ), ‖𝓕 ψ.toFun u‖ ≤ C / (1 + u ^ 2)
⊢ ∀ᵐ (a : ℝ), ‖𝓕 ψ.toFun (a / c)‖ ≤ (fun u => C / (1 + (u / c) ^ 2)) a
Tactic:
apply eventually_of_forall (fun x => h _)
State after:
no goals

State before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
C : ℝ
⊢ Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
Tactic:
simpa using (integrable_inv_one_add_sq.comp_div hc).const_mul C
State after:
no goals

State before:
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
l1 : ∀ (C : ℝ), Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
⊢ AEStronglyMeasurable (fun u => 𝓕 ψ.toFun (u / c)) volume
Tactic:
apply Continuous.aestronglyMeasurable
State after:
case hf
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
l1 : ∀ (C : ℝ), Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
⊢ Continuous fun u => 𝓕 ψ.toFun (u / c)

State before:
case hf
n : ℕ
A a b c d u x y t σ' : ℝ
ψ✝ Ψ : ℝ → ℂ
F G : ℂ → ℂ
f : ℕ → ℂ
𝕜 : Type
inst✝ : _root_.RCLike 𝕜
ψ : W21
hc : c ≠ 0
l1 : ∀ (C : ℝ), Integrable (fun u => C / (1 + (u / c) ^ 2)) volume
⊢ Continuous fun u => 𝓕 ψ.toFun (u / c)
Tactic:
continuity
State after:
no goals