analysis.fourier.poisson_summation ⟷ Mathlib.Analysis.Fourier.PoissonSummation

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -6,7 +6,7 @@ Authors: David Loeffler
 import Analysis.Fourier.AddCircle
 import Analysis.Fourier.FourierTransform
 import Analysis.PSeries
-import Analysis.SchwartzSpace
+import Analysis.Distribution.SchwartzSpace
 import MeasureTheory.Measure.Lebesgue.Integral
 
 #align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"e8e130de9dba4ed6897183c3193c752ffadbcc77"

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -163,7 +163,7 @@ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
     ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) :=
     by
     intro x hx y hy
-    rw [max_lt_iff] at hx 
+    rw [max_lt_iff] at hx
     have hxR : 0 < x + R := by
       rcases le_or_lt 0 R with (h | h)
       · exact add_pos_of_pos_of_nonneg hx.1 h
@@ -186,9 +186,9 @@ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   simp only [is_O, is_O_with, eventually_at_top] at hc' ⊢
   obtain ⟨d, hd⟩ := hc'
   refine' ⟨c * (1 / 2) ^ (-b), ⟨max (1 + max 0 (-2 * R)) (d - R), fun x hx => _⟩⟩
-  rw [ge_iff_le, max_le_iff] at hx 
+  rw [ge_iff_le, max_le_iff] at hx
   have hx' : max 0 (-2 * R) < x := by linarith
-  rw [max_lt_iff] at hx' 
+  rw [max_lt_iff] at hx'
   rw [norm_norm,
     ContinuousMap.norm_le _
       (mul_nonneg (mul_nonneg hc.le <| rpow_nonneg_of_nonneg one_half_pos.le _) (norm_nonneg _))]
@@ -213,7 +213,7 @@ theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   have h2 := (isBigO_norm_Icc_restrict_atTop hb h1 (-S) (-R)).comp_tendsto tendsto_neg_at_bot_at_top
   have : (fun x : ℝ => |x| ^ (-b)) ∘ Neg.neg = fun x : ℝ => |x| ^ (-b) := by ext1 x;
     simp only [Function.comp_apply, abs_neg]
-  rw [this] at h2 
+  rw [this] at h2
   refine' (is_O_of_le _ fun x => _).trans h2
   -- equality holds, but less work to prove `≤` alone
   rw [norm_norm, Function.comp_apply, norm_norm, ContinuousMap.norm_le _ (norm_nonneg _)]
@@ -232,7 +232,7 @@ theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
       |x| ^ (-b) :=
   by
   obtain ⟨r, hr⟩ := K.is_compact.bounded.subset_ball 0
-  rw [closed_ball_eq_Icc, zero_add, zero_sub] at hr 
+  rw [closed_ball_eq_Icc, zero_add, zero_sub] at hr
   have :
     ∀ x : ℝ,
       ‖(f.comp (ContinuousMap.addRight x)).restrict K‖ ≤ ‖f.restrict (Icc (x - r) (x + r))‖ :=

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,11 +3,11 @@ Copyright (c) 2023 David Loeffler. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 -/
-import Mathbin.Analysis.Fourier.AddCircle
-import Mathbin.Analysis.Fourier.FourierTransform
-import Mathbin.Analysis.PSeries
-import Mathbin.Analysis.SchwartzSpace
-import Mathbin.MeasureTheory.Measure.Lebesgue.Integral
+import Analysis.Fourier.AddCircle
+import Analysis.Fourier.FourierTransform
+import Analysis.PSeries
+import Analysis.SchwartzSpace
+import MeasureTheory.Measure.Lebesgue.Integral
 
 #align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"e8e130de9dba4ed6897183c3193c752ffadbcc77"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,11 +2,6 @@
 Copyright (c) 2023 David Loeffler. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
-
-! This file was ported from Lean 3 source module analysis.fourier.poisson_summation
-! leanprover-community/mathlib commit e8e130de9dba4ed6897183c3193c752ffadbcc77
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Fourier.AddCircle
 import Mathbin.Analysis.Fourier.FourierTransform
@@ -14,6 +9,8 @@ import Mathbin.Analysis.PSeries
 import Mathbin.Analysis.SchwartzSpace
 import Mathbin.MeasureTheory.Measure.Lebesgue.Integral
 
+#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"e8e130de9dba4ed6897183c3193c752ffadbcc77"
+
 /-!
 # Poisson's summation formula
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/6285167a053ad0990fc88e56c48ccd9fae6550eb

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 
 ! This file was ported from Lean 3 source module analysis.fourier.poisson_summation
-! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
+! leanprover-community/mathlib commit e8e130de9dba4ed6897183c3193c752ffadbcc77
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -17,6 +17,9 @@ import Mathbin.MeasureTheory.Measure.Lebesgue.Integral
 /-!
 # Poisson's summation formula
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 We prove Poisson's summation formula `∑ (n : ℤ), f n = ∑ (n : ℤ), 𝓕 f n`, where `𝓕 f` is the
 Fourier transform of `f`, under the following hypotheses:
 * `f` is a continuous function `ℝ → ℂ`.

mathlib commit https://github.com/leanprover-community/mathlib/commit/f2ad3645af9effcdb587637dc28a6074edc813f9

@@ -56,6 +56,7 @@ attribute [local instance] Real.fact_zero_lt_one
 
 open ContinuousMap
 
+#print Real.fourierCoeff_tsum_comp_add /-
 /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function
 `∑' n : ℤ, f (x + n)` is the value at `m` of the Fourier transform of `f`. -/
 theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
@@ -120,7 +121,9 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
       push_cast
       ring
 #align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
+-/
 
+#print Real.tsum_eq_tsum_fourierIntegral /-
 /-- **Poisson's summation formula**, most general form. -/
 theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
     (h_norm :
@@ -140,11 +143,13 @@ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
     rw [← Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
     rfl
 #align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
+-/
 
 section RpowDecay
 
 variable {E : Type _} [NormedAddCommGroup E]
 
+#print isBigO_norm_Icc_restrict_atTop /-
 /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
 theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
@@ -194,7 +199,9 @@ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   · apply abs_of_nonneg; linarith [y.2.1]
   · exact abs_of_pos hx'.1
 #align is_O_norm_Icc_restrict_at_top isBigO_norm_Icc_restrict_atTop
+-/
 
+#print isBigO_norm_Icc_restrict_atBot /-
 theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
     (hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
     IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) :=
@@ -216,7 +223,9 @@ theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   · exact ⟨by linarith [hx.2], by linarith [hx.1]⟩
   · rw [ContinuousMap.restrict_apply_mk, ContinuousMap.comp_apply, ContinuousMap.coe_mk, neg_neg]
 #align is_O_norm_Icc_restrict_at_bot isBigO_norm_Icc_restrict_atBot
+-/
 
+#print isBigO_norm_restrict_cocompact /-
 theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
     (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
     IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
@@ -242,7 +251,9 @@ theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
   · refine' (is_O_of_le at_top _).trans (isBigO_norm_Icc_restrict_atTop hb hf.2 (-r) r)
     simp_rw [norm_norm]; exact this
 #align is_O_norm_restrict_cocompact isBigO_norm_restrict_cocompact
+-/
 
+#print Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable /-
 /-- **Poisson's summation formula**, assuming that `f` decays as
 `|x| ^ (-b)` for some `1 < b` and its Fourier transform is summable. -/
 theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ → ℂ} (hc : Continuous f)
@@ -256,7 +267,9 @@ theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ →
           Int.tendsto_coe_cofinite))
     hFf
 #align real.tsum_eq_tsum_fourier_integral_of_rpow_decay_of_summable Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable
+-/
 
+#print Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay /-
 /-- **Poisson's summation formula**, assuming that both `f` and its Fourier transform decay as
 `|x| ^ (-b)` for some `1 < b`. (This is the one-dimensional case of Corollary VII.2.6 of Stein and
 Weiss, *Introduction to Fourier analysis on Euclidean spaces*.) -/
@@ -266,11 +279,13 @@ theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay {f : ℝ → ℂ} (hc :
   Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable hc hb hf
     (summable_of_isBigO (Real.summable_abs_int_rpow hb) (hFf.comp_tendsto Int.tendsto_coe_cofinite))
 #align real.tsum_eq_tsum_fourier_integral_of_rpow_decay Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay
+-/
 
 end RpowDecay
 
 section Schwartz
 
+#print SchwartzMap.tsum_eq_tsum_fourierIntegral /-
 /-- **Poisson's summation formula** for Schwartz functions. -/
 theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
     ∑' n : ℤ, f n = ∑' n : ℤ, g n :=
@@ -282,6 +297,7 @@ theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hf
     Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay f.continuous one_lt_two
       (f.is_O_cocompact_rpow (-2)) (by simpa only [hfg] using g.is_O_cocompact_rpow (-2))
 #align schwartz_map.tsum_eq_tsum_fourier_integral SchwartzMap.tsum_eq_tsum_fourierIntegral
+-/
 
 end Schwartz
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3

@@ -125,7 +125,7 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
 theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
     (h_norm :
       ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖)
-    (h_sum : Summable fun n : ℤ => 𝓕 f n) : (∑' n : ℤ, f n) = ∑' n : ℤ, 𝓕 f n :=
+    (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = ∑' n : ℤ, 𝓕 f n :=
   by
   let F : C(UnitAddCircle, ℂ) :=
     ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩
@@ -247,7 +247,7 @@ theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
 `|x| ^ (-b)` for some `1 < b` and its Fourier transform is summable. -/
 theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ → ℂ} (hc : Continuous f)
     {b : ℝ} (hb : 1 < b) (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
-    (hFf : Summable fun n : ℤ => 𝓕 f n) : (∑' n : ℤ, f n) = ∑' n : ℤ, 𝓕 f n :=
+    (hFf : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = ∑' n : ℤ, 𝓕 f n :=
   Real.tsum_eq_tsum_fourierIntegral
     (fun K =>
       summable_of_isBigO (Real.summable_abs_int_rpow hb)
@@ -262,8 +262,7 @@ theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ →
 Weiss, *Introduction to Fourier analysis on Euclidean spaces*.) -/
 theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay {f : ℝ → ℂ} (hc : Continuous f) {b : ℝ}
     (hb : 1 < b) (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
-    (hFf : IsBigO (cocompact ℝ) (𝓕 f) fun x : ℝ => |x| ^ (-b)) :
-    (∑' n : ℤ, f n) = ∑' n : ℤ, 𝓕 f n :=
+    (hFf : IsBigO (cocompact ℝ) (𝓕 f) fun x : ℝ => |x| ^ (-b)) : ∑' n : ℤ, f n = ∑' n : ℤ, 𝓕 f n :=
   Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable hc hb hf
     (summable_of_isBigO (Real.summable_abs_int_rpow hb) (hFf.comp_tendsto Int.tendsto_coe_cofinite))
 #align real.tsum_eq_tsum_fourier_integral_of_rpow_decay Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay
@@ -274,7 +273,7 @@ section Schwartz
 
 /-- **Poisson's summation formula** for Schwartz functions. -/
 theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
-    (∑' n : ℤ, f n) = ∑' n : ℤ, g n :=
+    ∑' n : ℤ, f n = ∑' n : ℤ, g n :=
   by
   -- We know that Schwartz functions are `O(‖x ^ (-b)‖)` for *every* `b`; for this argument we take
   -- `b = 2` and work with that.

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -119,7 +119,6 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
       congr 2
       push_cast
       ring
-    
 #align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
 
 /-- **Poisson's summation formula**, most general form. -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -133,7 +133,7 @@ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
   have : Summable (fourierCoeff F) := by
     convert h_sum
     exact funext fun n => Real.fourierCoeff_tsum_comp_add h_norm n
-  convert(has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
+  convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
   · have := (has_sum_apply (summable_of_locally_summable_norm h_norm).HasSum 0).tsum_eq
     simpa only [coe_mk, ← QuotientAddGroup.mk_zero, periodic.lift_coe, zsmul_one, comp_apply,
       coe_add_right, zero_add] using this

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -97,7 +97,7 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
       exact funext fun n => neK _ _
     _ = ∑' n : ℤ, ∫ x in 0 ..1, (e * f).comp (ContinuousMap.addRight n) x :=
       by
-      simp only [ContinuousMap.comp_apply, mul_comp] at eadd⊢
+      simp only [ContinuousMap.comp_apply, mul_comp] at eadd ⊢
       simp_rw [eadd]
     -- Rearrange sum of interval integrals into an integral over `ℝ`.
         _ =
@@ -106,7 +106,7 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
       suffices : integrable ⇑(e * f); exact this.has_sum_interval_integral_comp_add_int.tsum_eq
       apply integrable_of_summable_norm_Icc
       convert hf ⟨Icc 0 1, is_compact_Icc⟩
-      simp_rw [ContinuousMap.comp_apply, mul_comp] at eadd⊢
+      simp_rw [ContinuousMap.comp_apply, mul_comp] at eadd ⊢
       simp_rw [eadd]
       exact funext fun n => neK ⟨Icc 0 1, is_compact_Icc⟩ _
     -- Minor tidying to finish
@@ -159,7 +159,7 @@ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
     ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) :=
     by
     intro x hx y hy
-    rw [max_lt_iff] at hx
+    rw [max_lt_iff] at hx 
     have hxR : 0 < x + R := by
       rcases le_or_lt 0 R with (h | h)
       · exact add_pos_of_pos_of_nonneg hx.1 h
@@ -179,12 +179,12 @@ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
     exact rpow_le_rpow (mul_pos one_half_pos hx.1).le (by linarith) hb.le
   -- Now the main proof.
   obtain ⟨c, hc, hc'⟩ := hf.exists_pos
-  simp only [is_O, is_O_with, eventually_at_top] at hc'⊢
+  simp only [is_O, is_O_with, eventually_at_top] at hc' ⊢
   obtain ⟨d, hd⟩ := hc'
   refine' ⟨c * (1 / 2) ^ (-b), ⟨max (1 + max 0 (-2 * R)) (d - R), fun x hx => _⟩⟩
-  rw [ge_iff_le, max_le_iff] at hx
+  rw [ge_iff_le, max_le_iff] at hx 
   have hx' : max 0 (-2 * R) < x := by linarith
-  rw [max_lt_iff] at hx'
+  rw [max_lt_iff] at hx' 
   rw [norm_norm,
     ContinuousMap.norm_le _
       (mul_nonneg (mul_nonneg hc.le <| rpow_nonneg_of_nonneg one_half_pos.le _) (norm_nonneg _))]
@@ -207,7 +207,7 @@ theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   have h2 := (isBigO_norm_Icc_restrict_atTop hb h1 (-S) (-R)).comp_tendsto tendsto_neg_at_bot_at_top
   have : (fun x : ℝ => |x| ^ (-b)) ∘ Neg.neg = fun x : ℝ => |x| ^ (-b) := by ext1 x;
     simp only [Function.comp_apply, abs_neg]
-  rw [this] at h2
+  rw [this] at h2 
   refine' (is_O_of_le _ fun x => _).trans h2
   -- equality holds, but less work to prove `≤` alone
   rw [norm_norm, Function.comp_apply, norm_norm, ContinuousMap.norm_le _ (norm_nonneg _)]
@@ -224,7 +224,7 @@ theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
       |x| ^ (-b) :=
   by
   obtain ⟨r, hr⟩ := K.is_compact.bounded.subset_ball 0
-  rw [closed_ball_eq_Icc, zero_add, zero_sub] at hr
+  rw [closed_ball_eq_Icc, zero_add, zero_sub] at hr 
   have :
     ∀ x : ℝ,
       ‖(f.comp (ContinuousMap.addRight x)).restrict K‖ ≤ ‖f.restrict (Icc (x - r) (x + r))‖ :=
@@ -236,7 +236,7 @@ theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
     exact ⟨by linarith [(hr hy).1], by linarith [(hr hy).2]⟩
     simp_rw [ContinuousMap.restrict_apply, ContinuousMap.comp_apply, ContinuousMap.coe_addRight,
       Subtype.coe_mk]
-  simp_rw [cocompact_eq, is_O_sup] at hf⊢
+  simp_rw [cocompact_eq, is_O_sup] at hf ⊢
   constructor
   · refine' (is_O_of_le at_bot _).trans (isBigO_norm_Icc_restrict_atBot hb hf.1 (-r) r)
     simp_rw [norm_norm]; exact this

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -50,7 +50,7 @@ open Set hiding restrict_apply
 
 open TopologicalSpace Filter MeasureTheory Asymptotics
 
-open Real BigOperators Filter FourierTransform
+open scoped Real BigOperators Filter FourierTransform
 
 attribute [local instance] Real.fact_zero_lt_one
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -72,8 +72,7 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
     simp_rw [norm_eq_supr_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
   have eadd : ∀ n : ℤ, e.comp (ContinuousMap.addRight n) = e :=
     by
-    intro n
-    ext1 x
+    intro n; ext1 x
     have : periodic e 1 := periodic.comp (fun x => AddCircle.coe_add_period 1 x) _
     simpa only [mul_one] using this.int_mul n x
   -- Now the main argument. First unwind some definitions.
@@ -193,8 +192,7 @@ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   have A : ∀ x : ℝ, 0 ≤ |x| ^ (-b) := fun x => by positivity
   rwa [mul_assoc, mul_le_mul_left hc, norm_of_nonneg (A _), norm_of_nonneg (A _)]
   convert claim x (by linarith only [hx.1]) y.1 y.2.1
-  · apply abs_of_nonneg
-    linarith [y.2.1]
+  · apply abs_of_nonneg; linarith [y.2.1]
   · exact abs_of_pos hx'.1
 #align is_O_norm_Icc_restrict_at_top isBigO_norm_Icc_restrict_atTop
 
@@ -205,12 +203,9 @@ theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   have h1 : is_O at_top (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ^ (-b) :=
     by
     convert hf.comp_tendsto tendsto_neg_at_top_at_bot
-    ext1 x
-    simp only [Function.comp_apply, abs_neg]
+    ext1 x; simp only [Function.comp_apply, abs_neg]
   have h2 := (isBigO_norm_Icc_restrict_atTop hb h1 (-S) (-R)).comp_tendsto tendsto_neg_at_bot_at_top
-  have : (fun x : ℝ => |x| ^ (-b)) ∘ Neg.neg = fun x : ℝ => |x| ^ (-b) :=
-    by
-    ext1 x
+  have : (fun x : ℝ => |x| ^ (-b)) ∘ Neg.neg = fun x : ℝ => |x| ^ (-b) := by ext1 x;
     simp only [Function.comp_apply, abs_neg]
   rw [this] at h2
   refine' (is_O_of_le _ fun x => _).trans h2
@@ -244,11 +239,9 @@ theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
   simp_rw [cocompact_eq, is_O_sup] at hf⊢
   constructor
   · refine' (is_O_of_le at_bot _).trans (isBigO_norm_Icc_restrict_atBot hb hf.1 (-r) r)
-    simp_rw [norm_norm]
-    exact this
+    simp_rw [norm_norm]; exact this
   · refine' (is_O_of_le at_top _).trans (isBigO_norm_Icc_restrict_atTop hb hf.2 (-r) r)
-    simp_rw [norm_norm]
-    exact this
+    simp_rw [norm_norm]; exact this
 #align is_O_norm_restrict_cocompact isBigO_norm_restrict_cocompact
 
 /-- **Poisson's summation formula**, assuming that `f` decays as

mathlib commit https://github.com/leanprover-community/mathlib/commit/f51de8769c34652d82d1c8e5f8f18f8374782bed

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 
 ! This file was ported from Lean 3 source module analysis.fourier.poisson_summation
-! leanprover-community/mathlib commit 4fa54b337f7d52805480306db1b1439c741848c8
+! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -12,6 +12,7 @@ import Mathbin.Analysis.Fourier.AddCircle
 import Mathbin.Analysis.Fourier.FourierTransform
 import Mathbin.Analysis.PSeries
 import Mathbin.Analysis.SchwartzSpace
+import Mathbin.MeasureTheory.Measure.Lebesgue.Integral
 
 /-!
 # Poisson's summation formula

mathlib commit https://github.com/leanprover-community/mathlib/commit/f8c79b0a623404854a2902b836eac32156fd7712

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 
 ! This file was ported from Lean 3 source module analysis.fourier.poisson_summation
-! leanprover-community/mathlib commit 0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
+! leanprover-community/mathlib commit 4fa54b337f7d52805480306db1b1439c741848c8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -12,7 +12,6 @@ import Mathbin.Analysis.Fourier.AddCircle
 import Mathbin.Analysis.Fourier.FourierTransform
 import Mathbin.Analysis.PSeries
 import Mathbin.Analysis.SchwartzSpace
-import Mathbin.Analysis.SpecialFunctions.Pow.Tactic
 
 /-!
 # Poisson's summation formula

mathlib commit https://github.com/leanprover-community/mathlib/commit/0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 
 ! This file was ported from Lean 3 source module analysis.fourier.poisson_summation
-! leanprover-community/mathlib commit 38f16f960f5006c6c0c2bac7b0aba5273188f4e5
+! leanprover-community/mathlib commit 0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -12,6 +12,7 @@ import Mathbin.Analysis.Fourier.AddCircle
 import Mathbin.Analysis.Fourier.FourierTransform
 import Mathbin.Analysis.PSeries
 import Mathbin.Analysis.SchwartzSpace
+import Mathbin.Analysis.SpecialFunctions.Pow.Tactic
 
 /-!
 # Poisson's summation formula

mathlib commit https://github.com/leanprover-community/mathlib/commit/039ef89bef6e58b32b62898dd48e9d1a4312bb65

@@ -148,9 +148,9 @@ variable {E : Type _} [NormedAddCommGroup E]
 
 /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
-theorem isO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
-    (hf : IsO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
-    IsO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) :=
+theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
+    (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
+    IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) :=
   by
   -- First establish an explicit estimate on decay of inverse powers.
   -- This is logically independent of the rest of the proof, but of no mathematical interest in
@@ -195,18 +195,18 @@ theorem isO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   · apply abs_of_nonneg
     linarith [y.2.1]
   · exact abs_of_pos hx'.1
-#align is_O_norm_Icc_restrict_at_top isO_norm_Icc_restrict_atTop
+#align is_O_norm_Icc_restrict_at_top isBigO_norm_Icc_restrict_atTop
 
-theorem isO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
-    (hf : IsO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
-    IsO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) :=
+theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
+    (hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
+    IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) :=
   by
   have h1 : is_O at_top (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ^ (-b) :=
     by
     convert hf.comp_tendsto tendsto_neg_at_top_at_bot
     ext1 x
     simp only [Function.comp_apply, abs_neg]
-  have h2 := (isO_norm_Icc_restrict_atTop hb h1 (-S) (-R)).comp_tendsto tendsto_neg_at_bot_at_top
+  have h2 := (isBigO_norm_Icc_restrict_atTop hb h1 (-S) (-R)).comp_tendsto tendsto_neg_at_bot_at_top
   have : (fun x : ℝ => |x| ^ (-b)) ∘ Neg.neg = fun x : ℝ => |x| ^ (-b) :=
     by
     ext1 x
@@ -220,11 +220,11 @@ theorem isO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   refine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ ⟨-x, _⟩)
   · exact ⟨by linarith [hx.2], by linarith [hx.1]⟩
   · rw [ContinuousMap.restrict_apply_mk, ContinuousMap.comp_apply, ContinuousMap.coe_mk, neg_neg]
-#align is_O_norm_Icc_restrict_at_bot isO_norm_Icc_restrict_atBot
+#align is_O_norm_Icc_restrict_at_bot isBigO_norm_Icc_restrict_atBot
 
-theorem isO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
-    (hf : IsO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
-    IsO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
+theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
+    (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
+    IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
       |x| ^ (-b) :=
   by
   obtain ⟨r, hr⟩ := K.is_compact.bounded.subset_ball 0
@@ -242,23 +242,23 @@ theorem isO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
       Subtype.coe_mk]
   simp_rw [cocompact_eq, is_O_sup] at hf⊢
   constructor
-  · refine' (is_O_of_le at_bot _).trans (isO_norm_Icc_restrict_atBot hb hf.1 (-r) r)
+  · refine' (is_O_of_le at_bot _).trans (isBigO_norm_Icc_restrict_atBot hb hf.1 (-r) r)
     simp_rw [norm_norm]
     exact this
-  · refine' (is_O_of_le at_top _).trans (isO_norm_Icc_restrict_atTop hb hf.2 (-r) r)
+  · refine' (is_O_of_le at_top _).trans (isBigO_norm_Icc_restrict_atTop hb hf.2 (-r) r)
     simp_rw [norm_norm]
     exact this
-#align is_O_norm_restrict_cocompact isO_norm_restrict_cocompact
+#align is_O_norm_restrict_cocompact isBigO_norm_restrict_cocompact
 
 /-- **Poisson's summation formula**, assuming that `f` decays as
 `|x| ^ (-b)` for some `1 < b` and its Fourier transform is summable. -/
 theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ → ℂ} (hc : Continuous f)
-    {b : ℝ} (hb : 1 < b) (hf : IsO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
+    {b : ℝ} (hb : 1 < b) (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
     (hFf : Summable fun n : ℤ => 𝓕 f n) : (∑' n : ℤ, f n) = ∑' n : ℤ, 𝓕 f n :=
   Real.tsum_eq_tsum_fourierIntegral
     (fun K =>
-      summable_of_isO (Real.summable_abs_int_rpow hb)
-        ((isO_norm_restrict_cocompact (ContinuousMap.mk _ hc) (zero_lt_one.trans hb) hf
+      summable_of_isBigO (Real.summable_abs_int_rpow hb)
+        ((isBigO_norm_restrict_cocompact (ContinuousMap.mk _ hc) (zero_lt_one.trans hb) hf
               K).comp_tendsto
           Int.tendsto_coe_cofinite))
     hFf
@@ -268,10 +268,11 @@ theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ →
 `|x| ^ (-b)` for some `1 < b`. (This is the one-dimensional case of Corollary VII.2.6 of Stein and
 Weiss, *Introduction to Fourier analysis on Euclidean spaces*.) -/
 theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay {f : ℝ → ℂ} (hc : Continuous f) {b : ℝ}
-    (hb : 1 < b) (hf : IsO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
-    (hFf : IsO (cocompact ℝ) (𝓕 f) fun x : ℝ => |x| ^ (-b)) : (∑' n : ℤ, f n) = ∑' n : ℤ, 𝓕 f n :=
+    (hb : 1 < b) (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
+    (hFf : IsBigO (cocompact ℝ) (𝓕 f) fun x : ℝ => |x| ^ (-b)) :
+    (∑' n : ℤ, f n) = ∑' n : ℤ, 𝓕 f n :=
   Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable hc hb hf
-    (summable_of_isO (Real.summable_abs_int_rpow hb) (hFf.comp_tendsto Int.tendsto_coe_cofinite))
+    (summable_of_isBigO (Real.summable_abs_int_rpow hb) (hFf.comp_tendsto Int.tendsto_coe_cofinite))
 #align real.tsum_eq_tsum_fourier_integral_of_rpow_decay Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay
 
 end RpowDecay

mathlib commit https://github.com/leanprover-community/mathlib/commit/290a7ba01fbcab1b64757bdaa270d28f4dcede35

@@ -238,7 +238,7 @@ theorem isO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
     rintro ⟨y, hy⟩
     refine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ ⟨y + x, _⟩)
     exact ⟨by linarith [(hr hy).1], by linarith [(hr hy).2]⟩
-    simp_rw [ContinuousMap.restrict_apply, ContinuousMap.comp_apply, ContinuousMap.coe_add_right,
+    simp_rw [ContinuousMap.restrict_apply, ContinuousMap.comp_apply, ContinuousMap.coe_addRight,
       Subtype.coe_mk]
   simp_rw [cocompact_eq, is_O_sup] at hf⊢
   constructor

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce7e9d53d4bbc38065db3b595cd5bd73c323bc1d

@@ -133,7 +133,7 @@ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
   have : Summable (fourierCoeff F) := by
     convert h_sum
     exact funext fun n => Real.fourierCoeff_tsum_comp_add h_norm n
-  convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
+  convert(has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
   · have := (has_sum_apply (summable_of_locally_summable_norm h_norm).HasSum 0).tsum_eq
     simpa only [coe_mk, ← QuotientAddGroup.mk_zero, periodic.lift_coe, zsmul_one, comp_apply,
       coe_add_right, zero_add] using this

mathlib commit https://github.com/leanprover-community/mathlib/commit/38f16f960f5006c6c0c2bac7b0aba5273188f4e5

@@ -4,12 +4,14 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
 
 ! This file was ported from Lean 3 source module analysis.fourier.poisson_summation
-! leanprover-community/mathlib commit 3353f3371120058977ce1e20bf7fc8986c0fb042
+! leanprover-community/mathlib commit 38f16f960f5006c6c0c2bac7b0aba5273188f4e5
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
 import Mathbin.Analysis.Fourier.AddCircle
 import Mathbin.Analysis.Fourier.FourierTransform
+import Mathbin.Analysis.PSeries
+import Mathbin.Analysis.SchwartzSpace
 
 /-!
 # Poisson's summation formula
@@ -19,10 +21,19 @@ Fourier transform of `f`, under the following hypotheses:
 * `f` is a continuous function `ℝ → ℂ`.
 * The sum `∑ (n : ℤ), 𝓕 f n` is convergent.
 * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ K }` is convergent.
+See `real.tsum_eq_tsum_fourier_integral` for this formulation.
+
+These hypotheses are potentially a little awkward to apply, so we also provide the less general but
+easier-to-use result `real.tsum_eq_tsum_fourier_integral_of_rpow_decay`, in which we assume `f` and
+`𝓕 f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
+`schwartz_map.tsum_eq_tsum_fourier_integral`, where we assume that both `f` and `𝓕 f` are Schwartz
+functions.
 
 ## TODO
 
-* Show that the conditions on `f` are automatically satisfied for Schwartz functions.
+At the moment `schwartz_map.tsum_eq_tsum_fourier_integral` requires separate proofs that both `f`
+and `𝓕 f` are Schwartz functions. In fact, `𝓕 f` is automatically Schwartz if `f` is; and once
+we have this lemma in the library, we should adjust the hypotheses here accordingly.
 -/
 
 
@@ -30,11 +41,13 @@ noncomputable section
 
 open Function hiding comp_apply
 
-open Complex Real
+open Complex hiding abs_of_nonneg
+
+open Real
 
 open Set hiding restrict_apply
 
-open TopologicalSpace Filter MeasureTheory
+open TopologicalSpace Filter MeasureTheory Asymptotics
 
 open Real BigOperators Filter FourierTransform
 
@@ -109,7 +122,7 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
     
 #align real.fourier_coeff_tsum_comp_add Real.fourierCoeff_tsum_comp_add
 
-/-- **Poisson's summation formula**. -/
+/-- **Poisson's summation formula**, most general form. -/
 theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
     (h_norm :
       ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖)
@@ -129,3 +142,153 @@ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
     rfl
 #align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
 
+section RpowDecay
+
+variable {E : Type _} [NormedAddCommGroup E]
+
+/-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
+`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
+theorem isO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
+    (hf : IsO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
+    IsO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) :=
+  by
+  -- First establish an explicit estimate on decay of inverse powers.
+  -- This is logically independent of the rest of the proof, but of no mathematical interest in
+  -- itself, so it is proved using `async` rather than being formulated as a separate lemma.
+  have claim :
+    ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) :=
+    by
+    intro x hx y hy
+    rw [max_lt_iff] at hx
+    have hxR : 0 < x + R := by
+      rcases le_or_lt 0 R with (h | h)
+      · exact add_pos_of_pos_of_nonneg hx.1 h
+      · rw [← sub_lt_iff_lt_add, zero_sub]
+        refine' lt_trans _ hx.2
+        rwa [neg_mul, neg_lt_neg_iff, two_mul, add_lt_iff_neg_left]
+    have hy' : 0 < y := hxR.trans_le hy
+    have : y ^ (-b) ≤ (x + R) ^ (-b) :=
+      by
+      rw [rpow_neg hy'.le, rpow_neg hxR.le,
+        inv_le_inv (rpow_pos_of_pos hy' _) (rpow_pos_of_pos hxR _)]
+      exact rpow_le_rpow hxR.le hy hb.le
+    refine' this.trans _
+    rw [← mul_rpow one_half_pos.le hx.1.le, rpow_neg (mul_pos one_half_pos hx.1).le,
+      rpow_neg hxR.le]
+    refine' inv_le_inv_of_le (rpow_pos_of_pos (mul_pos one_half_pos hx.1) _) _
+    exact rpow_le_rpow (mul_pos one_half_pos hx.1).le (by linarith) hb.le
+  -- Now the main proof.
+  obtain ⟨c, hc, hc'⟩ := hf.exists_pos
+  simp only [is_O, is_O_with, eventually_at_top] at hc'⊢
+  obtain ⟨d, hd⟩ := hc'
+  refine' ⟨c * (1 / 2) ^ (-b), ⟨max (1 + max 0 (-2 * R)) (d - R), fun x hx => _⟩⟩
+  rw [ge_iff_le, max_le_iff] at hx
+  have hx' : max 0 (-2 * R) < x := by linarith
+  rw [max_lt_iff] at hx'
+  rw [norm_norm,
+    ContinuousMap.norm_le _
+      (mul_nonneg (mul_nonneg hc.le <| rpow_nonneg_of_nonneg one_half_pos.le _) (norm_nonneg _))]
+  refine' fun y => (hd y.1 (by linarith [hx.1, y.2.1])).trans _
+  have A : ∀ x : ℝ, 0 ≤ |x| ^ (-b) := fun x => by positivity
+  rwa [mul_assoc, mul_le_mul_left hc, norm_of_nonneg (A _), norm_of_nonneg (A _)]
+  convert claim x (by linarith only [hx.1]) y.1 y.2.1
+  · apply abs_of_nonneg
+    linarith [y.2.1]
+  · exact abs_of_pos hx'.1
+#align is_O_norm_Icc_restrict_at_top isO_norm_Icc_restrict_atTop
+
+theorem isO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
+    (hf : IsO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
+    IsO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) :=
+  by
+  have h1 : is_O at_top (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ^ (-b) :=
+    by
+    convert hf.comp_tendsto tendsto_neg_at_top_at_bot
+    ext1 x
+    simp only [Function.comp_apply, abs_neg]
+  have h2 := (isO_norm_Icc_restrict_atTop hb h1 (-S) (-R)).comp_tendsto tendsto_neg_at_bot_at_top
+  have : (fun x : ℝ => |x| ^ (-b)) ∘ Neg.neg = fun x : ℝ => |x| ^ (-b) :=
+    by
+    ext1 x
+    simp only [Function.comp_apply, abs_neg]
+  rw [this] at h2
+  refine' (is_O_of_le _ fun x => _).trans h2
+  -- equality holds, but less work to prove `≤` alone
+  rw [norm_norm, Function.comp_apply, norm_norm, ContinuousMap.norm_le _ (norm_nonneg _)]
+  rintro ⟨x, hx⟩
+  rw [ContinuousMap.restrict_apply_mk]
+  refine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ ⟨-x, _⟩)
+  · exact ⟨by linarith [hx.2], by linarith [hx.1]⟩
+  · rw [ContinuousMap.restrict_apply_mk, ContinuousMap.comp_apply, ContinuousMap.coe_mk, neg_neg]
+#align is_O_norm_Icc_restrict_at_bot isO_norm_Icc_restrict_atBot
+
+theorem isO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
+    (hf : IsO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
+    IsO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
+      |x| ^ (-b) :=
+  by
+  obtain ⟨r, hr⟩ := K.is_compact.bounded.subset_ball 0
+  rw [closed_ball_eq_Icc, zero_add, zero_sub] at hr
+  have :
+    ∀ x : ℝ,
+      ‖(f.comp (ContinuousMap.addRight x)).restrict K‖ ≤ ‖f.restrict (Icc (x - r) (x + r))‖ :=
+    by
+    intro x
+    rw [ContinuousMap.norm_le _ (norm_nonneg _)]
+    rintro ⟨y, hy⟩
+    refine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ ⟨y + x, _⟩)
+    exact ⟨by linarith [(hr hy).1], by linarith [(hr hy).2]⟩
+    simp_rw [ContinuousMap.restrict_apply, ContinuousMap.comp_apply, ContinuousMap.coe_add_right,
+      Subtype.coe_mk]
+  simp_rw [cocompact_eq, is_O_sup] at hf⊢
+  constructor
+  · refine' (is_O_of_le at_bot _).trans (isO_norm_Icc_restrict_atBot hb hf.1 (-r) r)
+    simp_rw [norm_norm]
+    exact this
+  · refine' (is_O_of_le at_top _).trans (isO_norm_Icc_restrict_atTop hb hf.2 (-r) r)
+    simp_rw [norm_norm]
+    exact this
+#align is_O_norm_restrict_cocompact isO_norm_restrict_cocompact
+
+/-- **Poisson's summation formula**, assuming that `f` decays as
+`|x| ^ (-b)` for some `1 < b` and its Fourier transform is summable. -/
+theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ → ℂ} (hc : Continuous f)
+    {b : ℝ} (hb : 1 < b) (hf : IsO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
+    (hFf : Summable fun n : ℤ => 𝓕 f n) : (∑' n : ℤ, f n) = ∑' n : ℤ, 𝓕 f n :=
+  Real.tsum_eq_tsum_fourierIntegral
+    (fun K =>
+      summable_of_isO (Real.summable_abs_int_rpow hb)
+        ((isO_norm_restrict_cocompact (ContinuousMap.mk _ hc) (zero_lt_one.trans hb) hf
+              K).comp_tendsto
+          Int.tendsto_coe_cofinite))
+    hFf
+#align real.tsum_eq_tsum_fourier_integral_of_rpow_decay_of_summable Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable
+
+/-- **Poisson's summation formula**, assuming that both `f` and its Fourier transform decay as
+`|x| ^ (-b)` for some `1 < b`. (This is the one-dimensional case of Corollary VII.2.6 of Stein and
+Weiss, *Introduction to Fourier analysis on Euclidean spaces*.) -/
+theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay {f : ℝ → ℂ} (hc : Continuous f) {b : ℝ}
+    (hb : 1 < b) (hf : IsO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
+    (hFf : IsO (cocompact ℝ) (𝓕 f) fun x : ℝ => |x| ^ (-b)) : (∑' n : ℤ, f n) = ∑' n : ℤ, 𝓕 f n :=
+  Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable hc hb hf
+    (summable_of_isO (Real.summable_abs_int_rpow hb) (hFf.comp_tendsto Int.tendsto_coe_cofinite))
+#align real.tsum_eq_tsum_fourier_integral_of_rpow_decay Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay
+
+end RpowDecay
+
+section Schwartz
+
+/-- **Poisson's summation formula** for Schwartz functions. -/
+theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
+    (∑' n : ℤ, f n) = ∑' n : ℤ, g n :=
+  by
+  -- We know that Schwartz functions are `O(‖x ^ (-b)‖)` for *every* `b`; for this argument we take
+  -- `b = 2` and work with that.
+  simp_rw [← hfg]
+  exact
+    Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay f.continuous one_lt_two
+      (f.is_O_cocompact_rpow (-2)) (by simpa only [hfg] using g.is_O_cocompact_rpow (-2))
+#align schwartz_map.tsum_eq_tsum_fourier_integral SchwartzMap.tsum_eq_tsum_fourierIntegral
+
+end Schwartz
+

mathlib commit https://github.com/leanprover-community/mathlib/commit/c23aca359d4be6975b2a577d5cdb9d77b82c7407

A PR analogous to #12338: reformatting proofs following the multiple goals linter of #12339.

@@ -174,9 +174,9 @@ theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   rintro ⟨x, hx⟩
   rw [ContinuousMap.restrict_apply_mk]
   refine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ ⟨-x, _⟩)
-  rw [ContinuousMap.restrict_apply_mk, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
-    ContinuousMap.coe_mk, neg_neg]
-  exact ⟨by linarith [hx.2], by linarith [hx.1]⟩
+  · rw [ContinuousMap.restrict_apply_mk, ContinuousMap.comp_apply, ContinuousMap.coe_mk,
+      ContinuousMap.coe_mk, neg_neg]
+    exact ⟨by linarith [hx.2], by linarith [hx.1]⟩
 set_option linter.uppercaseLean3 false in
 #align is_O_norm_Icc_restrict_at_bot isBigO_norm_Icc_restrict_atBot
 

This is a (rather boring) technical step in developing the theory of Hurwitz zeta functions: one needs to show that certain sums related to Jacobi theta series decay exponentially for large t.

@@ -49,8 +49,6 @@ open TopologicalSpace Filter MeasureTheory Asymptotics
 
 open scoped Real BigOperators Filter FourierTransform
 
-attribute [local instance] Real.fact_zero_lt_one
-
 open ContinuousMap
 
 /-- The key lemma for Poisson summation: the `m`-th Fourier coefficient of the periodic function

The file Analysis/Fourier/FourierTransform.lean predates the general approach to additive characters elsewhere in the library; this merges the two (getting rid of the rather kludgy notation e [x] in the process). I also rejigged some slow proofs, to make them compile slightly faster.

@@ -62,7 +62,7 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
   -- block, but I think it's more legible this way. We start with preliminaries about the integrand.
   let e : C(ℝ, ℂ) := (fourier (-m)).comp ⟨((↑) : ℝ → UnitAddCircle), continuous_quotient_mk'⟩
   have neK : ∀ (K : Compacts ℝ) (g : C(ℝ, ℂ)), ‖(e * g).restrict K‖ = ‖g.restrict K‖ := by
-    have : ∀ x : ℝ, ‖e x‖ = 1 := fun x => abs_coe_circle (AddCircle.toCircle (-m • x))
+    have (x : ℝ) : ‖e x‖ = 1 := abs_coe_circle (AddCircle.toCircle (-m • x))
     intro K g
     simp_rw [norm_eq_iSup_norm, restrict_apply, mul_apply, norm_mul, this, one_mul]
   have eadd : ∀ (n : ℤ), e.comp (ContinuousMap.addRight n) = e := by

This is a very large PR, but it has been reviewed piecemeal already in PRs to the bump/v4.7.0 branch as we update to intermediate nightlies.

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Kyle Miller <kmill31415@gmail.com> Co-authored-by: damiano <adomani@gmail.com>

@@ -73,7 +73,7 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
   calc
     fourierCoeff (Periodic.lift <| f.periodic_tsum_comp_add_zsmul 1) m =
         ∫ x in (0 : ℝ)..1, e x * (∑' n : ℤ, f.comp (ContinuousMap.addRight n)) x := by
-      simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, comp_apply,
+      simp_rw [fourierCoeff_eq_intervalIntegral _ m 0, div_one, one_smul, zero_add, e, comp_apply,
         coe_mk, Periodic.lift_coe, zsmul_one, smul_eq_mul]
     -- Transform sum in C(ℝ, ℂ) evaluated at x into pointwise sum of values.
     _ = ∫ x in (0:ℝ)..1, ∑' n : ℤ, (e * f.comp (ContinuousMap.addRight n)) x := by
@@ -120,7 +120,7 @@ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
   · simpa only [F, coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
       coe_addRight, zero_add]
        using (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum x).tsum_eq
-  · simp_rw [← Real.fourierCoeff_tsum_comp_add h_norm, smul_eq_mul, coe_mk]
+  · simp_rw [← Real.fourierCoeff_tsum_comp_add h_norm, smul_eq_mul, F, coe_mk]
 
 #align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
 

Also extend the notation 𝓕 for the Fourier transform to inner product spaces, introduce a notation 𝓕⁻ for the inverse Fourier transform, and generalize a few results from the real line to inner product spaces.

@@ -97,7 +97,7 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
       exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _
     -- Minor tidying to finish
     _ = 𝓕 f m := by
-      rw [fourierIntegral_eq_integral_exp_smul]
+      rw [fourierIntegral_real_eq_integral_exp_smul]
       congr 1 with x : 1
       rw [smul_eq_mul, comp_apply, coe_mk, coe_mk, ContinuousMap.toFun_eq_coe, fourier_coe_apply]
       congr 2

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -117,7 +117,7 @@ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
     convert h_sum
     exact Real.fourierCoeff_tsum_comp_add h_norm _
   convert (has_pointwise_sum_fourier_series_of_summable this x).tsum_eq.symm using 1
-  · simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
+  · simpa only [F, coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
       coe_addRight, zero_add]
        using (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum x).tsum_eq
   · simp_rw [← Real.fourierCoeff_tsum_comp_add h_norm, smul_eq_mul, coe_mk]

Use Lean 4 syntax fun x ↦ instead, matching the style guide. This is close to exhaustive for doc comments; mathlib has about 460 remaining uses of λ (not all in Lean 3 syntax).

@@ -129,7 +129,7 @@ section RpowDecay
 variable {E : Type*} [NormedAddCommGroup E]
 
 /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
-`λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
+`fun x ↦ ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
 theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
     (hf : f =O[atTop] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
     (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atTop] fun x : ℝ => |x| ^ (-b) := by

example use case: cocompact_le with integrable_iff_integrableAtFilter_cocompact from #10248 becomes a way to prove integrability from big-O estimates (e.g. #10258)

Co-authored-by: L Lllvvuu <git@llllvvuu.dev>

@@ -195,7 +195,7 @@ theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
     refine' (le_of_eq _).trans (ContinuousMap.norm_coe_le_norm _ ⟨y + x, _⟩)
     · simp_rw [ContinuousMap.restrict_apply, ContinuousMap.comp_apply, ContinuousMap.coe_addRight]
     · exact ⟨by linarith [(hr hy).1], by linarith [(hr hy).2]⟩
-  simp_rw [cocompact_eq, isBigO_sup] at hf ⊢
+  simp_rw [cocompact_eq_atBot_atTop, isBigO_sup] at hf ⊢
   constructor
   · refine' (isBigO_of_le atBot _).trans (isBigO_norm_Icc_restrict_atBot hb hf.1 (-r) r)
     simp_rw [norm_norm]; exact this

Add a more general version of the Jacobi theta function with a second variable, and prove the transformation law for this more general function rather than just for the one-variable version. Preparatory to functional equations for Dirichlet L-functions.

@@ -109,19 +109,19 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
 theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
     (h_norm :
       ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖)
-    (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by
+    (h_sum : Summable fun n : ℤ => 𝓕 f n) (x : ℝ) :
+    ∑' n : ℤ, f (x + n) = ∑' n : ℤ, 𝓕 f n * fourier n (x : UnitAddCircle) := by
   let F : C(UnitAddCircle, ℂ) :=
     ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩
   have : Summable (fourierCoeff F) := by
     convert h_sum
     exact Real.fourierCoeff_tsum_comp_add h_norm _
-  convert (has_pointwise_sum_fourier_series_of_summable this 0).tsum_eq.symm using 1
-  · have := (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum 0).tsum_eq
-    simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
-      coe_addRight, zero_add] using this
-  · congr 1 with n : 1
-    rw [← Real.fourierCoeff_tsum_comp_add h_norm n, fourier_eval_zero, smul_eq_mul, mul_one]
-    rfl
+  convert (has_pointwise_sum_fourier_series_of_summable this x).tsum_eq.symm using 1
+  · simpa only [coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
+      coe_addRight, zero_add]
+       using (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum x).tsum_eq
+  · simp_rw [← Real.fourierCoeff_tsum_comp_add h_norm, smul_eq_mul, coe_mk]
+
 #align real.tsum_eq_tsum_fourier_integral Real.tsum_eq_tsum_fourierIntegral
 
 section RpowDecay
@@ -135,7 +135,7 @@ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
     (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atTop] fun x : ℝ => |x| ^ (-b) := by
   -- First establish an explicit estimate on decay of inverse powers.
   -- This is logically independent of the rest of the proof, but of no mathematical interest in
-  -- itself, so it is proved using `async` rather than being formulated as a separate lemma.
+  -- itself, so it is proved in-line rather than being formulated as a separate lemma.
   have claim : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y →
       y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := fun x hx y hy ↦ by
     rw [max_lt_iff] at hx
@@ -207,15 +207,12 @@ set_option linter.uppercaseLean3 false in
 /-- **Poisson's summation formula**, assuming that `f` decays as
 `|x| ^ (-b)` for some `1 < b` and its Fourier transform is summable. -/
 theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ → ℂ} (hc : Continuous f)
-    {b : ℝ} (hb : 1 < b) (hf : f =O[cocompact ℝ] fun x : ℝ => |x| ^ (-b))
-    (hFf : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) :=
-  Real.tsum_eq_tsum_fourierIntegral
-    (fun K =>
-      summable_of_isBigO (Real.summable_abs_int_rpow hb)
-        ((isBigO_norm_restrict_cocompact (ContinuousMap.mk _ hc) (zero_lt_one.trans hb) hf
-              K).comp_tendsto
-          Int.tendsto_coe_cofinite))
-    hFf
+    {b : ℝ} (hb : 1 < b) (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
+    (hFf : Summable fun n : ℤ => 𝓕 f n) (x : ℝ) :
+    ∑' n : ℤ, f (x + n) = ∑' n : ℤ, 𝓕 f n * fourier n (x : UnitAddCircle) :=
+  Real.tsum_eq_tsum_fourierIntegral (fun K => summable_of_isBigO (Real.summable_abs_int_rpow hb)
+    ((isBigO_norm_restrict_cocompact ⟨_, hc⟩ (zero_lt_one.trans hb) hf K).comp_tendsto
+    Int.tendsto_coe_cofinite)) hFf x
 #align real.tsum_eq_tsum_fourier_integral_of_rpow_decay_of_summable Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable
 
 /-- **Poisson's summation formula**, assuming that both `f` and its Fourier transform decay as
@@ -223,10 +220,10 @@ theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ →
 Weiss, *Introduction to Fourier analysis on Euclidean spaces*.) -/
 theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay {f : ℝ → ℂ} (hc : Continuous f) {b : ℝ}
     (hb : 1 < b) (hf : f =O[cocompact ℝ] (|·| ^ (-b)))
-    (hFf : (𝓕 f) =O[cocompact ℝ] (|·| ^ (-b))) :
-    ∑' n : ℤ, f n = ∑' n : ℤ, 𝓕 f n :=
-  Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable hc hb hf
-    (summable_of_isBigO (Real.summable_abs_int_rpow hb) (hFf.comp_tendsto Int.tendsto_coe_cofinite))
+    (hFf : (𝓕 f) =O[cocompact ℝ] (|·| ^ (-b))) (x : ℝ) :
+    ∑' n : ℤ, f (x + n) = ∑' n : ℤ, 𝓕 f n * fourier n (x : UnitAddCircle) :=
+  Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable hc hb hf (summable_of_isBigO
+    (Real.summable_abs_int_rpow hb) (hFf.comp_tendsto Int.tendsto_coe_cofinite)) x
 #align real.tsum_eq_tsum_fourier_integral_of_rpow_decay Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay
 
 end RpowDecay
@@ -234,15 +231,13 @@ end RpowDecay
 section Schwartz
 
 /-- **Poisson's summation formula** for Schwartz functions. -/
-theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
-    ∑' n : ℤ, f n = (∑' n : ℤ, g n) := by
+theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 ⇑f = ⇑g) (x : ℝ) :
+    ∑' n : ℤ, f (x + n) = (∑' n : ℤ, g n * fourier n (x : UnitAddCircle)) := by
   -- We know that Schwartz functions are `O(‖x ^ (-b)‖)` for *every* `b`; for this argument we take
   -- `b = 2` and work with that.
-  simp_rw [← hfg]
-  rw [Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay f.continuous one_lt_two
-    (f.isBigO_cocompact_rpow (-2))]
-  rw [hfg]
-  exact g.isBigO_cocompact_rpow (-2)
+  simp only [← hfg, Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay f.continuous one_lt_two
+    (f.isBigO_cocompact_rpow (-2)) (hfg ▸ g.isBigO_cocompact_rpow (-2))]
+
 #align schwartz_map.tsum_eq_tsum_fourier_integral SchwartzMap.tsum_eq_tsum_fourierIntegral
 
 end Schwartz

@@ -131,30 +131,17 @@ variable {E : Type*} [NormedAddCommGroup E]
 /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/
 theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
-    (hf : IsBigO atTop f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
-    IsBigO atTop (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
+    (hf : f =O[atTop] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
+    (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atTop] fun x : ℝ => |x| ^ (-b) := by
   -- First establish an explicit estimate on decay of inverse powers.
   -- This is logically independent of the rest of the proof, but of no mathematical interest in
   -- itself, so it is proved using `async` rather than being formulated as a separate lemma.
-  have claim :
-    ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by
-    intro x hx y hy
+  have claim : ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y →
+      y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := fun x hx y hy ↦ by
     rw [max_lt_iff] at hx
     obtain ⟨hx1, hx2⟩ := hx
-    have hxR : 0 < x + R := by
-      rcases le_or_lt 0 R with (h | _)
-      · positivity
-      · linarith
-    have hy' : 0 < y := hxR.trans_le hy
-    have : y ^ (-b) ≤ (x + R) ^ (-b) := by
-      rw [rpow_neg, rpow_neg, inv_le_inv]
-      · gcongr
-      all_goals positivity
-    refine' this.trans _
-    rw [← mul_rpow, rpow_neg, rpow_neg]
-    · gcongr
-      linarith
-    all_goals positivity
+    rw [← mul_rpow] <;> try positivity
+    apply rpow_le_rpow_of_nonpos <;> linarith
   -- Now the main proof.
   obtain ⟨c, hc, hc'⟩ := hf.exists_pos
   simp only [IsBigO, IsBigOWith, eventually_atTop] at hc' ⊢
@@ -163,9 +150,7 @@ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   rw [ge_iff_le, max_le_iff] at hx
   have hx' : max 0 (-2 * R) < x := by linarith
   rw [max_lt_iff] at hx'
-  rw [norm_norm,
-    ContinuousMap.norm_le _
-      (mul_nonneg (mul_nonneg hc.le <| rpow_nonneg one_half_pos.le _) (norm_nonneg _))]
+  rw [norm_norm, ContinuousMap.norm_le _ (by positivity)]
   refine' fun y => (hd y.1 (by linarith [hx.1, y.2.1])).trans _
   have A : ∀ x : ℝ, 0 ≤ |x| ^ (-b) := fun x => by positivity
   rw [mul_assoc, mul_le_mul_left hc, norm_of_nonneg (A _), norm_of_nonneg (A _)]
@@ -176,9 +161,9 @@ set_option linter.uppercaseLean3 false in
 #align is_O_norm_Icc_restrict_at_top isBigO_norm_Icc_restrict_atTop
 
 theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
-    (hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
-    IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by
-  have h1 : IsBigO atTop (f.comp (ContinuousMap.mk _ continuous_neg)) fun x : ℝ => |x| ^ (-b) := by
+    (hf : f =O[atBot] fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
+    (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) =O[atBot] fun x : ℝ => |x| ^ (-b) := by
+  have h1 : (f.comp (ContinuousMap.mk _ continuous_neg)) =O[atTop] fun x : ℝ => |x| ^ (-b) := by
     convert hf.comp_tendsto tendsto_neg_atTop_atBot using 1
     ext1 x; simp only [Function.comp_apply, abs_neg]
   have h2 := (isBigO_norm_Icc_restrict_atTop hb h1 (-S) (-R)).comp_tendsto tendsto_neg_atBot_atTop
@@ -198,13 +183,11 @@ set_option linter.uppercaseLean3 false in
 #align is_O_norm_Icc_restrict_at_bot isBigO_norm_Icc_restrict_atBot
 
 theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
-    (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
-    IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
-      |x| ^ (-b) := by
+    (hf : f =O[cocompact ℝ] fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
+    (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) =O[cocompact ℝ] (|·| ^ (-b)) := by
   obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_closedBall 0
   rw [closedBall_eq_Icc, zero_add, zero_sub] at hr
-  have :
-    ∀ x : ℝ,
+  have : ∀ x : ℝ,
       ‖(f.comp (ContinuousMap.addRight x)).restrict K‖ ≤ ‖f.restrict (Icc (x - r) (x + r))‖ := by
     intro x
     rw [ContinuousMap.norm_le _ (norm_nonneg _)]
@@ -224,7 +207,7 @@ set_option linter.uppercaseLean3 false in
 /-- **Poisson's summation formula**, assuming that `f` decays as
 `|x| ^ (-b)` for some `1 < b` and its Fourier transform is summable. -/
 theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ → ℂ} (hc : Continuous f)
-    {b : ℝ} (hb : 1 < b) (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
+    {b : ℝ} (hb : 1 < b) (hf : f =O[cocompact ℝ] fun x : ℝ => |x| ^ (-b))
     (hFf : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) :=
   Real.tsum_eq_tsum_fourierIntegral
     (fun K =>
@@ -239,8 +222,8 @@ theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ →
 `|x| ^ (-b)` for some `1 < b`. (This is the one-dimensional case of Corollary VII.2.6 of Stein and
 Weiss, *Introduction to Fourier analysis on Euclidean spaces*.) -/
 theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay {f : ℝ → ℂ} (hc : Continuous f) {b : ℝ}
-    (hb : 1 < b) (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
-    (hFf : IsBigO (cocompact ℝ) (𝓕 f) fun x : ℝ => |x| ^ (-b)) :
+    (hb : 1 < b) (hf : f =O[cocompact ℝ] (|·| ^ (-b)))
+    (hFf : (𝓕 f) =O[cocompact ℝ] (|·| ^ (-b))) :
     ∑' n : ℤ, f n = ∑' n : ℤ, 𝓕 f n :=
   Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable hc hb hf
     (summable_of_isBigO (Real.summable_abs_int_rpow hb) (hFf.comp_tendsto Int.tendsto_coe_cofinite))

This better matches other lemma names.

From LeanAPAP

@@ -165,7 +165,7 @@ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
   rw [max_lt_iff] at hx'
   rw [norm_norm,
     ContinuousMap.norm_le _
-      (mul_nonneg (mul_nonneg hc.le <| rpow_nonneg_of_nonneg one_half_pos.le _) (norm_nonneg _))]
+      (mul_nonneg (mul_nonneg hc.le <| rpow_nonneg one_half_pos.le _) (norm_nonneg _))]
   refine' fun y => (hd y.1 (by linarith [hx.1, y.2.1])).trans _
   have A : ∀ x : ℝ, 0 ≤ |x| ^ (-b) := fun x => by positivity
   rw [mul_assoc, mul_le_mul_left hc, norm_of_nonneg (A _), norm_of_nonneg (A _)]

Following on from previous gcongr golfing PRs #4702 and #4784.

This is a replacement for #7901: this round of golfs, first introduced there, there exposed some performance issues in gcongr, hopefully fixed by #8731, and I am opening a new PR so that the performance can be checked against current master rather than master at the time of #7901.

@@ -140,22 +140,21 @@ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
     ∀ x : ℝ, max 0 (-2 * R) < x → ∀ y : ℝ, x + R ≤ y → y ^ (-b) ≤ (1 / 2) ^ (-b) * x ^ (-b) := by
     intro x hx y hy
     rw [max_lt_iff] at hx
+    obtain ⟨hx1, hx2⟩ := hx
     have hxR : 0 < x + R := by
-      rcases le_or_lt 0 R with (h | h)
-      · exact add_pos_of_pos_of_nonneg hx.1 h
-      · rw [← sub_lt_iff_lt_add, zero_sub]
-        refine' lt_trans _ hx.2
-        rwa [neg_mul, neg_lt_neg_iff, two_mul, add_lt_iff_neg_left]
+      rcases le_or_lt 0 R with (h | _)
+      · positivity
+      · linarith
     have hy' : 0 < y := hxR.trans_le hy
     have : y ^ (-b) ≤ (x + R) ^ (-b) := by
-      rw [rpow_neg hy'.le, rpow_neg hxR.le,
-        inv_le_inv (rpow_pos_of_pos hy' _) (rpow_pos_of_pos hxR _)]
-      exact rpow_le_rpow hxR.le hy hb.le
+      rw [rpow_neg, rpow_neg, inv_le_inv]
+      · gcongr
+      all_goals positivity
     refine' this.trans _
-    rw [← mul_rpow one_half_pos.le hx.1.le, rpow_neg (mul_pos one_half_pos hx.1).le,
-      rpow_neg hxR.le]
-    refine' inv_le_inv_of_le (rpow_pos_of_pos (mul_pos one_half_pos hx.1) _) _
-    exact rpow_le_rpow (mul_pos one_half_pos hx.1).le (by linarith) hb.le
+    rw [← mul_rpow, rpow_neg, rpow_neg]
+    · gcongr
+      linarith
+    all_goals positivity
   -- Now the main proof.
   obtain ⟨c, hc, hc'⟩ := hf.exists_pos
   simp only [IsBigO, IsBigOWith, eventually_atTop] at hc' ⊢

Use Bornology.IsBounded instead.

@@ -202,7 +202,7 @@ theorem isBigO_norm_restrict_cocompact (f : C(ℝ, E)) {b : ℝ} (hb : 0 < b)
     (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b)) (K : Compacts ℝ) :
     IsBigO (cocompact ℝ) (fun x => ‖(f.comp (ContinuousMap.addRight x)).restrict K‖) fun x =>
       |x| ^ (-b) := by
-  obtain ⟨r, hr⟩ := K.isCompact.bounded.subset_ball 0
+  obtain ⟨r, hr⟩ := K.isCompact.isBounded.subset_closedBall 0
   rw [closedBall_eq_Icc, zero_add, zero_sub] at hr
   have :
     ∀ x : ℝ,

Co-authored-by: Eric Wieser <wieser.eric@gmail.com>

@@ -92,7 +92,7 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
       suffices Integrable (e * f) from this.hasSum_intervalIntegral_comp_add_int.tsum_eq
       apply integrable_of_summable_norm_Icc
       convert hf ⟨Icc 0 1, isCompact_Icc⟩ using 1
-      simp_rw [ContinuousMap.comp_apply, mul_comp] at eadd ⊢
+      simp_rw [mul_comp] at eadd ⊢
       simp_rw [eadd]
       exact funext fun n => neK ⟨Icc 0 1, isCompact_Icc⟩ _
     -- Minor tidying to finish

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -126,7 +126,7 @@ theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
 
 section RpowDecay
 
-variable {E : Type _} [NormedAddCommGroup E]
+variable {E : Type*} [NormedAddCommGroup E]
 
 /-- If `f` is `O(x ^ (-b))` at infinity, then so is the function
 `λ x, ‖f.restrict (Icc (x + R) (x + S))‖` for any fixed `R` and `S`. -/

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,11 +2,6 @@
 Copyright (c) 2023 David Loeffler. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: David Loeffler
-
-! This file was ported from Lean 3 source module analysis.fourier.poisson_summation
-! leanprover-community/mathlib commit fd5edc43dc4f10b85abfe544b88f82cf13c5f844
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.Analysis.Fourier.AddCircle
 import Mathlib.Analysis.Fourier.FourierTransform
@@ -14,6 +9,8 @@ import Mathlib.Analysis.PSeries
 import Mathlib.Analysis.Distribution.SchwartzSpace
 import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
 
+#align_import analysis.fourier.poisson_summation from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
+
 /-!
 # Poisson's summation formula
 

Grepping for [^ .:{-] [^ :] and reviewing the results. Once I started I couldn't stop. :-)

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -83,7 +83,7 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
       simp_rw [coe_mul, Pi.mul_apply,
         ← ContinuousMap.tsum_apply (summable_of_locally_summable_norm hf), tsum_mul_left]
     -- Swap sum and integral.
-    _ =  ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
+    _ = ∑' n : ℤ, ∫ x in (0:ℝ)..1, (e * f.comp (ContinuousMap.addRight n)) x := by
       refine' (intervalIntegral.tsum_intervalIntegral_eq_of_summable_norm _).symm
       convert hf ⟨uIcc 0 1, isCompact_uIcc⟩ using 1
       exact funext fun n => neK _ _

@@ -11,7 +11,7 @@ Authors: David Loeffler
 import Mathlib.Analysis.Fourier.AddCircle
 import Mathlib.Analysis.Fourier.FourierTransform
 import Mathlib.Analysis.PSeries
-import Mathlib.Analysis.SchwartzSpace
+import Mathlib.Analysis.Distribution.SchwartzSpace
 import Mathlib.MeasureTheory.Measure.Lebesgue.Integral
 
 /-!

• Also remove most superfluous parentheses around big operators (∑, ∏ and variants).
• roughly the used regex: ([^a-zA-Zα-ωΑ-Ω'𝓝ℳ₀𝕂ₛ)]) \(([∑∏][^()∑∏]*,[^()∑∏:]*)\) ([⊂⊆=<≤]) replaced by $1 $2 $3

@@ -112,7 +112,7 @@ theorem Real.fourierCoeff_tsum_comp_add {f : C(ℝ, ℂ)}
 theorem Real.tsum_eq_tsum_fourierIntegral {f : C(ℝ, ℂ)}
     (h_norm :
       ∀ K : Compacts ℝ, Summable fun n : ℤ => ‖(f.comp <| ContinuousMap.addRight n).restrict K‖)
-    (h_sum : Summable fun n : ℤ => 𝓕 f n) : (∑' n : ℤ, f n) = (∑' n : ℤ, 𝓕 f n) := by
+    (h_sum : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) := by
   let F : C(UnitAddCircle, ℂ) :=
     ⟨(f.periodic_tsum_comp_add_zsmul 1).lift, continuous_coinduced_dom.mpr (map_continuous _)⟩
   have : Summable (fourierCoeff F) := by
@@ -229,7 +229,7 @@ set_option linter.uppercaseLean3 false in
 `|x| ^ (-b)` for some `1 < b` and its Fourier transform is summable. -/
 theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable {f : ℝ → ℂ} (hc : Continuous f)
     {b : ℝ} (hb : 1 < b) (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
-    (hFf : Summable fun n : ℤ => 𝓕 f n) : (∑' n : ℤ, f n) = (∑' n : ℤ, 𝓕 f n) :=
+    (hFf : Summable fun n : ℤ => 𝓕 f n) : ∑' n : ℤ, f n = (∑' n : ℤ, 𝓕 f n) :=
   Real.tsum_eq_tsum_fourierIntegral
     (fun K =>
       summable_of_isBigO (Real.summable_abs_int_rpow hb)
@@ -245,7 +245,7 @@ Weiss, *Introduction to Fourier analysis on Euclidean spaces*.) -/
 theorem Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay {f : ℝ → ℂ} (hc : Continuous f) {b : ℝ}
     (hb : 1 < b) (hf : IsBigO (cocompact ℝ) f fun x : ℝ => |x| ^ (-b))
     (hFf : IsBigO (cocompact ℝ) (𝓕 f) fun x : ℝ => |x| ^ (-b)) :
-    (∑' n : ℤ, f n) = ∑' n : ℤ, 𝓕 f n :=
+    ∑' n : ℤ, f n = ∑' n : ℤ, 𝓕 f n :=
   Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay_of_summable hc hb hf
     (summable_of_isBigO (Real.summable_abs_int_rpow hb) (hFf.comp_tendsto Int.tendsto_coe_cofinite))
 #align real.tsum_eq_tsum_fourier_integral_of_rpow_decay Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay
@@ -256,7 +256,7 @@ section Schwartz
 
 /-- **Poisson's summation formula** for Schwartz functions. -/
 theorem SchwartzMap.tsum_eq_tsum_fourierIntegral (f g : SchwartzMap ℝ ℂ) (hfg : 𝓕 f = g) :
-    (∑' n : ℤ, f n) = (∑' n : ℤ, g n) := by
+    ∑' n : ℤ, f n = (∑' n : ℤ, g n) := by
   -- We know that Schwartz functions are `O(‖x ^ (-b)‖)` for *every* `b`; for this argument we take
   -- `b = 2` and work with that.
   simp_rw [← hfg]

@@ -22,17 +22,17 @@ Fourier transform of `f`, under the following hypotheses:
 * `f` is a continuous function `ℝ → ℂ`.
 * The sum `∑ (n : ℤ), 𝓕 f n` is convergent.
 * For all compacts `K ⊂ ℝ`, the sum `∑ (n : ℤ), sup { ‖f(x + n)‖ | x ∈ K }` is convergent.
-See `real.tsum_eq_tsum_fourier_integral` for this formulation.
+See `Real.tsum_eq_tsum_fourierIntegral` for this formulation.
 
 These hypotheses are potentially a little awkward to apply, so we also provide the less general but
-easier-to-use result `real.tsum_eq_tsum_fourier_integral_of_rpow_decay`, in which we assume `f` and
+easier-to-use result `Real.tsum_eq_tsum_fourierIntegral_of_rpow_decay`, in which we assume `f` and
 `𝓕 f` both decay as `|x| ^ (-b)` for some `b > 1`, and the even more specific result
-`schwartz_map.tsum_eq_tsum_fourier_integral`, where we assume that both `f` and `𝓕 f` are Schwartz
+`SchwartzMap.tsum_eq_tsum_fourierIntegral`, where we assume that both `f` and `𝓕 f` are Schwartz
 functions.
 
 ## TODO
 
-At the moment `schwartz_map.tsum_eq_tsum_fourier_integral` requires separate proofs that both `f`
+At the moment `SchwartzMap.tsum_eq_tsum_fourierIntegral` requires separate proofs that both `f`
 and `𝓕 f` are Schwartz functions. In fact, `𝓕 f` is automatically Schwartz if `f` is; and once
 we have this lemma in the library, we should adjust the hypotheses here accordingly.
 -/
@@ -179,7 +179,6 @@ theorem isBigO_norm_Icc_restrict_atTop {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
 set_option linter.uppercaseLean3 false in
 #align is_O_norm_Icc_restrict_at_top isBigO_norm_Icc_restrict_atTop
 
-set_option autoImplicit false in
 theorem isBigO_norm_Icc_restrict_atBot {f : C(ℝ, E)} {b : ℝ} (hb : 0 < b)
     (hf : IsBigO atBot f fun x : ℝ => |x| ^ (-b)) (R S : ℝ) :
     IsBigO atBot (fun x : ℝ => ‖f.restrict (Icc (x + R) (x + S))‖) fun x : ℝ => |x| ^ (-b) := by

Co-authored-by: Ruben Van de Velde <65514131+Ruben-VandeVelde@users.noreply.github.com>