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Found 63 declarations mentioning MeasureTheory.convolution.

• MeasureTheory.convolution_mul 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {x : G} [NontriviallyNormedField 𝕜] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [Sub G] [NormedSpace ℝ 𝕜] {f g : G → 𝕜} : MeasureTheory.convolution f g (ContinuousLinearMap.mul 𝕜 𝕜) μ x = ∫ (t : G), f t * g (x - t) ∂μ
• MeasureTheory.convolution_lsmul 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {F : Type uF} [NormedAddCommGroup F] {x : G} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [Sub G] {f : G → 𝕜} {g : G → F} : MeasureTheory.convolution f g (ContinuousLinearMap.lsmul 𝕜 𝕜) μ x = ∫ (t : G), f t • g (x - t) ∂μ
• MeasureTheory.convolution_mul_swap 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {x : G} [NontriviallyNormedField 𝕜] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [MeasurableNeg G] [MeasurableAdd G] [NormedSpace ℝ 𝕜] {f g : G → 𝕜} : MeasureTheory.convolution f g (ContinuousLinearMap.mul 𝕜 𝕜) μ x = ∫ (t : G), f (x - t) * g t ∂μ
• MeasureTheory.dist_convolution_le 📋 Mathlib.Analysis.Convolution
  {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [SeminormedAddCommGroup G] [BorelSpace G] [SecondCountableTopology G] [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] [NormedSpace ℝ E'] [CompleteSpace E'] {f : G → ℝ} {x₀ : G} {R ε : ℝ} {z₀ : E'} (hε : 0 ≤ ε) (hf : Function.support f ⊆ Metric.ball 0 R) (hnf : ∀ (x : G), 0 ≤ f x) (hintf : ∫ (x : G), f x ∂μ = 1) (hmg : MeasureTheory.AEStronglyMeasurable g μ) (hg : ∀ x ∈ Metric.ball x₀ R, dist (g x) z₀ ≤ ε) : dist (MeasureTheory.convolution f g (ContinuousLinearMap.lsmul ℝ ℝ) μ x₀) z₀ ≤ ε
• MeasureTheory.convolution_lsmul_swap 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {F : Type uF} [NormedAddCommGroup F] {x : G} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [MeasurableNeg G] [MeasurableAdd G] {f : G → 𝕜} {g : G → F} : MeasureTheory.convolution f g (ContinuousLinearMap.lsmul 𝕜 𝕜) μ x = ∫ (t : G), f (x - t) • g t ∂μ
• MeasureTheory.convolution_tendsto_right 📋 Mathlib.Analysis.Convolution
  {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [SeminormedAddCommGroup G] [BorelSpace G] [SecondCountableTopology G] [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] [NormedSpace ℝ E'] [CompleteSpace E'] {ι : Type u_1} {g : ι → G → E'} {l : Filter ι} {x₀ : G} {z₀ : E'} {φ : ι → G → ℝ} {k : ι → G} (hnφ : ∀ᶠ (i : ι) in l, ∀ (x : G), 0 ≤ φ i x) (hiφ : ∀ᶠ (i : ι) in l, ∫ (x : G), φ i x ∂μ = 1) (hφ : Filter.Tendsto (fun n => Function.support (φ n)) l (nhds 0).smallSets) (hmg : ∀ᶠ (i : ι) in l, MeasureTheory.AEStronglyMeasurable (g i) μ) (hcg : Filter.Tendsto (Function.uncurry g) (l ×ˢ nhds x₀) (nhds z₀)) (hk : Filter.Tendsto k l (nhds x₀)) : Filter.Tendsto (fun i => MeasureTheory.convolution (φ i) (g i) (ContinuousLinearMap.lsmul ℝ ℝ) μ (k i)) l (nhds z₀)
• MeasureTheory.convolution_mono_right_of_nonneg 📋 Mathlib.Analysis.Convolution
  {G : Type uG} {x : G} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] {f g g' : G → ℝ} (hfg' : MeasureTheory.ConvolutionExistsAt f g' x (ContinuousLinearMap.lsmul ℝ ℝ) μ) (hf : ∀ (x : G), 0 ≤ f x) (hg : ∀ (x : G), g x ≤ g' x) (hg' : ∀ (x : G), 0 ≤ g' x) : MeasureTheory.convolution f g (ContinuousLinearMap.lsmul ℝ ℝ) μ x ≤ MeasureTheory.convolution f g' (ContinuousLinearMap.lsmul ℝ ℝ) μ x
• MeasureTheory.convolution_mono_right 📋 Mathlib.Analysis.Convolution
  {G : Type uG} {x : G} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [AddGroup G] {f g g' : G → ℝ} (hfg : MeasureTheory.ConvolutionExistsAt f g x (ContinuousLinearMap.lsmul ℝ ℝ) μ) (hfg' : MeasureTheory.ConvolutionExistsAt f g' x (ContinuousLinearMap.lsmul ℝ ℝ) μ) (hf : ∀ (x : G), 0 ≤ f x) (hg : ∀ (x : G), g x ≤ g' x) : MeasureTheory.convolution f g (ContinuousLinearMap.lsmul ℝ ℝ) μ x ≤ MeasureTheory.convolution f g' (ContinuousLinearMap.lsmul ℝ ℝ) μ x
• MeasureTheory.posConvolution_eq_convolution_indicator 📋 Mathlib.Analysis.Convolution
  {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [NormedSpace ℝ E] [NormedSpace ℝ E'] [NormedSpace ℝ F] (f : ℝ → E) (g : ℝ → E') (L : E →L[ℝ] E' →L[ℝ] F) (ν : MeasureTheory.Measure ℝ := by volume_tac) [MeasureTheory.NullSingletonClass ν] : MeasureTheory.posConvolution f g L ν = MeasureTheory.convolution ((Set.Ioi 0).indicator f) ((Set.Ioi 0).indicator g) L ν
• MeasureTheory.convolution 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedSpace ℝ F] [Sub G] (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F) (μ : MeasureTheory.Measure G := by volume_tac) : G → F
• MeasureTheory.convolution_zero 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] : MeasureTheory.convolution f 0 L μ = 0
• MeasureTheory.zero_convolution 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] : MeasureTheory.convolution 0 g L μ = 0
• HasCompactSupport.convolution 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [T2Space G] (hcf : HasCompactSupport f) (hcg : HasCompactSupport g) : HasCompactSupport (MeasureTheory.convolution f g L μ)
• MeasureTheory.support_convolution_subset_swap 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] : Function.support (MeasureTheory.convolution f g L μ) ⊆ Function.support g + Function.support f
• HasCompactSupport.continuous_convolution_right 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [BorelSpace G] (hcg : HasCompactSupport g) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : Continuous g) : Continuous (MeasureTheory.convolution f g L μ)
• MeasureTheory.support_convolution_subset 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddCommGroup G] : Function.support (MeasureTheory.convolution f g L μ) ⊆ Function.support f + Function.support g
• BddAbove.continuous_convolution_right_of_integrable 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [BorelSpace G] [FirstCountableTopology G] [SecondCountableTopologyEither G E'] (hbg : BddAbove (Set.range fun x => ‖g x‖)) (hf : MeasureTheory.Integrable f μ) (hg : Continuous g) : Continuous (MeasureTheory.convolution f g L μ)
• MeasureTheory.convolution_congr 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f f' : G → E} {g g' : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant] (h1 : f =ᵐ[μ] f') (h2 : g =ᵐ[μ] g') : MeasureTheory.convolution f g L μ = MeasureTheory.convolution f' g' L μ
• MeasureTheory.Integrable.integrable_convolution 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] [MeasurableAdd₂ G] [MeasurableNeg G] [MeasureTheory.SFinite μ] [μ.IsAddRightInvariant] (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) : MeasureTheory.Integrable (MeasureTheory.convolution f g L μ) μ
• HasCompactSupport.continuous_convolution_left 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [TopologicalSpace G] [IsTopologicalAddGroup G] [BorelSpace G] (hcf : HasCompactSupport f) (hf : Continuous f) (hg : MeasureTheory.LocallyIntegrable g μ) : Continuous (MeasureTheory.convolution f g L μ)
• BddAbove.continuous_convolution_left_of_integrable 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [TopologicalSpace G] [IsTopologicalAddGroup G] [BorelSpace G] [FirstCountableTopology G] [SecondCountableTopologyEither G E] (hbf : BddAbove (Set.range fun x => ‖f x‖)) (hf : Continuous f) (hg : MeasureTheory.Integrable g μ) : Continuous (MeasureTheory.convolution f g L μ)
• MeasureTheory.ConvolutionExistsAt.add_distrib 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f f' : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] {x : G} (hfg : MeasureTheory.ConvolutionExistsAt f g x L μ) (hfg' : MeasureTheory.ConvolutionExistsAt f' g x L μ) : MeasureTheory.convolution (f + f') g L μ x = MeasureTheory.convolution f g L μ x + MeasureTheory.convolution f' g L μ x
• MeasureTheory.ConvolutionExistsAt.distrib_add 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g g' : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] {x : G} (hfg : MeasureTheory.ConvolutionExistsAt f g x L μ) (hfg' : MeasureTheory.ConvolutionExistsAt f g' x L μ) : MeasureTheory.convolution f (g + g') L μ x = MeasureTheory.convolution f g L μ x + MeasureTheory.convolution f g' L μ x
• MeasureTheory.ConvolutionExists.add_distrib 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f f' : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] (hfg : MeasureTheory.ConvolutionExists f g L μ) (hfg' : MeasureTheory.ConvolutionExists f' g L μ) : MeasureTheory.convolution (f + f') g L μ = MeasureTheory.convolution f g L μ + MeasureTheory.convolution f' g L μ
• MeasureTheory.ConvolutionExists.distrib_add 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g g' : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] (hfg : MeasureTheory.ConvolutionExists f g L μ) (hfg' : MeasureTheory.ConvolutionExists f g' L μ) : MeasureTheory.convolution f (g + g') L μ = MeasureTheory.convolution f g L μ + MeasureTheory.convolution f g' L μ
• MeasureTheory.convolution_flip 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [MeasurableNeg G] [MeasurableAdd G] : MeasureTheory.convolution g f L.flip μ = MeasureTheory.convolution f g L μ
• MeasureTheory.continuousOn_convolution_right_with_param_comp 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} {P : Type uP} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [BorelSpace G] [TopologicalSpace P] {s : Set P} {v : P → G} (hv : ContinuousOn v s) {g : P → G → E'} {k : Set G} (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContinuousOn (↿g) (s ×ˢ Set.univ)) : ContinuousOn (fun x => MeasureTheory.convolution f (g x) L μ (v x)) s
• MeasureTheory.convolution_neg_of_neg_eq 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} {x : G} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [MeasurableNeg G] [MeasurableAdd G] (h1 : ∀ᵐ (x : G) ∂μ, f (-x) = f x) (h2 : ∀ᵐ (x : G) ∂μ, g (-x) = g x) : MeasureTheory.convolution f g L μ (-x) = MeasureTheory.convolution f g L μ x
• MeasureTheory.continuousOn_convolution_right_with_param 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} {P : Type uP} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] [TopologicalSpace G] [IsTopologicalAddGroup G] [BorelSpace G] [TopologicalSpace P] {g : P → G → E'} {s : Set P} {k : Set G} (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContinuousOn (↿g) (s ×ˢ Set.univ)) : ContinuousOn (fun q => MeasureTheory.convolution f (g q.1) L μ q.2) (s ×ˢ Set.univ)
• MeasureTheory.convolution_smul 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] [SMulCommClass ℝ 𝕜 F] {y : 𝕜} : MeasureTheory.convolution f (y • g) L μ = y • MeasureTheory.convolution f g L μ
• MeasureTheory.smul_convolution 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] {L : E →L[𝕜] E' →L[𝕜] F} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddGroup G] [SMulCommClass ℝ 𝕜 F] {y : 𝕜} : MeasureTheory.convolution (y • f) g L μ = y • MeasureTheory.convolution f g L μ
• MeasureTheory.convolution_def 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} {x : G} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [Sub G] : MeasureTheory.convolution f g L μ x = ∫ (t : G), (L (f t)) (g (x - t)) ∂μ
• MeasureTheory.convolution_eq_swap 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} {x : G} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [AddCommGroup G] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] [MeasurableNeg G] [MeasurableAdd G] : MeasureTheory.convolution f g L μ x = ∫ (t : G), (L (f (x - t))) (g t) ∂μ
• MeasureTheory.convolution_eq_right' 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [SeminormedAddCommGroup G] {x₀ : G} {R : ℝ} (hf : Function.support f ⊆ Metric.ball 0 R) (hg : ∀ x ∈ Metric.ball x₀ R, g x = g x₀) : MeasureTheory.convolution f g L μ x₀ = ∫ (t : G), (L (f t)) (g x₀) ∂μ
• MeasureTheory.convolution_precompR_apply 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {E'' : Type uE''} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup E''] [NormedAddCommGroup F] {f : G → E} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 E''] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [NormedAddCommGroup G] [BorelSpace G] {g : G → E'' →L[𝕜] E'} (hf : MeasureTheory.LocallyIntegrable f μ) (hcg : HasCompactSupport g) (hg : Continuous g) (x₀ : G) (x : E'') : (MeasureTheory.convolution f g (ContinuousLinearMap.precompR E'' L) μ x₀) x = MeasureTheory.convolution f (fun a => (g a) x) L μ x₀
• MeasureTheory.integral_convolution 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] {μ ν : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [CompleteSpace F] [AddGroup G] [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [μ.IsAddRightInvariant] [MeasurableAdd₂ G] [MeasurableNeg G] [NormedSpace ℝ E] [NormedSpace ℝ E'] [CompleteSpace E] [CompleteSpace E'] (hf : MeasureTheory.Integrable f ν) (hg : MeasureTheory.Integrable g μ) : ∫ (x : G), MeasureTheory.convolution f g L ν x ∂μ = (L (∫ (x : G), f x ∂ν)) (∫ (x : G), g x ∂μ)
• MeasureTheory.dist_convolution_le' 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 F] (L : E →L[𝕜] E' →L[𝕜] F) [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ F] [SeminormedAddCommGroup G] [BorelSpace G] [SecondCountableTopology G] [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] {x₀ : G} {R ε : ℝ} {z₀ : E'} (hε : 0 ≤ ε) (hif : MeasureTheory.Integrable f μ) (hf : Function.support f ⊆ Metric.ball 0 R) (hmg : MeasureTheory.AEStronglyMeasurable g μ) (hg : ∀ x ∈ Metric.ball x₀ R, dist (g x) z₀ ≤ ε) : dist (MeasureTheory.convolution f g L μ x₀) (∫ (t : G), (L (f t)) z₀ ∂μ) ≤ (‖L‖ * ∫ (x : G), ‖f x‖ ∂μ) * ε
• MeasureTheory.convolution_assoc 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {E'' : Type uE''} {F : Type uF} {F' : Type uF'} {F'' : Type uF''} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup E''] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 E''] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] {μ ν : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [CompleteSpace F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] [NormedSpace 𝕜 F'] [CompleteSpace F'] [NormedAddCommGroup F''] [NormedSpace ℝ F''] [NormedSpace 𝕜 F''] [CompleteSpace F''] {k : G → E''} (L₂ : F →L[𝕜] E'' →L[𝕜] F') (L₃ : E →L[𝕜] F'' →L[𝕜] F') (L₄ : E' →L[𝕜] E'' →L[𝕜] F'') [AddGroup G] [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [μ.IsAddRightInvariant] [MeasurableAdd₂ G] [ν.IsAddRightInvariant] [MeasurableNeg G] (hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)) {x₀ : G} (hf : MeasureTheory.AEStronglyMeasurable f ν) (hg : MeasureTheory.AEStronglyMeasurable g μ) (hk : MeasureTheory.AEStronglyMeasurable k μ) (hfg : ∀ᵐ (y : G) ∂μ, MeasureTheory.ConvolutionExistsAt f g y L ν) (hgk : ∀ᵐ (x : G) ∂ν, MeasureTheory.ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖k x‖) x (ContinuousLinearMap.mul ℝ ℝ) μ) (hfgk : MeasureTheory.ConvolutionExistsAt (fun x => ‖f x‖) (MeasureTheory.convolution (fun x => ‖g x‖) (fun x => ‖k x‖) (ContinuousLinearMap.mul ℝ ℝ) μ) x₀ (ContinuousLinearMap.mul ℝ ℝ) ν) : MeasureTheory.convolution (MeasureTheory.convolution f g L ν) k L₂ μ x₀ = MeasureTheory.convolution f (MeasureTheory.convolution g k L₄ μ) L₃ ν x₀
• MeasureTheory.convolution_assoc' 📋 Mathlib.Analysis.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {E'' : Type uE''} {F : Type uF} {F' : Type uF'} {F'' : Type uF''} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup E''] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace 𝕜 E''] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] {μ ν : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [CompleteSpace F] [NormedAddCommGroup F'] [NormedSpace ℝ F'] [NormedSpace 𝕜 F'] [CompleteSpace F'] [NormedAddCommGroup F''] [NormedSpace ℝ F''] [NormedSpace 𝕜 F''] [CompleteSpace F''] {k : G → E''} (L₂ : F →L[𝕜] E'' →L[𝕜] F') (L₃ : E →L[𝕜] F'' →L[𝕜] F') (L₄ : E' →L[𝕜] E'' →L[𝕜] F'') [AddGroup G] [MeasureTheory.SFinite μ] [MeasureTheory.SFinite ν] [μ.IsAddRightInvariant] [MeasurableAdd₂ G] [ν.IsAddRightInvariant] [MeasurableNeg G] (hL : ∀ (x : E) (y : E') (z : E''), (L₂ ((L x) y)) z = (L₃ x) ((L₄ y) z)) {x₀ : G} (hfg : ∀ᵐ (y : G) ∂μ, MeasureTheory.ConvolutionExistsAt f g y L ν) (hgk : ∀ᵐ (x : G) ∂ν, MeasureTheory.ConvolutionExistsAt g k x L₄ μ) (hi : MeasureTheory.Integrable (Function.uncurry fun x y => (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)) : MeasureTheory.convolution (MeasureTheory.convolution f g L ν) k L₂ μ x₀ = MeasureTheory.convolution f (MeasureTheory.convolution g k L₄ μ) L₃ ν x₀
• HasCompactSupport.contDiff_convolution_right 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) {n : ℕ∞} (hcg : HasCompactSupport g) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiff 𝕜 (↑n) g) : ContDiff 𝕜 (↑n) (MeasureTheory.convolution f g L μ)
• HasCompactSupport.contDiff_convolution_left 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [μ.IsAddLeftInvariant] [μ.IsNegInvariant] {n : ℕ∞} (hcf : HasCompactSupport f) (hf : ContDiff 𝕜 (↑n) f) (hg : MeasureTheory.LocallyIntegrable g μ) : ContDiff 𝕜 (↑n) (MeasureTheory.convolution f g L μ)
• MeasureTheory.contDiffOn_convolution_right_with_param_comp 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} {P : Type uP} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {μ : MeasureTheory.Measure G} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F) {s : Set P} {v : P → G} (hv : ContDiffOn 𝕜 (↑n) v s) {f : G → E} {g : P → G → E'} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ Set.univ)) : ContDiffOn 𝕜 (↑n) (fun x => MeasureTheory.convolution f (g x) L μ (v x)) s
• MeasureTheory.contDiffOn_convolution_right_with_param 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} {P : Type uP} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {μ : MeasureTheory.Measure G} {f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ Set.univ)) : ContDiffOn 𝕜 (↑n) (fun q => MeasureTheory.convolution f (g q.1) L μ q.2) (s ×ˢ Set.univ)
• MeasureTheory.contDiffOn_convolution_right_with_param_aux 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {E : Type uE} [NormedAddCommGroup E] [RCLike 𝕜] [NormedSpace 𝕜 E] {G E' F P : Type uP} [NormedAddCommGroup E'] [NormedAddCommGroup F] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {f : G → E} {n : ℕ∞} (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ Set.univ)) : ContDiffOn 𝕜 (↑n) (fun q => MeasureTheory.convolution f (g q.1) L μ q.2) (s ×ˢ Set.univ)
• MeasureTheory.contDiffOn_convolution_left_with_param_comp 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} {P : Type uP} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] [μ.IsNegInvariant] (L : E' →L[𝕜] E →L[𝕜] F) {s : Set P} {n : ℕ∞} {v : P → G} (hv : ContDiffOn 𝕜 (↑n) v s) {f : G → E} {g : P → G → E'} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ Set.univ)) : ContDiffOn 𝕜 (↑n) (fun x => MeasureTheory.convolution (g x) f L μ (v x)) s
• MeasureTheory.contDiffOn_convolution_left_with_param 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} {P : Type uP} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {μ : MeasureTheory.Measure G} [μ.IsAddLeftInvariant] [μ.IsNegInvariant] (L : E' →L[𝕜] E →L[𝕜] F) {f : G → E} {n : ℕ∞} {g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 (↑n) (↿g) (s ×ˢ Set.univ)) : ContDiffOn 𝕜 (↑n) (fun q => MeasureTheory.convolution (g q.1) f L μ q.2) (s ×ˢ Set.univ)
• HasCompactSupport.hasDerivAt_convolution_right 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] {f₀ : 𝕜 → E} {g₀ : 𝕜 → E'} (L : E →L[𝕜] E' →L[𝕜] F) {μ : MeasureTheory.Measure 𝕜} [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] (hf : MeasureTheory.LocallyIntegrable f₀ μ) (hcg : HasCompactSupport g₀) (hg : ContDiff 𝕜 1 g₀) (x₀ : 𝕜) : HasDerivAt (MeasureTheory.convolution f₀ g₀ L μ) (MeasureTheory.convolution f₀ (deriv g₀) L μ x₀) x₀
• HasCompactSupport.hasDerivAt_convolution_left 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] {f₀ : 𝕜 → E} {g₀ : 𝕜 → E'} (L : E →L[𝕜] E' →L[𝕜] F) {μ : MeasureTheory.Measure 𝕜} [μ.IsAddLeftInvariant] [MeasureTheory.SFinite μ] [μ.IsNegInvariant] (hcf : HasCompactSupport f₀) (hf : ContDiff 𝕜 1 f₀) (hg : MeasureTheory.LocallyIntegrable g₀ μ) (x₀ : 𝕜) : HasDerivAt (MeasureTheory.convolution f₀ g₀ L μ) (MeasureTheory.convolution (deriv f₀) g₀ L μ x₀) x₀
• HasCompactSupport.hasFDerivAt_convolution_right 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [MeasureTheory.SFinite μ] [μ.IsAddLeftInvariant] (hcg : HasCompactSupport g) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiff 𝕜 1 g) (x₀ : G) : HasFDerivAt (MeasureTheory.convolution f g L μ) (MeasureTheory.convolution f (fderiv 𝕜 g) (ContinuousLinearMap.precompR G L) μ x₀) x₀
• HasCompactSupport.hasFDerivAt_convolution_left 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} {g : G → E'} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [MeasureTheory.SFinite μ] [μ.IsAddLeftInvariant] [μ.IsNegInvariant] (hcf : HasCompactSupport f) (hf : ContDiff 𝕜 1 f) (hg : MeasureTheory.LocallyIntegrable g μ) (x₀ : G) : HasFDerivAt (MeasureTheory.convolution f g L μ) (MeasureTheory.convolution (fderiv 𝕜 f) g (ContinuousLinearMap.precompL G L) μ x₀) x₀
• MeasureTheory.hasFDerivAt_convolution_right_with_param 📋 Mathlib.Analysis.Calculus.ContDiff.Convolution
  {𝕜 : Type u𝕜} {G : Type uG} {E : Type uE} {E' : Type uE'} {F : Type uF} {P : Type uP} [NormedAddCommGroup E] [NormedAddCommGroup E'] [NormedAddCommGroup F] {f : G → E} [RCLike 𝕜] [NormedSpace 𝕜 E] [NormedSpace 𝕜 E'] [NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G] [NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [NormedAddCommGroup P] [NormedSpace 𝕜 P] {μ : MeasureTheory.Measure G} (L : E →L[𝕜] E' →L[𝕜] F) {g : P → G → E'} {s : Set P} {k : Set G} (hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ (p : P) (x : G), p ∈ s → x ∉ k → g p x = 0) (hf : MeasureTheory.LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 1 (↿g) (s ×ˢ Set.univ)) (q₀ : P × G) (hq₀ : q₀.1 ∈ s) : HasFDerivAt (fun q => MeasureTheory.convolution f (g q.1) L μ q.2) (MeasureTheory.convolution f (fun x => fderiv 𝕜 ↿g (q₀.1, x)) (ContinuousLinearMap.precompR (P × G) L) μ q₀.2) q₀
• ContDiffBump.normed_convolution_eq_right 📋 Mathlib.Analysis.Calculus.BumpFunction.Convolution
  {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace E'] {φ : ContDiffBump 0} [BorelSpace G] [FiniteDimensional ℝ G] [MeasureTheory.IsLocallyFiniteMeasure μ] [μ.IsOpenPosMeasure] {x₀ : G} (hg : ∀ x ∈ Metric.ball x₀ φ.rOut, g x = g x₀) : MeasureTheory.convolution (φ.normed μ) g (ContinuousLinearMap.lsmul ℝ ℝ) μ x₀ = g x₀
• ContDiffBump.convolution_tendsto_right_of_continuous 📋 Mathlib.Analysis.Calculus.BumpFunction.Convolution
  {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace E'] [BorelSpace G] [FiniteDimensional ℝ G] [μ.IsAddHaarMeasure] {ι : Type u_1} {φ : ι → ContDiffBump 0} {l : Filter ι} (hφ : Filter.Tendsto (fun i => (φ i).rOut) l (nhds 0)) (hg : Continuous g) (x₀ : G) : Filter.Tendsto (fun i => MeasureTheory.convolution ((φ i).normed μ) g (ContinuousLinearMap.lsmul ℝ ℝ) μ x₀) l (nhds (g x₀))
• ContDiffBump.dist_normed_convolution_le 📋 Mathlib.Analysis.Calculus.BumpFunction.Convolution
  {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace E'] {φ : ContDiffBump 0} [BorelSpace G] [FiniteDimensional ℝ G] [μ.IsAddHaarMeasure] {x₀ : G} {ε : ℝ} (hmg : MeasureTheory.AEStronglyMeasurable g μ) (hg : ∀ x ∈ Metric.ball x₀ φ.rOut, dist (g x) (g x₀) ≤ ε) : dist (MeasureTheory.convolution (φ.normed μ) g (ContinuousLinearMap.lsmul ℝ ℝ) μ x₀) (g x₀) ≤ ε
• ContDiffBump.convolution_eq_right 📋 Mathlib.Analysis.Calculus.BumpFunction.Convolution
  {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace E'] {φ : ContDiffBump 0} [HasContDiffBump G] {x₀ : G} (hg : ∀ x ∈ Metric.ball x₀ φ.rOut, g x = g x₀) : MeasureTheory.convolution (↑φ) g (ContinuousLinearMap.lsmul ℝ ℝ) μ x₀ = MeasureTheory.integral μ ↑φ • g x₀
• ContDiffBump.convolution_tendsto_right 📋 Mathlib.Analysis.Calculus.BumpFunction.Convolution
  {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace E'] [BorelSpace G] [FiniteDimensional ℝ G] [μ.IsAddHaarMeasure] {ι : Type u_1} {φ : ι → ContDiffBump 0} {g : ι → G → E'} {k : ι → G} {x₀ : G} {z₀ : E'} {l : Filter ι} (hφ : Filter.Tendsto (fun i => (φ i).rOut) l (nhds 0)) (hig : ∀ᶠ (i : ι) in l, MeasureTheory.AEStronglyMeasurable (g i) μ) (hcg : Filter.Tendsto (Function.uncurry g) (l ×ˢ nhds x₀) (nhds z₀)) (hk : Filter.Tendsto k l (nhds x₀)) : Filter.Tendsto (fun i => MeasureTheory.convolution ((φ i).normed μ) (g i) (ContinuousLinearMap.lsmul ℝ ℝ) μ (k i)) l (nhds z₀)
• ContDiffBump.ae_convolution_tendsto_right_of_locallyIntegrable 📋 Mathlib.Analysis.Calculus.BumpFunction.Convolution
  {G : Type uG} {E' : Type uE'} [NormedAddCommGroup E'] {g : G → E'} [MeasurableSpace G] {μ : MeasureTheory.Measure G} [NormedSpace ℝ E'] [NormedAddCommGroup G] [NormedSpace ℝ G] [CompleteSpace E'] [BorelSpace G] [FiniteDimensional ℝ G] [μ.IsAddHaarMeasure] {ι : Type u_1} {φ : ι → ContDiffBump 0} {l : Filter ι} {K : ℝ} (hφ : Filter.Tendsto (fun i => (φ i).rOut) l (nhds 0)) (h'φ : ∀ᶠ (i : ι) in l, (φ i).rOut ≤ K * (φ i).rIn) (hg : MeasureTheory.LocallyIntegrable g μ) : ∀ᵐ (x₀ : G) ∂μ, Filter.Tendsto (fun i => MeasureTheory.convolution ((φ i).normed μ) g (ContinuousLinearMap.lsmul ℝ ℝ) μ x₀) l (nhds (g x₀))
• Real.fourier_smul_convolution_eq 📋 Mathlib.Analysis.Fourier.Convolution
  {E : Type u_3} {F₁ : Type u_5} [NormedAddCommGroup E] [NormedAddCommGroup F₁] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [CompleteSpace F₁] [NormedSpace ℂ F₁] {f₁ : E → ℂ} {f₂ : E → F₁} (hf₁ : MeasureTheory.Integrable f₁ MeasureTheory.volume) (hf₂ : MeasureTheory.Integrable f₂ MeasureTheory.volume) (ξ : E) : FourierTransform.fourier (MeasureTheory.convolution f₁ f₂ (ContinuousLinearMap.lsmul ℂ ℂ) MeasureTheory.volume) ξ = FourierTransform.fourier f₁ ξ • FourierTransform.fourier f₂ ξ
• Real.fourier_mul_convolution_eq 📋 Mathlib.Analysis.Fourier.Convolution
  {R : Type u_2} {E : Type u_3} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [NormedRing R] [NormedSpace ℂ R] [IsScalarTower ℂ R R] [SMulCommClass ℂ R R] [CompleteSpace R] {f₁ f₂ : E → R} (hf₁ : MeasureTheory.Integrable f₁ MeasureTheory.volume) (hf₂ : MeasureTheory.Integrable f₂ MeasureTheory.volume) (ξ : E) : FourierTransform.fourier (MeasureTheory.convolution f₁ f₂ (ContinuousLinearMap.mul ℂ R) MeasureTheory.volume) ξ = FourierTransform.fourier f₁ ξ * FourierTransform.fourier f₂ ξ
• Real.fourier_bilin_convolution_eq 📋 Mathlib.Analysis.Fourier.Convolution
  {E : Type u_3} {F₁ : Type u_5} {F₂ : Type u_6} {F₃ : Type u_7} [NormedAddCommGroup E] [NormedAddCommGroup F₁] [NormedAddCommGroup F₂] [NormedAddCommGroup F₃] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [NormedSpace ℂ F₃] [CompleteSpace F₁] [CompleteSpace F₂] [CompleteSpace F₃] [NormedSpace ℂ F₁] [NormedSpace ℂ F₂] (B : F₁ →L[ℂ] F₂ →L[ℂ] F₃) {f₁ : E → F₁} {f₂ : E → F₂} (hf₁ : MeasureTheory.Integrable f₁ MeasureTheory.volume) (hf₂ : MeasureTheory.Integrable f₂ MeasureTheory.volume) (ξ : E) : FourierTransform.fourier (MeasureTheory.convolution f₁ f₂ B MeasureTheory.volume) ξ = (B (FourierTransform.fourier f₁ ξ)) (FourierTransform.fourier f₂ ξ)
• Real.fourier_bilin_convolution_eq_integral 📋 Mathlib.Analysis.Fourier.Convolution
  {𝕜 : Type u_1} {E : Type u_3} {F₁ : Type u_5} {F₂ : Type u_6} {F₃ : Type u_7} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedAddCommGroup F₁] [NormedAddCommGroup F₂] [NormedAddCommGroup F₃] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [NormedSpace 𝕜 F₁] [NormedSpace 𝕜 F₂] [NormedSpace 𝕜 F₃] [NormedSpace ℂ F₃] (B : F₁ →L[𝕜] F₂ →L[𝕜] F₃) {f₁ : E → F₁} {f₂ : E → F₂} (hf₁ : MeasureTheory.Integrable f₁ MeasureTheory.volume) (hf₂ : MeasureTheory.Integrable f₂ MeasureTheory.volume) (ξ : E) : FourierTransform.fourier (MeasureTheory.convolution f₁ f₂ B MeasureTheory.volume) ξ = ∫ (y : E) (x : E), Real.fourierChar (-inner ℝ (y + x) ξ) • (B (f₁ x)) (f₂ y)
• SchwartzMap.convolution_apply 📋 Mathlib.Analysis.Fourier.Convolution
  {E : Type u_3} {F₁ : Type u_5} {F₂ : Type u_6} {F₃ : Type u_7} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [NormedAddCommGroup F₁] [NormedSpace ℂ F₁] [NormedAddCommGroup F₂] [NormedSpace ℂ F₂] [NormedAddCommGroup F₃] [NormedSpace ℂ F₃] [CompleteSpace F₃] [CompleteSpace F₁] [CompleteSpace F₂] (B : F₁ →L[ℂ] F₂ →L[ℂ] F₃) (f : SchwartzMap E F₁) (g : SchwartzMap E F₂) (x : E) : (((SchwartzMap.convolution B) f) g) x = MeasureTheory.convolution (⇑f) (⇑g) B MeasureTheory.volume x
• SchwartzMap.fourier_convolution_apply 📋 Mathlib.Analysis.Fourier.Convolution
  {E : Type u_3} {F₁ : Type u_5} {F₂ : Type u_6} {F₃ : Type u_7} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [NormedAddCommGroup F₁] [NormedSpace ℂ F₁] [NormedAddCommGroup F₂] [NormedSpace ℂ F₂] [NormedAddCommGroup F₃] [NormedSpace ℂ F₃] [CompleteSpace F₃] [CompleteSpace F₁] [CompleteSpace F₂] (B : F₁ →L[ℂ] F₂ →L[ℂ] F₃) (f : SchwartzMap E F₁) (g : SchwartzMap E F₂) (x : E) : (FourierTransform.fourier (((SchwartzMap.convolution B) f) g)) x = FourierTransform.fourier (MeasureTheory.convolution (⇑f) (⇑g) B MeasureTheory.volume) x

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. By subexpression:
   🔍 _ * (_ ^ _)
   finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

   The pattern can also be non-linear, as in
   🔍 Real.sqrt ?a * Real.sqrt ?a

   If the pattern has parameters, they are matched in any order. Both of these will find List.map:
   🔍 (?a -> ?b) -> List ?a -> List ?b
   🔍 List ?a -> (?a -> ?b) -> List ?b

4. By main conclusion:
   🔍 |- tsum _ = _ * tsum _
   finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

   As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
   🔍 |- _ < _ → tsum _ < tsum _
   will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59