feat: port Analysis.Complex.UpperHalfPlane.FunctionsBoundedAtInfty (#5531)

Author: Parcly-Taxel

Mathlib.lean

@@ -558,6 +558,7 @@ import Mathlib.Analysis.Complex.RemovableSingularity
 import Mathlib.Analysis.Complex.Schwarz
 import Mathlib.Analysis.Complex.UnitDisc.Basic
 import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
+import Mathlib.Analysis.Complex.UpperHalfPlane.FunctionsBoundedAtInfty
 import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
 import Mathlib.Analysis.ConstantSpeed
 import Mathlib.Analysis.Convex.Basic

Mathlib/Analysis/Complex/UpperHalfPlane/FunctionsBoundedAtInfty.lean

@@ -0,0 +1,110 @@
+/-
+Copyright (c) 2022 Chris Birkbeck. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Chris Birkbeck, David Loeffler
+
+! This file was ported from Lean 3 source module analysis.complex.upper_half_plane.functions_bounded_at_infty
+! leanprover-community/mathlib commit f2ce6086713c78a7f880485f7917ea547a215982
+! Please do not edit these lines, except to modify the commit id
+! if you have ported upstream changes.
+-/
+import Mathlib.Algebra.Module.Submodule.Basic
+import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
+import Mathlib.Order.Filter.ZeroAndBoundedAtFilter
+
+/-!
+# Bounded at infinity
+
+For complex valued functions on the upper half plane, this file defines the filter `atImInfty`
+required for defining when functions are bounded at infinity and zero at infinity.
+Both of which are relevant for defining modular forms.
+
+-/
+
+
+open Complex Filter
+
+open scoped Topology UpperHalfPlane
+
+noncomputable section
+
+namespace UpperHalfPlane
+
+/-- Filter for approaching `i∞`. -/
+def atImInfty :=
+  Filter.atTop.comap UpperHalfPlane.im
+#align upper_half_plane.at_im_infty UpperHalfPlane.atImInfty
+
+theorem atImInfty_basis : atImInfty.HasBasis (fun _ => True) fun i : ℝ => im ⁻¹' Set.Ici i :=
+  Filter.HasBasis.comap UpperHalfPlane.im Filter.atTop_basis
+#align upper_half_plane.at_im_infty_basis UpperHalfPlane.atImInfty_basis
+
+theorem atImInfty_mem (S : Set ℍ) : S ∈ atImInfty ↔ ∃ A : ℝ, ∀ z : ℍ, A ≤ im z → z ∈ S := by
+  simp only [atImInfty, Filter.mem_comap', Filter.mem_atTop_sets, ge_iff_le, Set.mem_setOf_eq,
+    UpperHalfPlane.coe_im]
+  refine' ⟨fun ⟨a, h⟩ => ⟨a, fun z hz => h (im z) hz rfl⟩, _⟩
+  rintro ⟨A, h⟩
+  refine' ⟨A, fun b hb x hx => h x _⟩
+  rwa [hx]
+#align upper_half_plane.at_im_infty_mem UpperHalfPlane.atImInfty_mem
+
+/-- A function ` f : ℍ → α` is bounded at infinity if it is bounded along `atImInfty`. -/
+def IsBoundedAtImInfty {α : Type _} [Norm α] (f : ℍ → α) : Prop :=
+  BoundedAtFilter atImInfty f
+#align upper_half_plane.is_bounded_at_im_infty UpperHalfPlane.IsBoundedAtImInfty
+
+/-- A function ` f : ℍ → α` is zero at infinity it is zero along `atImInfty`. -/
+def IsZeroAtImInfty {α : Type _} [Zero α] [TopologicalSpace α] (f : ℍ → α) : Prop :=
+  ZeroAtFilter atImInfty f
+#align upper_half_plane.is_zero_at_im_infty UpperHalfPlane.IsZeroAtImInfty
+
+theorem zero_form_isBoundedAtImInfty {α : Type _} [NormedField α] :
+    IsBoundedAtImInfty (0 : ℍ → α) :=
+  const_boundedAtFilter atImInfty (0 : α)
+#align upper_half_plane.zero_form_is_bounded_at_im_infty UpperHalfPlane.zero_form_isBoundedAtImInfty
+
+/-- Module of functions that are zero at infinity. -/
+def zeroAtImInftySubmodule (α : Type _) [NormedField α] : Submodule α (ℍ → α) :=
+  zeroAtFilterSubmodule atImInfty
+#align upper_half_plane.zero_at_im_infty_submodule UpperHalfPlane.zeroAtImInftySubmodule
+
+/-- Subalgebra of functions that are bounded at infinity. -/
+def boundedAtImInftySubalgebra (α : Type _) [NormedField α] : Subalgebra α (ℍ → α) :=
+  boundedFilterSubalgebra atImInfty
+#align upper_half_plane.bounded_at_im_infty_subalgebra UpperHalfPlane.boundedAtImInftySubalgebra
+
+nonrec theorem IsBoundedAtImInfty.mul {f g : ℍ → ℂ} (hf : IsBoundedAtImInfty f)
+    (hg : IsBoundedAtImInfty g) : IsBoundedAtImInfty (f * g) := by
+  simpa only [Pi.one_apply, mul_one, norm_eq_abs] using hf.mul hg
+#align upper_half_plane.is_bounded_at_im_infty.mul UpperHalfPlane.IsBoundedAtImInfty.mul
+
+theorem bounded_mem (f : ℍ → ℂ) :
+    IsBoundedAtImInfty f ↔ ∃ M A : ℝ, ∀ z : ℍ, A ≤ im z → abs (f z) ≤ M := by
+  simp [IsBoundedAtImInfty, BoundedAtFilter, Asymptotics.isBigO_iff, Filter.Eventually,
+    atImInfty_mem]
+#align upper_half_plane.bounded_mem UpperHalfPlane.bounded_mem
+
+theorem zero_at_im_infty (f : ℍ → ℂ) :
+    IsZeroAtImInfty f ↔ ∀ ε : ℝ, 0 < ε → ∃ A : ℝ, ∀ z : ℍ, A ≤ im z → abs (f z) ≤ ε := by
+  rw [IsZeroAtImInfty, ZeroAtFilter, tendsto_iff_forall_eventually_mem]
+  constructor
+  · simp_rw [Filter.Eventually, atImInfty_mem]
+    intro h ε hε
+    simpa using h (Metric.closedBall (0 : ℂ) ε) (Metric.closedBall_mem_nhds (0 : ℂ) hε)
+  · simp_rw [Metric.mem_nhds_iff]
+    intro h s hs
+    simp_rw [Filter.Eventually, atImInfty_mem]
+    obtain ⟨ε, h1, h2⟩ := hs
+    have h11 : 0 < ε / 2 := by linarith
+    obtain ⟨A, hA⟩ := h (ε / 2) h11
+    use A
+    intro z hz
+    have hzs : f z ∈ s := by
+      apply h2
+      simp only [mem_ball_zero_iff, norm_eq_abs]
+      apply lt_of_le_of_lt (hA z hz)
+      linarith
+    apply hzs
+#align upper_half_plane.zero_at_im_infty UpperHalfPlane.zero_at_im_infty
+
+end UpperHalfPlane