{"id":"lem:snormEssSup_eq_sSup_snorm","label":"snormEssSup_eq_sSup_snorm","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f\\) be measurable and the measure \\(\\mu \\) be \\(\\sigma \\)-finite. Then \\[ \\| f\\| _{L^\\infty } = \\sup _{\\| g\\| _{L^1} \\leq 1, \\ g \\text{ simple}} \\| fg \\| _{L^1}. \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.snormEssSup_eq_sSup_snorm","latex_link":"chap-interpolation.html#lem:snormEssSup_eq_sSup_snorm","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.snormEssSup_eq_sSup_snorm","lean_decl_name":"MeasureTheory.snormEssSup_eq_sSup_snorm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/fpvandoorn/BonnAnalysis/f3acd7b6e64e9949e697d595d495e01dc4c3dbf9/BonnAnalysis/ComplexInterpolation.lean","lean4_code":"lemma snormEssSup_eq_sSup_snorm [SigmaFinite μ] (f : α → ℂ) (hf : Measurable f)\n(hf' : sSup {x |∃ g : SimpleFunc α ℂ, snorm g 1 μ ≤ 1 ∧\nx = snorm (f * (g : α → ℂ)) 1 μ} < ⊤):\nsnormEssSup f μ = sSup {x |∃ g : SimpleFunc α ℂ, snorm g 1 μ ≤ 1\n∧ x = snorm (f * (g : α → ℂ)) 1 μ} := by\n   apply le_antisymm ?_\n   .  apply sSup_le\n      intro s ⟨g, hg₁, hg₂⟩\n      rw [hg₂, mul_comm]\n      simp [snorm, snorm']\n      apply le_trans (lintegral_norm_mul_le_one_top _ (by measurability))\n      apply mul_le_of_le_one_of_le hg₁ (by simp [snorm])\n   .  set M := sSup {x |∃ g : SimpleFunc α ℂ, snorm g 1 μ ≤ 1 ∧\n       x = snorm (f * (g : α → ℂ)) 1 μ}\n      apply essSup_le_of_ae_le\n      simp only [EventuallyLE, eventually_iff, mem_ae_iff]\n      by_contra h\n\n      have aux1 : ∃ B : Set α, MeasurableSet B ∧ B ⊆ {x | ‖f x‖₊ ≤ M}ᶜ ∧ 0 < μ B ∧ μ B < ⊤ := by\n        apply Measure.exists_subset_measure_lt_top (MeasurableSet.compl _) ((ne_eq _ _) ▸ h).bot_lt\n        convert hf.ennnorm measurableSet_Iic\n      rcases aux1 with ⟨B, hB⟩\n\n      let g : SimpleFunc α ℂ := {\n        toFun := B.indicator (fun _ ↦ ((μ B)⁻¹.toReal : ℂ))\n        measurableSet_fiber' := by\n          intro x\n          simp [indicator_preimage]\n          rcases eq_or_ne x ((μ B)⁻¹.toReal : ℂ) with (H | H)\n          simp [H]\n          apply MeasurableSet.ite hB.1 (by simp only [MeasurableSet.univ, M])\n          apply measurableSet_preimage measurable_zero (MeasurableSet.singleton _)\n          apply MeasurableSet.ite hB.1 _ (measurableSet_preimage measurable_zero\n          (MeasurableSet.singleton _))\n          rw [preimage_const_of_not_mem]\n          apply MeasurableSet.empty\n          rw [mem_singleton_iff]\n          exact H.symm\n        finite_range' := by\n          have : range (B.indicator fun _ ↦ ((μ B)⁻¹.toReal : ℂ))\n          ⊆ {0, ((μ B)⁻¹.toReal : ℂ)}  := by\n            intro x\n            rw [Set.mem_range_indicator]\n            simp\n            rintro (h | h)\n            exact Or.intro_left _ h.1\n            exact Or.intro_right _ h.2.symm\n          apply Set.Finite.subset _ this\n          simp only [finite_singleton, Finite.insert]\n      }\n\n      have aux_μB : (μ B)⁻¹ =  ‖(μ B)⁻¹.toReal‖₊  := by\n        have : (μ B)⁻¹ < ⊤ := by simp [hB.2.2.1]\n        generalize (μ B)⁻¹ = y at this -- better solution???\n        cases y\n        simp only [lt_self_iff_false] at this\n        simp only [coe_toReal, nnnorm_eq]\n\n      have : ¬ (M < snorm (f * (g : α → ℂ)) 1 μ) := by\n        simp\n        apply le_sSup\n        use g\n        constructor\n        simp [snorm, snorm', lintegral_indicator_const]\n        have : ∫⁻ (a : α), (B.indicator (fun _ ↦ (μ B)⁻¹) a) ∂ μ ≤ 1 := by\n          rw [lintegral_indicator_const hB.1,\n              ENNReal.inv_mul_cancel hB.2.2.1.ne.symm hB.2.2.2.ne]\n        convert this\n        rename α => x\n        by_cases hx : x ∈ B\n        simp [indicator_of_mem hx, aux_μB.symm]\n        simp [indicator_of_not_mem hx]\n        rfl\n\n      apply this\n      calc M = ∫⁻ _ in B, (μ B)⁻¹ * M ∂ μ := by {\n        rw [setLIntegral_const, mul_assoc, mul_comm M, ← mul_assoc,\n        ENNReal.inv_mul_cancel hB.2.2.1.ne.symm hB.2.2.2.ne, one_mul]\n      }\n      _ < ∫⁻ x in B, (μ B)⁻¹ * ‖f x‖₊ ∂ μ := by\n          apply setLIntegral_strict_mono hB.1 hB.2.2.1.ne.symm (hf.ennnorm.const_mul _)\n          rw [setLIntegral_const]\n          apply mul_ne_top (mul_ne_top (ENNReal.inv_ne_top.mpr\n          hB.2.2.1.ne.symm) hf'.ne) hB.2.2.2.ne\n          apply eventually_of_forall\n          intro x hx\n          rw [ENNReal.mul_lt_mul_left (ENNReal.inv_ne_zero.mpr hB.2.2.2.ne)\n          (ENNReal.inv_ne_top.mpr hB.2.2.1.ne.symm)]\n          convert hB.2.1 hx\n          simp only [mem_compl_iff, mem_setOf_eq, not_le]\n      _ = ∫⁻ x : α, B.indicator (fun a ↦ ‖f a‖₊ * ‖(fun _ ↦ ((μ B)⁻¹.toReal : ℂ)) a‖₊) x ∂μ := by\n          rw [lintegral_indicator _ hB.1]\n          simp [mul_comm]\n          congr with x; congr\n      _ = snorm (f * B.indicator fun _ ↦ ((μ B)⁻¹.toReal : ℂ)) 1 μ := by\n          simp [snorm, snorm']\n          congr with x\n          by_cases hx : x ∈ B\n          simp only [indicator_of_mem hx, nnnorm_real]\n          simp only [indicator_of_not_mem hx, nnnorm_zero, ENNReal.coe_zero, mul_zero]","line":1025,"end_line":1124,"proof_status":"formalized"}
{"id":"a0000000211","label":"a0000000211","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(\\lambda {\\gt} -d\\), Let \\(H_\\lambda \\) be the tempered distribution associated to \\(|x|^\\lambda \\in L^1_{loc}\\). If \\(-d {\\lt} \\lambda {\\lt} 0\\) then \\[ \\widehat{H_\\lambda } = c_\\lambda H_{-d-\\lambda } \\]","doc_url":null,"latex_link":"sect0001.html#a0000000211","lean_link":null,"lean_decl_name":"a0000000211","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000168","label":"a0000000168","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Then \\(C^\\infty (\\mathbb {R}^d)\\) is dense in \\(\\mathcal{D}'(\\mathbb {R}^d)\\).","doc_url":null,"latex_link":"sect0001.html#a0000000168","lean_link":null,"lean_decl_name":"a0000000168","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:hausdorff_young","label":"hausdorff_young","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["thm:riesz_interpolation"],"nlp_text":"Let \\(X=[0,2\\pi ]\\) with normalized Lebesgue measure \\(\\frac{d\\theta }{2\\pi }\\) and let \\(Y=\\mathbb {Z}\\) with counting measure.Consider the operator \\(T: f \\mapsto \\{ a_n\\} _{n\\in \\mathbb {Z}}\\) where \\[ a_n = \\frac{1}{2\\pi } \\int _0^{2\\pi } f(\\theta ) e^{-in\\theta } d\\theta \\]","doc_url":null,"latex_link":"chap-interpolation.html#lem:hausdorff_young","lean_link":null,"lean_decl_name":"hausdorff_young","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:fourier-inversion-L1","label":"fourier-inversion-L1","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier-multiplication","thm:kernel-approximation"],"nlp_text":"Let \\(f\\in L^1(V,E)\\). If \\(\\widehat f\\in L^1(V,E)\\), then \\(\\mathcal F^{-1}\\mathcal Ff=f\\).","doc_url":null,"latex_link":"chap-plancherel.html#thm:fourier-inversion-L1","lean_link":null,"lean_decl_name":"fourier-inversion-L1","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:fourier-gaussian","label":"fourier-gaussian","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier-prop"],"nlp_text":"Let \\(x\\in V\\) and \\(\\delta {\\gt}0\\). Define the modulated Gaussian \\[ u_{x,\\delta }(y):V\\to \\mathbb {C},\\quad y\\mapsto e^{-\\delta \\pi |y|^2}e^{2\\pi i x y}. \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/fourierIntegral_gaussian_innerProductSpace'","latex_link":"chap-plancherel.html#lem:fourier-gaussian","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/fourierIntegral_gaussian_innerProductSpace'","lean_decl_name":"fourierIntegral_gaussian_innerProductSpace'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/b387c548d1dc9cea5c5fcb71e5b3370cca1a04cc/Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean","lean4_code":"theorem _root_.fourierIntegral_gaussian_innerProductSpace' (hb : 0 < b.re) (x w : V) :\n    𝓕 (fun v ↦ cexp (- b * ‖v‖^2 + 2 * π * Complex.I * ⟪x, v⟫)) w =\n      (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * ‖x - w‖ ^ 2 / b) := by\n  simp only [neg_mul, fourierIntegral_eq', ofReal_neg, ofReal_mul, ofReal_ofNat,\n    smul_eq_mul, ← Complex.exp_add, real_inner_comm w]\n  convert integral_cexp_neg_mul_sq_norm_add hb (2 * π * Complex.I) (x - w) using 3 with v\n  · congr 1\n    simp [inner_sub_left]\n    ring\n  · have : b ≠ 0 := by contrapose! hb; rw [hb, zero_re]\n    field_simp [mul_pow]\n    ring","line":369,"end_line":380,"proof_status":"formalized"}
{"id":"thm:plancherel","label":"plancherel","background_color":"#A3D6FF","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier-gaussian","lem:fourier-multiplication","thm:kernel-approximation"],"nlp_text":"Suppose that \\(f : V \\to E\\) is in \\(L^1(V,E)\\cap L^2(V,E)\\) and let \\(\\widehat{f}\\) be the Fourier transform of \\(f\\). Then \\(\\widehat{f},\\check{f}\\in L^2(V,E)\\) and \\[ \\| \\widehat{f}\\| _{L^2} = \\| f\\| _{L^2}=\\| \\check f\\| _{L^2}. \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.memℒp_fourierIntegral","latex_link":"chap-plancherel.html#thm:plancherel","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.memℒp_fourierIntegral","lean_decl_name":"MeasureTheory.memℒp_fourierIntegral","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/fpvandoorn/BonnAnalysis/f3acd7b6e64e9949e697d595d495e01dc4c3dbf9/BonnAnalysis/Plancherel.lean","lean4_code":"theorem memℒp_fourierIntegral {f : V → ℂ} (hf : Integrable f) (h2f : Memℒp f 2) :\n    Memℒp (𝓕 f) 2 := by\n  unfold Memℒp; constructor\n  · apply Continuous.aestronglyMeasurable\n    exact VectorFourier.fourierIntegral_continuous continuous_fourierChar\n        (by simp only [innerₗ_apply]; exact continuous_inner) hf\n  · rw [snorm_fourierIntegral hf h2f]\n    exact h2f.2","line":473,"end_line":480,"proof_status":"statement_formalized"}
{"id":"lem:fourier12-cauchy","label":"fourier12-cauchy","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["thm:plancherel","lem:L12-dense"],"nlp_text":"Let \\(f\\in L^2(V,E)\\) and \\((f_n)_n\\subset L^1(V,E)\\cap L^2(V,E)\\) a sequence with \\(f_n\\xrightarrow [L^2]{}f\\). Then \\((\\widehat f_n)_n\\) is a Cauchy sequence, hence converges in \\(L^2(V,E)\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#lem:fourier12-cauchy","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:invFourier-L2","label":"invFourier-L2","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier12-cauchy"],"nlp_text":"Define analogously \\(\\mathcal F^{-1}f:=\\check f:=\\lim _{n\\to \\infty }\\check f_n\\), if \\(f_n\\xrightarrow [L^2]{}f\\in L^2(V,E)\\) with \\((f_n)_n\\subset L^1(V,E)\\cap L^2(V,E)\\). By the same arguments as above, this is well-defined.","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#def:invFourier-L2","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:fourier-L2","label":"fourier-L2","background_color":"white","border_color":"black","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier12-cauchy"],"nlp_text":"Let \\(f\\in L^2(V,E)\\) and take a sequence \\((f_n)_n\\subset L^1(V,E)\\cap L^2(V,E)\\) with \\(f_n\\xrightarrow [L^2]{}f\\). Set \\[ \\mathcal Ff:=\\widehat f:=\\lim _{n\\to \\infty }\\widehat{f_n}, \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#def:fourier-L2","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"definitions","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:fourier-bounded","label":"fourier-bounded","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["def:fourier-transform"],"nlp_text":"Let \\(f\\in L^1(V,E)\\). Then its Fourier transform \\(\\widehat f\\) is well-defined and bounded. In particular, the Fourier transform defines a map \\(\\mathcal F:L^1(V,E)\\to L^\\infty (V,E)\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_convergent_iff","latex_link":"chap-plancherel.html#lem:fourier-bounded","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_convergent_iff","lean_decl_name":"VectorFourier.fourierIntegral_convergent_iff","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Fourier/FourierTransform.lean","lean4_code":"theorem fourierIntegral_convergent_iff (he : Continuous e)\n    (hL : Continuous fun p : V × W ↦ L p.1 p.2) {f : V → E} (w : W) :\n    Integrable (fun v : V ↦ e (-L v w) • f v) μ ↔ Integrable f μ := by\n  -- first prove one-way implication\n  have aux {g : V → E} (hg : Integrable g μ) (x : W) :\n      Integrable (fun v : V ↦ e (-L v x) • g v) μ := by\n    have c : Continuous fun v ↦ e (-L v x) := he.comp (hL.comp (.prodMk_left _)).neg\n    simp_rw [← integrable_norm_iff (c.aestronglyMeasurable.smul hg.1), Circle.norm_smul]\n    exact hg.norm\n  -- then use it for both directions\n  refine ⟨fun hf ↦ ?_, fun hf ↦ aux hf w⟩\n  have := aux hf (-w)\n  simp_rw [← mul_smul (e _) (e _) (f _), ← e.map_add_eq_mul, LinearMap.map_neg, neg_add_cancel,\n    e.map_zero_eq_one, one_smul] at this -- the `(e _)` speeds up elaboration considerably\n  exact this\n\n","line":124,"end_line":139,"proof_status":"formalized"}
{"id":"a0000000173","label":"a0000000173","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For a multiindex \\(\\alpha \\) and a distribution \\(F\\) define the distribution \\[ \\partial ^\\alpha F (\\phi ) = (-1)^{|\\alpha |}(F \\partial ^\\alpha \\phi ) \\]","doc_url":null,"latex_link":"sect0001.html#a0000000173","lean_link":null,"lean_decl_name":"a0000000173","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000191","label":"a0000000191","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The fourier transformation is a continuous bijection \\begin{align*} \\mathcal{S}& \\to \\mathcal{S}\\\\ \\phi & \\mapsto \\hat\\phi = (\\xi \\mapsto \\int _{\\mathbb {R}^d} \\phi (x) e^{-2\\pi i x \\xi } \\ \\mathrm{d} x) \\end{align*}","doc_url":null,"latex_link":"sect0001.html#a0000000191","lean_link":null,"lean_decl_name":"a0000000191","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000195","label":"a0000000195","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Define \\[ \\hat F (\\phi ) = F (\\hat\\phi ) \\]","doc_url":null,"latex_link":"sect0001.html#a0000000195","lean_link":null,"lean_decl_name":"a0000000195","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:fourier-cont","label":"fourier-cont","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["def:fourier-transform"],"nlp_text":"Let \\(f\\in L^1(V,E)\\). Then \\(\\widehat f\\) is continuous.","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_continuous","latex_link":"chap-plancherel.html#lem:fourier-cont","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_continuous","lean_decl_name":"VectorFourier.fourierIntegral_continuous","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Fourier/FourierTransform.lean","lean4_code":"theorem fourierIntegral_continuous [FirstCountableTopology W] (he : Continuous e)\n    (hL : Continuous fun p : V × W ↦ L p.1 p.2) {f : V → E} (hf : Integrable f μ) :\n    Continuous (fourierIntegral e μ L f) := by\n  apply continuous_of_dominated\n  · exact fun w ↦ ((fourierIntegral_convergent_iff he hL w).2 hf).1\n  · exact fun w ↦ ae_of_all _ fun v ↦ le_of_eq (Circle.norm_smul _ _)\n  · exact hf.norm\n  · refine ae_of_all _ fun v ↦ (he.comp ?_).smul continuous_const\n    exact (hL.comp (.prodMk_right _)).neg\n\nend Continuous\n\nsection Fubini\n\nvariable [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V]\n  [TopologicalSpace W] [MeasurableSpace W] [BorelSpace W]\n  {e : AddChar 𝕜 𝕊} {μ : Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜}\n  {ν : Measure W} [SigmaFinite μ] [SigmaFinite ν] [SecondCountableTopology V]\n\nvariable [CompleteSpace E] [CompleteSpace F]\n\n/-- The Fourier transform satisfies `∫ 𝓕 f * g = ∫ f * 𝓕 g`, i.e., it is self-adjoint.\nVersion where the multiplication is replaced by a general bilinear form `M`. -/\n","line":151,"end_line":173,"proof_status":"formalized"}
{"id":"lem:fourier-multiplication","label":"fourier-multiplication","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier-cont"],"nlp_text":"Let \\(f,g\\in L^1(V,E)\\). Then \\[ \\int _WM(\\widehat f(w),g(w))\\, d\\nu (w)=\\int _VM(f(v),\\widehat g(v))\\, d\\mu (v). \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.integral_bilin_fourierIntegral_eq_flip","latex_link":"chap-plancherel.html#lem:fourier-multiplication","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.integral_bilin_fourierIntegral_eq_flip","lean_decl_name":"VectorFourier.integral_bilin_fourierIntegral_eq_flip","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Fourier/FourierTransform.lean","lean4_code":"theorem integral_bilin_fourierIntegral_eq_flip\n    {f : V → E} {g : W → F} (M : E →L[ℂ] F →L[ℂ] G) (he : Continuous e)\n    (hL : Continuous fun p : V × W ↦ L p.1 p.2) (hf : Integrable f μ) (hg : Integrable g ν) :\n    ∫ ξ, M (fourierIntegral e μ L f ξ) (g ξ) ∂ν =\n      ∫ x, M (f x) (fourierIntegral e ν L.flip g x) ∂μ := by\n  by_cases hG : CompleteSpace G; swap; · simp [integral, hG]\n  calc\n  _ = ∫ ξ, M.flip (g ξ) (∫ x, e (-L x ξ) • f x ∂μ) ∂ν := rfl\n  _ = ∫ ξ, (∫ x, M.flip (g ξ) (e (-L x ξ) • f x) ∂μ) ∂ν := by\n    congr with ξ\n    apply (ContinuousLinearMap.integral_comp_comm _ _).symm\n    exact (fourierIntegral_convergent_iff he hL _).2 hf\n  _ = ∫ x, (∫ ξ, M.flip (g ξ) (e (-L x ξ) • f x) ∂ν) ∂μ := by\n    rw [integral_integral_swap]\n    have : Integrable (fun (p : W × V) ↦ ‖M‖ * (‖g p.1‖ * ‖f p.2‖)) (ν.prod μ) :=\n      (hg.norm.mul_prod hf.norm).const_mul _\n    apply this.mono\n    · -- This proof can be golfed but becomes very slow; breaking it up into steps\n      -- speeds up compilation.\n      change AEStronglyMeasurable (fun p : W × V ↦ (M (e (-(L p.2) p.1) • f p.2) (g p.1))) _\n      have A : AEStronglyMeasurable (fun (p : W × V) ↦ e (-L p.2 p.1) • f p.2) (ν.prod μ) := by\n        refine (Continuous.aestronglyMeasurable ?_).smul hf.1.comp_snd\n        exact he.comp (hL.comp continuous_swap).neg\n      have A' : AEStronglyMeasurable (fun p ↦ (g p.1, e (-(L p.2) p.1) • f p.2) : W × V → F × E)\n        (Measure.prod ν μ) := hg.1.comp_fst.prodMk A\n      have B : Continuous (fun q ↦ M q.2 q.1 : F × E → G) := M.flip.continuous₂\n      apply B.comp_aestronglyMeasurable A' -- `exact` works, but `apply` is 10x faster!\n    · filter_upwards with ⟨ξ, x⟩\n      rw [Function.uncurry_apply_pair, Submonoid.smul_def, (M.flip (g ξ)).map_smul,\n        ← Submonoid.smul_def, Circle.norm_smul, ContinuousLinearMap.flip_apply,\n        norm_mul, norm_norm M, norm_mul, norm_norm, norm_norm, mul_comm (‖g _‖), ← mul_assoc]\n      exact M.le_opNorm₂ (f x) (g ξ)\n  _ = ∫ x, (∫ ξ, M (f x) (e (-L.flip ξ x) • g ξ) ∂ν) ∂μ := by\n      simp only [ContinuousLinearMap.flip_apply, ContinuousLinearMap.map_smul_of_tower,\n      ContinuousLinearMap.coe_smul', Pi.smul_apply, LinearMap.flip_apply]\n  _ = ∫ x, M (f x) (∫ ξ, e (-L.flip ξ x) • g ξ ∂ν) ∂μ := by\n    congr with x\n    apply ContinuousLinearMap.integral_comp_comm\n    apply (fourierIntegral_convergent_iff he _ _).2 hg\n    exact hL.comp continuous_swap\n\n/-- The Fourier transform satisfies `∫ 𝓕 f * g = ∫ f * 𝓕 g`, i.e., it is self-adjoint. -/\n","line":174,"end_line":215,"proof_status":"formalized"}
{"id":"thm:fourier-inversion","label":"fourier-inversion","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:fourier-multiplication","thm:kernel-approximation"],"nlp_text":"Let \\(f:V\\to E\\) be integrable and continuous. Assume \\(\\widehat f\\) is integrable as well. Then \\[ \\mathcal F^{-1}\\mathcal F f=f. \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.Integrable.fourier_inversion","latex_link":"chap-plancherel.html#thm:fourier-inversion","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.Integrable.fourier_inversion","lean_decl_name":"MeasureTheory.Integrable.fourier_inversion","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/b387c548d1dc9cea5c5fcb71e5b3370cca1a04cc/Mathlib/Analysis/Fourier/Inversion.lean","lean4_code":"theorem MeasureTheory.Integrable.fourier_inversion\n    (hf : Integrable f) (h'f : Integrable (𝓕 f)) {v : V}\n    (hv : ContinuousAt f v) : 𝓕⁻ (𝓕 f) v = f v :=\n  tendsto_nhds_unique (Real.tendsto_integral_gaussian_smul hf h'f v)\n    (Real.tendsto_integral_gaussian_smul' hf hv)","line":159,"end_line":163,"proof_status":"formalized"}
{"id":"thm:fourier2-properties","label":"fourier2-properties","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["thm:fourier-inversion","def:invFourier-L2","lem:fourier2-welldef"],"nlp_text":"Plancherel’s Theorem, the inversion formula, and the properties of Lemma 2.5 hold for the Fourier transform on \\(L^2(V,E)\\) as well.","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#thm:fourier2-properties","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:fourier-prop","label":"fourier-prop","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["def:fourier-transform"],"nlp_text":"Lef \\(f,g\\in L^1(V,E)\\), \\(t\\in \\mathbb {R}\\) and \\(a,b\\in \\mathbb {C}\\). The Fourier transform satisfies the following elementary properties: \\(\\mathcal F(af+bg)=a\\mathcal Ff+b\\mathcal Fg\\)(Linearity) \\(\\mathcal F(f(x-t)) = e^{-2\\pi i ty}\\mathcal F f(y)\\)(Shifting) \\(\\mathcal F(f(tx)) = \\frac1{|t|}\\mathcal F f(\\frac yt)\\)(Scaling) If \\(E\\) admits a conjugation, then \\(\\mathcal F(\\overline{f(x)}) = \\overline{\\mathcal Ff(-y)}\\)(Conjugation) Define the convolution of \\(f\\) and \\(g\\) w.r.t. a bilinear map \\(M:E\\times E\\to F\\) as \\[ (f\\ast _Mg)(w):=\\int _VM(f(v),g(w-v))\\, d\\mu (v). \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_add","latex_link":"chap-plancherel.html#lem:fourier-prop","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral_add","lean_decl_name":"VectorFourier.fourierIntegral_add","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Fourier/FourierTransform.lean","lean4_code":"theorem fourierIntegral_add (he : Continuous e) (hL : Continuous fun p : V × W ↦ L p.1 p.2)\n    {f g : V → E} (hf : Integrable f μ) (hg : Integrable g μ) :\n    fourierIntegral e μ L (f + g) = fourierIntegral e μ L f + fourierIntegral e μ L g := by\n  ext1 w\n  dsimp only [Pi.add_apply, fourierIntegral]\n  simp_rw [smul_add]\n  rw [integral_add]\n  · exact (fourierIntegral_convergent_iff he hL w).2 hf\n  · exact (fourierIntegral_convergent_iff he hL w).2 hg\n\n/-- The Fourier integral of an `L^1` function is a continuous function. -/\n","line":140,"end_line":150,"proof_status":"formalized"}
{"id":"lem:fourier2-welldef","label":"fourier2-welldef","background_color":"white","border_color":"black","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["def:fourier-L2"],"nlp_text":"This is well-defined: By Lemma 2.15, the limit exists. Further it does not depend on the choice of sequence \\((f_n)_n\\). If \\(f\\in L^1(V,E)\\cap L^2(V,E)\\), this definition agrees with the Fourier transform on \\(L^1(V,E)\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#lem:fourier2-welldef","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"theorems and lemmas","status_from_color":"incomplete","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:fourier-is-l2-linear","label":"fourier-is-l2-linear","background_color":"white","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["thm:fourier2-properties"],"nlp_text":"The Fourier transform induces a continuous linear map \\(L^2(V,E) \\to L^2(V,E)\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.fourierIntegralL2","latex_link":"chap-plancherel.html#thm:fourier-is-l2-linear","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.fourierIntegralL2","lean_decl_name":"MeasureTheory.fourierIntegralL2","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/fpvandoorn/BonnAnalysis/f3acd7b6e64e9949e697d595d495e01dc4c3dbf9/BonnAnalysis/Plancherel.lean","lean4_code":"def fourierIntegralL2 : (V →₂[volume] ℂ) →L[ℝ] (V →₂[volume] ℂ) :=\n  sorry","line":619,"end_line":620,"proof_status":"formalized"}
{"id":"lem:weierstrass-kernel","label":"weierstrass-kernel","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(K_\\delta (v)=\\delta ^{-n/2}e^{-\\pi |v|^2/\\delta }\\) as in Lemma 2.6. This is a good kernel, called the Weierstrass kernel, satisfying \\[ \\int _{V}K_\\delta (x)\\, dx= 1\\quad \\text{and}\\quad \\int _{|x|{\\gt}\\eta }K_\\delta (x)\\, dx\\xrightarrow {\\delta \\to 0}0\\quad \\text{for all }\\eta {\\gt}0. \\]","doc_url":null,"latex_link":"chap-plancherel.html#lem:weierstrass-kernel","lean_link":null,"lean_decl_name":"weierstrass-kernel","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:kernel-approximation","label":"kernel-approximation","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:weierstrass-kernel"],"nlp_text":"Let \\(f:V\\to E\\) be integrable. Let \\(K_\\delta \\) be the Weierstrass kernel from Lemma 2.7, or indeed any family of functions satisfying the conditions of Lemma 2.7. Then \\[ (K_\\delta \\ast f)(x):=\\int _V K_\\delta (y)f(x-y)\\, d\\mu (y)\\xrightarrow {\\delta \\to 0}f(x) \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/Real.tendsto_integral_gaussian_smul'","latex_link":"chap-plancherel.html#thm:kernel-approximation","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/Real.tendsto_integral_gaussian_smul'","lean_decl_name":"Real.tendsto_integral_gaussian_smul'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/leanprover-community/mathlib4/b387c548d1dc9cea5c5fcb71e5b3370cca1a04cc/Mathlib/Analysis/Fourier/Inversion.lean","lean4_code":"lemma tendsto_integral_gaussian_smul' (hf : Integrable f) {v : V} (h'f : ContinuousAt f v) :\n    Tendsto (fun (c : ℝ) ↦\n      ∫ w : V, ((π * c : ℂ) ^ (finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * c * ‖v - w‖ ^ 2)) • f w)\n    atTop (𝓝 (f v)) := by\n  let φ : V → ℝ := fun w ↦ π ^ (finrank ℝ V / 2 : ℝ) * Real.exp (-π^2 * ‖w‖^2)\n  have A : Tendsto (fun (c : ℝ) ↦ ∫ w : V, (c ^ finrank ℝ V * φ (c • (v - w))) • f w)\n      atTop (𝓝 (f v)) := by\n    apply tendsto_integral_comp_smul_smul_of_integrable'\n    · exact fun x ↦ by positivity\n    · rw [integral_mul_left, GaussianFourier.integral_rexp_neg_mul_sq_norm (by positivity)]\n      nth_rewrite 2 [← pow_one π]\n      rw [← rpow_natCast, ← rpow_natCast, ← rpow_sub pi_pos, ← rpow_mul pi_nonneg,\n        ← rpow_add pi_pos]\n      ring_nf\n      exact rpow_zero _\n    · have A : Tendsto (fun (w : V) ↦ π^2 * ‖w‖^2) (cobounded V) atTop := by\n        rw [tendsto_const_mul_atTop_of_pos (by positivity)]\n        apply (tendsto_pow_atTop two_ne_zero).comp tendsto_norm_cobounded_atTop\n      have B := tendsto_rpow_mul_exp_neg_mul_atTop_nhds_zero (finrank ℝ V / 2) 1\n        zero_lt_one |>.comp A |>.const_mul (π ^ (-finrank ℝ V / 2 : ℝ))\n      rw [mul_zero] at B\n      convert B using 2 with x\n      simp only [neg_mul, one_mul, Function.comp_apply, ← mul_assoc, ← rpow_natCast, φ]\n      congr 1\n      rw [mul_rpow (by positivity) (by positivity), ← rpow_mul pi_nonneg,\n        ← rpow_mul (norm_nonneg _), ← mul_assoc, ← rpow_add pi_pos, mul_comm]\n      congr <;> ring\n    · exact hf\n    · exact h'f\n  have B : Tendsto\n      (fun (c : ℝ) ↦ ∫ w : V, ((c^(1/2:ℝ)) ^ finrank ℝ V * φ ((c^(1/2:ℝ)) • (v - w))) • f w)\n      atTop (𝓝 (f v)) :=\n    A.comp (tendsto_rpow_atTop (by norm_num))\n  apply B.congr'\n  filter_upwards [Ioi_mem_atTop 0] with c (hc : 0 < c)\n  congr with w\n  rw [← coe_smul]\n  congr\n  rw [ofReal_mul, ofReal_mul, ofReal_exp, ← mul_assoc]\n  congr\n  · rw [mul_cpow_ofReal_nonneg pi_nonneg hc.le, ← rpow_natCast, ← rpow_mul hc.le, mul_comm,\n      ofReal_cpow pi_nonneg, ofReal_cpow hc.le]\n    simp [div_eq_inv_mul]\n  · norm_cast\n    simp only [one_div, norm_smul, Real.norm_eq_abs, mul_pow, _root_.sq_abs, neg_mul, neg_inj,\n      ← rpow_natCast, ← rpow_mul hc.le, mul_assoc]\n    norm_num","line":104,"end_line":150,"proof_status":"formalized"}
{"id":"lem:threelines","label":"threelines","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(S\\) be the strip \\(S:=\\{ z \\in \\mathbb {C}\\ | \\ 0 {\\lt} \\mathrm{Re} \\, z {\\lt} 1 \\} \\). Let \\(f: \\overline{S} \\to \\mathbb {C}\\) be a function that is holomorphic on \\(S\\) and continuous and bounded on \\(\\overline{S}\\).Assume \\(M_0, M_1\\) are positive real numbers such that for all values of \\(y\\) in \\(\\mathbb {R}\\), we have \\[ | \\phi (iy) | \\leq M_0 \\ \\qquad | \\phi (1+iy) | \\leq M_1 \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/DiffContOnCl.norm_le_pow_mul_pow","latex_link":"chap-interpolation.html#lem:threelines","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/DiffContOnCl.norm_le_pow_mul_pow","lean_decl_name":"DiffContOnCl.norm_le_pow_mul_pow","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/fpvandoorn/BonnAnalysis/f3acd7b6e64e9949e697d595d495e01dc4c3dbf9/BonnAnalysis/ComplexInterpolation.lean","lean4_code":"theorem DiffContOnCl.norm_le_pow_mul_pow {a b : ℝ} {f : ℂ → ℂ} (hab: a<b)\n    (hf : DiffContOnCl ℂ f { z | z.re ∈ Ioo a b})\n    (h2f : IsBounded (f '' { z | z.re ∈ Icc a b}))\n    {M₀ M₁ : ℝ} (hM₀ : 0 < M₀) (hM₁ : 0 < M₁)\n    (h₀f : ∀ y : ℝ, ‖f (a + I * y)‖ ≤ M₀) (h₁f : ∀ y : ℝ, ‖f (b + I * y)‖ ≤ M₁)\n    {x y t s : ℝ} (ht : 0 ≤ t) (hs : 0 ≤ s) (hx : x = t * a + s * b) (hts : t + s = 1) :\n    ‖f (x + I * y)‖ ≤ M₀ ^ (1-((t-1)*a+s*b)/(b-a)) * M₁ ^ (((t-1)*a+s*b)/(b-a)) := by{\n\n      have hb_sub_a: b - a ≠ 0 := ne_of_gt (by simp [hab])\n\n      have hts'' : s = 1-t := eq_sub_of_add_eq (add_comm t s ▸ hts)\n\n      have hax: x ≥ a := by{\n        simp [hx]\n        rw [eq_sub_of_add_eq hts]\n        ring_nf\n        have : -(s * a) + s * b = s * (b-a) := by ring\n        rw [this]\n        simp only [le_add_iff_nonneg_left, ge_iff_le]\n        rw [mul_nonneg_iff]\n        exact Or.inl ⟨hs, by simp; exact le_of_lt hab⟩\n      }\n\n      -- Essentially same as above with minor tweaks\n      have hxb: b ≥ x := by{\n        simp [hx]\n        rw [hts'']\n        ring_nf\n        have : t * a - t * b = t * (a-b) := by ring\n        rw [this]\n        simp\n        rw [mul_nonpos_iff]\n        exact Or.inl ⟨ht, by simp; exact le_of_lt hab⟩\n      }\n\n      let g : ℂ → ℂ := fun z ↦ f (a + z * (b-a))\n      have hg: DiffContOnCl ℂ g { z | z.re ∈ Ioo 0 1} := by{\n        let h : ℂ → ℂ := fun z ↦ a + z *(b-a)\n        have hcomp: g = f ∘ h := rfl\n        rw [hcomp]\n        apply DiffContOnCl.comp (s:={ z | z.re ∈ Ioo a b}) hf ?_ ?_\n        · simp [h]\n          have : (fun (x:ℂ) ↦ x * (↑b - ↑a) ) = (fun (x:ℂ) ↦ x • ((b:ℂ) - (a:ℂ))) := rfl\n          exact DiffContOnCl.const_add (this ▸ DiffContOnCl.smul_const DiffContOnCl.id _) _\n        · simp [h, MapsTo]\n          refine fun z hz₀ hz₁ ↦ ⟨?_, ?_⟩\n          · apply Real.mul_pos hz₀\n            simp [hab]\n          · calc\n            a + z.re * (b-a) < a + 1 *(b - a) := by gcongr; simp [hab]\n            _ = b := by simp\n      }\n\n      have h2g: IsBounded (g '' { z | z.re ∈ Icc 0 1}) := by{\n        simp only [g]\n        apply IsBounded.subset h2f\n        intro z hz\n        obtain ⟨w, hw₁, hw₂⟩ := hz\n        simp\n        use ↑a + w * (↑b - ↑a)\n        simp\n        simp at hw₁\n        constructor\n        · constructor\n          · rw [mul_nonneg_iff]; left\n            constructor\n            · exact hw₁.1\n            · simp; exact le_of_lt hab\n          · calc\n            a + w.re * (b-a) ≤ a + 1 * (b-a) := by gcongr; simp; exact le_of_lt hab; exact hw₁.2\n            _ = b := by ring\n        · exact hw₂\n      }\n\n      have h₀g : ∀ y : ℝ, ‖g (I * y)‖ ≤ M₀ := by{\n        simp only [g]\n        simp\n        intro y\n        specialize h₀f (y* (b-a))\n        simp at h₀f\n        rw [mul_assoc]\n        exact h₀f\n      }\n\n      have h₁g : ∀ y : ℝ, ‖g (1 + I * y)‖ ≤ M₁ := by{\n        simp only [g]\n        simp\n        intro y\n        specialize h₁f (y * (b-a))\n        simp at h₁f\n        have : ↑a + (1 + I * ↑y) * (↑b - ↑a) = ↑b + I * (↑y * (↑b - ↑a)) := by ring\n        rw [this]\n        exact h₁f\n      }\n\n      let t':= (x-a)/(b-a)\n      let s':= 1 - t'\n      have ht' : 0 ≤ t' := by\n        simp only [t']\n        rw [div_nonneg_iff]\n        refine Or.inl ⟨by simp [hax], by simp; exact le_of_lt hab⟩\n\n      have hs' : 0 ≤ s' := by {\n        simp only [s', t']\n        simp\n        rw [div_le_one]\n        · simp [hxb]\n        · simp [hab]\n      }\n      have hts' : t' + s' = 1 := by simp [s']\n\n      have hgoal := DiffContOnCl.norm_le_pow_mul_pow₀₁ hg h2g hM₀ hM₁ h₀g h₁g ht' hs' hts' (y:=y/(b-a))\n\n      simp [g] at hgoal\n      simp only [t'] at hgoal\n      simp at hgoal\n      rw [add_mul, mul_comm_div, div_self (by norm_cast), mul_assoc, mul_comm_div, div_self (by norm_cast), ← add_assoc] at hgoal\n      simp at hgoal\n\n      have ht'₁: t'=((t - 1) * a + s * b)/(b-a) := by{\n        simp only [t', hx]\n        ring\n      }\n      simp only [s'] at hgoal\n      rw [← ht'₁]\n      exact hgoal\n    }","line":704,"end_line":830,"proof_status":"formalized"}
{"id":"thm:riesz_interpolation","label":"riesz_interpolation","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["lem:threelines"],"nlp_text":"Let \\((X, \\mu )\\) and \\((Y, \\nu )\\) be measure spaces and consider all \\(L^p\\) spaces to be complex valued.Suppose \\(T\\) is a linear map \\(L^{p_0}(X) + L^{p_1}(X) \\to L^{q_0}(Y) + L^{q_1}(Y)\\) that restricts to bounded operators \\(L^{p_0} \\to L^{q_0}\\) and \\(L^{p_1} \\to L^{q_1}\\). Let \\(M_0, M_1\\) be the respective bounds, i.e., \\[ \\begin{cases} ||Tf||_{L^{q_0}} \\leq M_0 ||f||_{L^{p_0}} \\\\ ||Tf||_{L^{q_1}} \\leq M_1 ||f||_{L^{p_1}} \\end{cases} \\]","doc_url":null,"latex_link":"chap-interpolation.html#thm:riesz_interpolation","lean_link":null,"lean_decl_name":"riesz_interpolation","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:hausdorff_young_dual","label":"hausdorff_young_dual","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":["thm:riesz_interpolation"],"nlp_text":"For \\(1\\leq p \\leq 2\\) and \\(q\\) conjugate exponent to \\(p\\), we have \\[ ||T' \\{ a_n\\} ||_{L^q} \\leq ||\\{ a_n\\} ||_{L^p} \\]","doc_url":null,"latex_link":"chap-interpolation.html#lem:hausdorff_young_dual","lean_link":null,"lean_decl_name":"hausdorff_young_dual","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000225","label":"a0000000225","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(F\\) is a fundamental solution of \\(L = \\frac{\\partial }{\\partial t} - \\Delta _x\\).","doc_url":null,"latex_link":"sect0001.html#a0000000225","lean_link":null,"lean_decl_name":"a0000000225","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000164","label":"a0000000164","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f,g \\in L^1_{loc}(\\Omega )\\) such that \\(\\Lambda _f = \\Lambda _g\\). Then \\(f = g\\) almost everywhere.","doc_url":null,"latex_link":"sect0001.html#a0000000164","lean_link":null,"lean_decl_name":"a0000000164","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000189","label":"a0000000189","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"All \\(\\partial ^\\alpha F\\) are tempered. Let \\(\\psi \\in C^\\infty \\) be slowly increasing, i.e. for each \\(\\alpha \\) exists \\(N_\\alpha \\) such that \\(\\partial _x^\\alpha \\psi (x) = O(|x|^{N_\\alpha })\\). Then \\(\\psi F\\), defined by \\((\\psi F)(\\phi )(F(\\psi \\phi )\\) is tempered.","doc_url":null,"latex_link":"sect0001.html#a0000000189","lean_link":null,"lean_decl_name":"a0000000189","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:snorm_eq_sSup_snorm","label":"snorm_eq_sSup_snorm","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(p\\) and \\(q\\) be real conjugate exponents. Let \\(f\\) be measurable. Then \\[ \\| f\\| _{L^q} = \\sup _{\\| g\\| _{L^p} \\leq 1, \\ g \\text{ simple}} \\| fg \\| _{L^1}. \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.snorm_eq_sSup_snorm","latex_link":"chap-interpolation.html#lem:snorm_eq_sSup_snorm","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/MeasureTheory.snorm_eq_sSup_snorm","lean_decl_name":"MeasureTheory.snorm_eq_sSup_snorm","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"source_repo":"remote_github","source_path":"https://raw.githubusercontent.com/fpvandoorn/BonnAnalysis/f3acd7b6e64e9949e697d595d495e01dc4c3dbf9/BonnAnalysis/ComplexInterpolation.lean","lean4_code":"lemma snorm_eq_sSup_snorm (p q : ℝ≥0) (hpq : NNReal.IsConjExponent p q) (f : α → ℂ)\n(hf : Measurable f) (hf' : snorm f p μ ≠ ∞) (hf0 : snorm f p μ ≠ 0):\nsnorm f p μ = sSup {snorm (f * (g.1 : α → ℂ)) 1 μ | g : Lp.simpleLe1 ℂ q μ} := by\n  apply le_antisymm ?_\n  . apply sSup_le\n    rintro b ⟨g, hg⟩\n    rw [← hg]\n    calc snorm (f * (g.1 : α → ℂ)) 1 μ = ∫⁻ a, ‖f a‖₊ * ‖g.1 a‖₊ ∂μ := by simp [snorm, snorm']\n    _ = ∫⁻ a, ((‖f ·‖₊) * (‖(g.1 : α → ℂ) ·‖₊)) a ∂μ := lintegral_congr (by\n      simp only [Pi.mul_apply, ENNReal.coe_mul, implies_true])\n    _ ≤ snorm f p μ * snorm g.1 q μ  := by\n      simp only [snorm, coe_toReal, snorm', ENNReal.coe_eq_zero,\n      hpq.ne_zero, ↓reduceIte, coe_ne_top, hpq.symm.ne_zero]\n      apply ENNReal.lintegral_mul_le_Lp_mul_Lq _ (NNReal.IsConjExponent.coe hpq\n      ) hf.aemeasurable.ennnorm (AEMeasurable.ennnorm (SimpleFunc.aemeasurable _))\n    _ ≤ snorm f p μ := mul_le_of_le_one_right (by positivity) g.2\n  . rcases eq_or_ne (snorm f p μ) 0 with hf0 | hf0\n    . simp [hf0]\n    . set g : α → ℝ≥0∞ :=\n      ENNReal.ofNNReal ∘ fun a ↦ ‖f a‖₊ ^ ((p : ℝ) - 1) * (snorm f p μ).toNNReal ^ (1 - (p : ℝ))\n      have g_meas : Measurable g:=\n        ((hf.nnnorm.pow_const _).mul_const _).coe_nnreal_ennreal\n      have g_norm : (∫⁻ (a : α), g a ^ (q : ℝ) ∂μ) ^ (q : ℝ)⁻¹ = 1 := by\n        simp only [g, Function.comp_apply, ENNReal.coe_mul]\n        calc (∫⁻ (a : α), ↑(‖f a‖₊ ^ ((p : ℝ) - 1) * (snorm f p μ).toNNReal ^\n        (1 - (p : ℝ))) ^ (q : ℝ) ∂μ) ^ (q : ℝ)⁻¹ = (∫⁻ (a : α), (↑‖f a‖₊ ^ (((p : ℝ) - 1) * q)\n        * (snorm f p μ).toNNReal ^ ((1 - (p : ℝ)) * q)) ∂μ) ^ (q : ℝ)⁻¹ := by {\n          congr 1; apply lintegral_congr (by\n          intro a; simp; rw [ENNReal.mul_rpow_of_nonneg (hz := by norm_num),\n        ← ENNReal.coe_rpow_of_nonneg _ (by norm_num; exact hpq.one_le),\n        ← ENNReal.coe_rpow_of_ne_zero (by rw [ENNReal.toNNReal_ne_zero]; exact ⟨hf0, hf'⟩),\n        ← ENNReal.rpow_mul, ← ENNReal.rpow_mul])}\n        _ = 1 := by\n          rw [lintegral_mul_const'' _ ((AEMeasurable.pow_const hf.aemeasurable.ennnorm) _),\n          (isConjExponent_coe.mpr hpq).sub_one_mul_conj, ENNReal.coe_toNNReal hf',\n          ← snorm_eq_lintegral_rpow_nnnorm' _ _ hpq.ne_zero, ← ENNReal.rpow_add _ _ hf0 hf',\n          sub_mul (1 : ℝ), (isConjExponent_coe.mpr hpq).mul_eq_add, one_mul,\n          sub_add_cancel_right, add_right_neg, ENNReal.rpow_zero, ENNReal.one_rpow]\n      have f_norm : ∫⁻ (a : α), ‖f a‖₊ * (g a) ∂ μ = snorm f p μ := by\n        simp only [g, Function.comp_apply]\n        calc ∫⁻ (a : α), ↑‖f a‖₊ * (↑(‖f a‖₊ ^ ((p : ℝ) - 1))\n          * ↑((snorm f p μ).toNNReal ^ (1 - (p : ℝ)))) ∂μ\n          = ∫⁻ (a : α), ↑‖f a‖₊ ^ (p : ℝ) * ↑((snorm f p μ).toNNReal ^ (1 - (p : ℝ))) ∂μ :=\n              lintegral_congr (by\n              intro _; rw [← mul_assoc]; congr\n              rw [← ENNReal.rpow_one ↑‖f _‖₊, ← ENNReal.coe_rpow_of_nonneg _ (by\n                norm_num; exact hpq.one_le),\n                ← ENNReal.rpow_add_of_nonneg _ _ (by norm_num) (by norm_num; exact hpq.one_le),\n                ENNReal.rpow_one, add_sub_cancel]\n              )\n        _ = (∫⁻ (a : α), ↑‖f a‖₊ ^ (p : ℝ) ∂ μ) * ↑((snorm f p μ).toNNReal ^ (1 - (p : ℝ))) := by\n          rw [lintegral_mul_const'' _ ((AEMeasurable.pow_const hf.aemeasurable.ennnorm) _)]\n        _ = snorm f p μ := by\n          rw [← snorm_eq_lintegral_rpow_nnnorm' _ _ hpq.ne_zero,\n              ← ENNReal.coe_rpow_of_ne_zero (ENNReal.toNNReal_ne_zero.mpr ⟨hf0, hf'⟩),\n              ENNReal.coe_toNNReal hf', ← ENNReal.rpow_add _ _ hf0 hf',\n              add_sub_cancel, ENNReal.rpow_one]\n      calc snorm f p μ = ∫⁻ (a : α), ‖f a‖₊ * (g a) ∂ μ := f_norm.symm\n      _ = ⨆ n, ∫⁻ a, ↑‖f a‖₊  * (eapprox g n a) ∂μ := by\n        apply mul_lintegral_eq_iSup_mul_eapprox_lintegral (f := fun a ↦ (‖f a‖₊ : ℝ≥0∞))\n          hf.ennnorm g_meas\n      _ ≤ sSup {∫⁻ (a : α), ‖f a‖₊ * (g.1 a) ∂ μ | g : {f : SimpleFunc α ℝ≥0∞ //\n      (∫⁻ a, (f a) ^ (q : ℝ) ∂ μ) ^ (q : ℝ)⁻¹ ≤ 1} } := by\n        apply iSup_le; intro n; apply le_sSup\n        simp only [Subtype.exists, exists_prop, mem_setOf_eq]\n        use eapprox g n\n        exact ⟨by\n          apply le_trans ?_ g_norm.le\n          gcongr\n          rw [← iSup_eapprox_apply _ g_meas]\n          apply le_iSup _ n, rfl⟩\n      _ ≤ sSup {∫⁻ (a : α), ‖f a‖₊ * (g.1 a) ∂ μ | g : {f : SimpleFunc α ℝ≥0 //\n       (∫⁻ a, (f a) ^ (q : ℝ) ∂ μ) ^ (q : ℝ)⁻¹ ≤ 1} } := by\n        gcongr\n        rintro x ⟨h, hh⟩\n        have ae := ae_lt_top_of_LpNorm_ne_top hpq.symm.ne_zero (SimpleFunc.measurable _)\n          (ne_top_of_le_ne_top one_ne_top h.2)\n        have : (∫⁻ (a : α), ↑(h.1 a).toNNReal ^ (q : ℝ) ∂μ) ^ (q : ℝ)⁻¹\n          = (∫⁻ (a : α), (h.1 a) ^ (q : ℝ) ∂μ) ^ (q : ℝ)⁻¹ := by\n          congr 1\n          apply lintegral_congr_ae\n          rw [Filter.EventuallyEq, Filter.Eventually, ← Filter.exists_mem_subset_iff]\n          use {a | h.1 a < ⊤}\n          exact ⟨ae, by simp; intro a ha; rw [ENNReal.coe_toNNReal ha.ne_top]⟩\n        use ⟨SimpleFunc.map ENNReal.toNNReal h.1, by simp [this, h.2]⟩\n        simp [← hh]\n        apply lintegral_congr_ae\n        rw [Filter.EventuallyEq, Filter.Eventually, ← Filter.exists_mem_subset_iff]\n        use {a | h.1 a < ⊤}\n        exact ⟨ae, by simp; intro a ha; rw [ENNReal.coe_toNNReal ha.ne_top]⟩\n      _ ≤ sSup {snorm (f * (g.1 : α → ℂ)) 1 μ | g : Lp.simpleLe1 ℂ q μ} := by\n        gcongr; rintro x ⟨h, hh⟩\n        use toLpSimpLe1 q hpq.symm.ne_zero _ h.2\n        convert hh\n        simp [snorm, snorm', toLpSimpLe1]","line":913,"end_line":1007,"proof_status":"formalized"}
{"id":"a0000000151","label":"a0000000151","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(F \\in \\mathcal{D}'(\\Omega ) , \\psi \\in D\\). The following two distributions coincide: The distribution determined by the smooth function \\(x \\mapsto F(\\tau _x (\\psi ^R))\\). The distribution \\(\\phi \\mapsto F(\\psi ^R * \\phi )\\). We write this distribution as \\(F * \\psi \\).","doc_url":null,"latex_link":"sect0001.html#a0000000151","lean_link":null,"lean_decl_name":"a0000000151","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000176","label":"a0000000176","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"For \\(F \\in D', \\phi \\in D \\) , We have \\[ \\partial ^\\alpha (F * \\phi ) = (\\partial ^\\alpha F) * \\phi = F * \\partial ^\\alpha \\phi \\]","doc_url":null,"latex_link":"sect0001.html#a0000000176","lean_link":null,"lean_decl_name":"a0000000176","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:cont","label":"cont","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Convolution \\(F * \\psi \\) is continuous in both variables.","doc_url":null,"latex_link":"sect0001.html#lemma:cont","lean_link":null,"lean_decl_name":"cont","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000185","label":"a0000000185","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We have a continuous inclusions \\(\\mathcal{D}\\subset \\mathcal{S}\\), hence \\(\\mathcal{S}' \\subset \\mathcal{D}'(\\mathbb {R}^d)\\). Moreover, this inclusion is dense. Hence being tempered is a property of distributions.","doc_url":null,"latex_link":"sect0001.html#a0000000185","lean_link":null,"lean_decl_name":"a0000000185","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"thm:maximum_modulus","label":"maximum_modulus","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(U\\) be a connected open set in a complex normed space \\(E\\). Let \\(f:E\\to F\\) be a function that complex differentiable on \\(U\\) and continuous on \\(\\bar{U}\\).If \\(|f(z)|\\) takes its maximum on a point \\(u\\in U\\), then it must be constant on \\(\\bar{U}\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/Analysis.Complex.norm_eqOn_closure_of_isPreconnected_of_isMaxOn","latex_link":"chap-interpolation.html#thm:maximum_modulus","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/Analysis.Complex.norm_eqOn_closure_of_isPreconnected_of_isMaxOn","lean_decl_name":"Analysis.Complex.norm_eqOn_closure_of_isPreconnected_of_isMaxOn","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Complex/AbsMax.lean","lean4_code":"theorem norm_eqOn_closure_of_isPreconnected_of_isMaxOn {f : E → F} {U : Set E} {c : E}\n    (hc : IsPreconnected U) (ho : IsOpen U) (hd : DiffContOnCl ℂ f U) (hcU : c ∈ U)\n    (hm : IsMaxOn (norm ∘ f) U c) : EqOn (norm ∘ f) (const E ‖f c‖) (closure U) :=\n  (norm_eqOn_of_isPreconnected_of_isMaxOn hc ho hd.differentiableOn hcU hm).of_subset_closure\n    hd.continuousOn.norm continuousOn_const subset_closure Subset.rfl\n\nsection StrictConvex\n\n/-!\n### The case of a strictly convex codomain\n\nIf the codomain `F` is a strictly convex space, then we can claim equalities like `f w = f z`\ninstead of `‖f w‖ = ‖f z‖`.\n\nInstead of repeating the proof starting with lemmas about integrals, we apply a corresponding lemma\nabove twice: for `f` and for `(f · + f c)`.  Then we have `‖f w‖ = ‖f z‖` and\n`‖f w + f z‖ = ‖f z + f z‖`, thus `‖f w + f z‖ = ‖f w‖ + ‖f z‖`. This is only possible if\n`f w = f z`, see `eq_of_norm_eq_of_norm_add_eq`.\n-/\n\nvariable [StrictConvexSpace ℝ F]\n\n/-- **Maximum modulus principle** on a connected set. Let `U` be a (pre)connected open set in a\ncomplex normed space.  Let `f : E → F` be a function that is complex differentiable on `U`. Suppose\nthat `‖f x‖` takes its maximum value on `U` at `c ∈ U`. Then `f x = f c` for all `x ∈ U`.\n\nTODO: change assumption from `IsMaxOn` to `IsLocalMax`. -/\n","line":246,"end_line":272,"proof_status":"formalized"}
{"id":"lem:L12-dense","label":"L12-dense","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(L^1(V,E)\\cap L^2(V,E)\\) is dense in \\(L^2(V,E)\\).","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","latex_link":"chap-plancherel.html#lem:L12-dense","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/","lean_decl_name":"","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":false,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"def:fourier-transform","label":"fourier-transform","background_color":"#B0ECA3","border_color":"green","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f\\in L^1(V,E)\\). Its Fourier transform (w.r.t. \\(L\\)) is the function \\(\\mathcal Ff=\\widehat f:W\\to E\\) given by \\[ \\mathcal Ff(w):=\\widehat f(w):=\\int _V e^{-2\\pi i L(v,w)}f(v)\\, d\\mu . \\]","doc_url":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral","latex_link":"chap-plancherel.html#def:fourier-transform","lean_link":"https://fpvandoorn.github.io/BonnAnalysis/docs/find/#doc/VectorFourier.fourierIntegral","lean_decl_name":"VectorFourier.fourierIntegral","kind_description":"definitions","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":true,"source_repo":"mathlib","source_path":"./Mathlib/Analysis/Fourier/FourierTransform.lean","lean4_code":"def fourierIntegral (e : AddChar 𝕜 𝕊) (μ : Measure V) (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E)\n    (w : W) : E :=\n  ∫ v, e (-L v w) • f v ∂μ\n\n","line":77,"end_line":80,"proof_status":"formalized"}
{"id":"a0000000139","label":"a0000000139","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"\\(\\mathcal{D}(\\Omega ) = C_c^\\infty (\\Omega )\\) is the set of test functions together with a topology determined by its converging sequences : \\(\\phi _n \\to \\phi \\) in \\(\\mathcal{D}(\\Omega )\\) if there exists a compact subset \\(K \\subset \\Omega \\) such that \\(\\operatorname{Supp}(\\phi _n) \\subset K\\). For all multiindices \\(\\alpha \\) we have \\(\\partial _\\alpha \\phi _n \\to \\partial _\\alpha \\phi \\) in uniformly. Convention: if \\(x \\in \\mathbb {R}^d \\setminus \\Omega \\), then \\(\\phi (x) := 0\\).\\(\\mathcal{D}'(\\Omega )\\) is the topological dual space, i.e. the space of continuous linear functionals \\(D(\\Omega ) \\to \\mathbb {C}\\) with the weak-*-convergence, i.e. pointwise convergence.","doc_url":null,"latex_link":"sect0001.html#a0000000139","lean_link":null,"lean_decl_name":"a0000000139","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000182","label":"a0000000182","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Consider the increasing sequence of norms on \\(C^\\infty (\\Omega )\\) defined by \\[ \\| \\phi \\| _N = \\sup \\{ | x^\\beta (\\partial _x^\\alpha \\phi (x)) | \\mid x \\in \\mathbb {R}^d , |\\alpha |,\\| \\beta \\| \\le N\\} \\]","doc_url":null,"latex_link":"sect0001.html#a0000000182","lean_link":null,"lean_decl_name":"a0000000182","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000201","label":"a0000000201","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"A fundamental solution of \\(L\\) is a distribution \\(F\\) such that \\(L(F) = \\delta \\)","doc_url":null,"latex_link":"sect0001.html#a0000000201","lean_link":null,"lean_decl_name":"a0000000201","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000207","label":"a0000000207","background_color":"#A3D6FF","border_color":"blue","kind":"box","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The characteristic polynomial of \\(L\\) is \\[ P(\\xi ) = \\sum _{|\\alpha |\\le m} a_\\alpha (2 \\pi i \\xi )^\\alpha \\]","doc_url":null,"latex_link":"sect0001.html#a0000000207","lean_link":null,"lean_decl_name":"a0000000207","kind_description":"definitions","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:slowlyInc","label":"slowlyInc","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(f \\in L^1_{loc}(\\mathbb {R}^d)\\) such that there exists \\(N \\ge 0\\) with \\[ \\int _{|x| {\\lt} R} |f(x)| \\ \\mathrm{d} x = O(R^N) , \\text{ as } R \\to \\infty \\]","doc_url":null,"latex_link":"sect0001.html#lemma:slowlyInc","lean_link":null,"lean_decl_name":"slowlyInc","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000193","label":"a0000000193","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"We have \\[ \\Lambda _{\\hat\\psi }(\\phi ) = \\int _{\\mathbb {R}^d} \\hat\\psi (x) \\phi (x) \\ \\mathrm{d} x = \\int _{\\mathbb {R}^d} \\psi (x) \\hat\\phi (x) \\ \\mathrm{d} x = \\Lambda _{\\psi }(\\hat\\phi ) \\]","doc_url":null,"latex_link":"sect0001.html#a0000000193","lean_link":null,"lean_decl_name":"a0000000193","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lemma:distribCommInt","label":"distribCommInt","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\(\\phi \\in C_c^{\\infty }(\\Omega \\times \\Omega )\\). Then for any \\(F \\in D'(\\Omega )\\) we have \\[ F \\left(\\int _\\Omega \\phi (x,\\_ ) \\ \\mathrm{d} x \\right) = \\int _\\Omega F(\\phi (x ,\\_ )) \\ \\mathrm{d} x \\]","doc_url":null,"latex_link":"sect0001.html#lemma:distribCommInt","lean_link":null,"lean_decl_name":"distribCommInt","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000188","label":"a0000000188","background_color":"white","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(F \\in \\mathcal{D}'\\) has compact support then it is tempered: Choose \\(\\eta \\in \\mathcal{D}\\) such that \\(\\eta |_U = 1\\) on some neighborhood \\(U \\supset \\operatorname{Supp}(F)\\). Then \\(F(\\eta \\phi ) = F(\\phi )\\) for all \\(\\phi \\in \\mathcal{D}\\) and \\(\\phi \\mapsto F (\\eta \\phi )\\) defines a continuous functional on \\(\\mathcal{S}\\). It does not depend on the choice of \\(\\eta \\), because given another such \\(\\eta '\\) and any \\(\\phi \\in \\mathcal{S}\\) we have \\(\\operatorname{Supp}((\\eta - \\eta ') \\cdot \\phi ) \\cap \\operatorname{Supp}(F) = \\varnothing \\).","doc_url":null,"latex_link":"sect0001.html#a0000000188","lean_link":null,"lean_decl_name":"a0000000188","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"proposition:deltaInClosure","label":"deltaInClosure","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"There exists a sequence \\(\\psi _n \\in C_c^{\\infty }(\\Omega )\\) such that \\(\\Lambda \\psi _n \\to \\delta _0\\) in \\(D'(\\Omega )\\).","doc_url":null,"latex_link":"sect0001.html#proposition:deltaInClosure","lean_link":null,"lean_decl_name":"deltaInClosure","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000214","label":"a0000000214","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"If \\(d = 2\\), the function \\(F := 1/(2\\pi ) \\log |x| \\in L^1_{loc}\\) is a fundamental solution of \\(\\Delta \\)","doc_url":null,"latex_link":"sect0001.html#a0000000214","lean_link":null,"lean_decl_name":"a0000000214","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"a0000000202","label":"a0000000202","background_color":"#A3D6FF","border_color":"blue","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"The operator \\[ f \\mapsto T(f) := F * f \\]","doc_url":null,"latex_link":"sect0001.html#a0000000202","lean_link":null,"lean_decl_name":"a0000000202","kind_description":"theorems and lemmas","status_from_color":"ready_to_formalize","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}
{"id":"lem:hoelder'","label":"hoelder'","background_color":"#1CAC78","border_color":"green","kind":"ellipse","dep_graph_url":"https://florisvandoorn.com/BonnAnalysis/blueprint/dep_graph_document.html","dependencies":[],"nlp_text":"Let \\((X, \\mu )\\) be a measure space and \\(0{\\lt} p_0 {\\lt}p_1\\leq \\infty \\), \\(0 \\le p \\le \\infty \\). Assume we have \\(t \\ge 0\\) such that \\[ \\frac{1}{p} = \\frac{1-t}{p_0} + \\frac{t}{p_1} \\]","doc_url":null,"latex_link":"chap-interpolation.html#lem:hoelder'","lean_link":null,"lean_decl_name":"hoelder'","kind_description":"theorems and lemmas","status_from_color":"formalized","lookup_in_mathlib4v4.22.0":true,"found_in_mathlib4v4.22.0":false,"proof_status":"nlp_only"}