| import Mathlib |
| import OAI.Geometry.SmoothYau.Smoothness.NormIteratedFDerivCompLinear |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set Filter MeasureTheory ProbabilityTheory |
| open scoped Topology ContDiff ENNReal |
| namespace YauCounterexamples |
| lemma finite_sum_tsupport_subset {E I : Type*} [TopologicalSpace E] |
| {V : Set E} (s : Finset I) (f : I → E → ℝ) |
| (hf : ∀ i ∈ s, tsupport (f i) ⊆ V) : |
| tsupport (fun x => ∑ i ∈ s, f i x) ⊆ V := by |
| classical |
| induction s using Finset.induction_on with |
| | empty => simp |
| | @insert a s ha ih => |
| simp only [Finset.sum_insert ha] |
| exact (tsupport_add _ _).trans (union_subset (hf a (Finset.mem_insert_self _ _)) |
| (ih (fun i hi => hf i (Finset.mem_insert_of_mem hi)))) |
|
|
| lemma finiteWaveSuperposition_tsupport {E I : Type*} [TopologicalSpace E] [Fintype I] |
| {O : Set E} {U : I → Fin 3 → E → ℂ} |
| (hU : ∀ i ℓ, tsupport (U i ℓ) ⊆ O) (γ : I → Fin 3 → ℂ) : |
| tsupport (finiteWaveSuperposition (fun _ => (0 : ℝ)) U γ) ⊆ O := by |
| classical |
| have he : finiteWaveSuperposition (fun _ => (0 : ℝ)) U γ = |
| fun x => ∑ i, ∑ ℓ, (γ i ℓ*U i ℓ x).re := by |
| funext x; exact zero_add _ |
| rw [he] |
| apply finite_sum_tsupport_subset |
| intro i _ |
| apply finite_sum_tsupport_subset |
| intro ℓ _ |
| exact (tsupport_comp_subset (g := Complex.re) rfl (fun x => γ i ℓ*U i ℓ x)).trans |
| (tsupport_mul_subset_right.trans (hU i ℓ)) |
| end YauCounterexamples |
| end |
|
|
| end OAI |
|
|
