fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
_root_.Continuous.convolution_integrand_fst [ContinuousSub G] (hg : Continuous g) (t : G) :
Continuous fun x => L (f t) (g (x - t)) :=
L.continuous₂.comp₂ continuous_const <| hg.comp <| continuous_id.sub continuous_const | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.Continuous.convolution_integrand_fst | null |
_root_.HasCompactSupport.convolution_integrand_bound_left (hcf : HasCompactSupport f)
(hf : Continuous f) {x t : G} {s : Set G} (hx : x ∈ s) :
‖L (f (x - t)) (g t)‖ ≤
(-tsupport f + s).indicator (fun t => (‖L‖ * ⨆ i, ‖f i‖) * ‖g t‖) t := by
convert hcf.convolution_integrand_bound_right L.flip hf hx usin... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.convolution_integrand_bound_left | null |
ConvolutionExistsAt [Sub G] (f : G → E) (g : G → E') (x : G) (L : E →L[𝕜] E' →L[𝕜] F)
(μ : Measure G := by volume_tac) : Prop :=
Integrable (fun t => L (f t) (g (x - t))) μ | def | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | ConvolutionExistsAt | The convolution of `f` and `g` exists at `x` when the function `t ↦ L (f t) (g (x - t))` is
integrable. There are various conditions on `f` and `g` to prove this. |
ConvolutionExists [Sub G] (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F)
(μ : Measure G := by volume_tac) : Prop :=
∀ x : G, ConvolutionExistsAt f g x L μ | def | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | ConvolutionExists | The convolution of `f` and `g` exists when the function `t ↦ L (f t) (g (x - t))` is integrable
for all `x : G`. There are various conditions on `f` and `g` to prove this. |
ConvolutionExistsAt.integrable [Sub G] {x : G} (h : ConvolutionExistsAt f g x L μ) :
Integrable (fun t => L (f t) (g (x - t))) μ :=
h | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | ConvolutionExistsAt.integrable | null |
AEStronglyMeasurable.convolution_integrand' [MeasurableAdd₂ G]
[MeasurableNeg G] (hf : AEStronglyMeasurable f ν)
(hg : AEStronglyMeasurable g <| map (fun p : G × G => p.1 - p.2) (μ.prod ν)) :
AEStronglyMeasurable (fun p : G × G => L (f p.2) (g (p.1 - p.2))) (μ.prod ν) :=
L.aestronglyMeasurable_comp₂ hf.co... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | AEStronglyMeasurable.convolution_integrand' | null |
AEStronglyMeasurable.convolution_integrand_snd'
(hf : AEStronglyMeasurable f μ) {x : G}
(hg : AEStronglyMeasurable g <| map (fun t => x - t) μ) :
AEStronglyMeasurable (fun t => L (f t) (g (x - t))) μ :=
L.aestronglyMeasurable_comp₂ hf <| hg.comp_measurable <| measurable_id.const_sub x | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | AEStronglyMeasurable.convolution_integrand_snd' | null |
AEStronglyMeasurable.convolution_integrand_swap_snd' {x : G}
(hf : AEStronglyMeasurable f <| map (fun t => x - t) μ) (hg : AEStronglyMeasurable g μ) :
AEStronglyMeasurable (fun t => L (f (x - t)) (g t)) μ :=
L.aestronglyMeasurable_comp₂ (hf.comp_measurable <| measurable_id.const_sub x) hg | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | AEStronglyMeasurable.convolution_integrand_swap_snd' | null |
_root_.BddAbove.convolutionExistsAt' {x₀ : G} {s : Set G}
(hbg : BddAbove ((fun i => ‖g i‖) '' ((fun t => -t + x₀) ⁻¹' s))) (hs : MeasurableSet s)
(h2s : (support fun t => L (f t) (g (x₀ - t))) ⊆ s) (hf : IntegrableOn f s μ)
(hmg : AEStronglyMeasurable g <| map (fun t => x₀ - t) (μ.restrict s)) :
Convol... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.BddAbove.convolutionExistsAt' | A sufficient condition to prove that `f ⋆[L, μ] g` exists.
We assume that `f` is integrable on a set `s` and `g` is bounded and ae strongly measurable
on `x₀ - s` (note that both properties hold if `g` is continuous with compact support). |
ConvolutionExistsAt.of_norm' {x₀ : G}
(h : ConvolutionExistsAt (fun x => ‖f x‖) (fun x => ‖g x‖) x₀ (mul ℝ ℝ) μ)
(hmf : AEStronglyMeasurable f μ) (hmg : AEStronglyMeasurable g <| map (fun t => x₀ - t) μ) :
ConvolutionExistsAt f g x₀ L μ := by
refine (h.const_mul ‖L‖).mono'
(hmf.convolution_integrand_s... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | ConvolutionExistsAt.of_norm' | If `‖f‖ *[μ] ‖g‖` exists, then `f *[L, μ] g` exists. |
AEStronglyMeasurable.convolution_integrand_snd (hf : AEStronglyMeasurable f μ)
(hg : AEStronglyMeasurable g μ) (x : G) :
AEStronglyMeasurable (fun t => L (f t) (g (x - t))) μ :=
hf.convolution_integrand_snd' L <|
hg.mono_ac <| (quasiMeasurePreserving_sub_left_of_right_invariant μ x).absolutelyContinuous | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | AEStronglyMeasurable.convolution_integrand_snd | null |
AEStronglyMeasurable.convolution_integrand_swap_snd
(hf : AEStronglyMeasurable f μ) (hg : AEStronglyMeasurable g μ) (x : G) :
AEStronglyMeasurable (fun t => L (f (x - t)) (g t)) μ :=
(hf.mono_ac
(quasiMeasurePreserving_sub_left_of_right_invariant μ
x).absolutelyContinuous).convolution_inte... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | AEStronglyMeasurable.convolution_integrand_swap_snd | null |
ConvolutionExistsAt.of_norm {x₀ : G}
(h : ConvolutionExistsAt (fun x => ‖f x‖) (fun x => ‖g x‖) x₀ (mul ℝ ℝ) μ)
(hmf : AEStronglyMeasurable f μ) (hmg : AEStronglyMeasurable g μ) :
ConvolutionExistsAt f g x₀ L μ :=
h.of_norm' L hmf <|
hmg.mono_ac (quasiMeasurePreserving_sub_left_of_right_invariant μ x₀... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | ConvolutionExistsAt.of_norm | If `‖f‖ *[μ] ‖g‖` exists, then `f *[L, μ] g` exists. |
AEStronglyMeasurable.convolution_integrand (hf : AEStronglyMeasurable f ν)
(hg : AEStronglyMeasurable g μ) :
AEStronglyMeasurable (fun p : G × G => L (f p.2) (g (p.1 - p.2))) (μ.prod ν) :=
hf.convolution_integrand' L <|
hg.mono_ac (quasiMeasurePreserving_sub_of_right_invariant μ ν).absolutelyContinuous | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | AEStronglyMeasurable.convolution_integrand | null |
Integrable.convolution_integrand (hf : Integrable f ν) (hg : Integrable g μ) :
Integrable (fun p : G × G => L (f p.2) (g (p.1 - p.2))) (μ.prod ν) := by
have h_meas : AEStronglyMeasurable (fun p : G × G => L (f p.2) (g (p.1 - p.2))) (μ.prod ν) :=
hf.aestronglyMeasurable.convolution_integrand L hg.aestronglyMea... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | Integrable.convolution_integrand | null |
Integrable.ae_convolution_exists (hf : Integrable f ν) (hg : Integrable g μ) :
∀ᵐ x ∂μ, ConvolutionExistsAt f g x L ν :=
((integrable_prod_iff <|
hf.aestronglyMeasurable.convolution_integrand L hg.aestronglyMeasurable).mp <|
hf.convolution_integrand L hg).1 | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | Integrable.ae_convolution_exists | null |
_root_.HasCompactSupport.convolutionExistsAt {x₀ : G}
(h : HasCompactSupport fun t => L (f t) (g (x₀ - t))) (hf : LocallyIntegrable f μ)
(hg : Continuous g) : ConvolutionExistsAt f g x₀ L μ := by
let u := (Homeomorph.neg G).trans (Homeomorph.addRight x₀)
let v := (Homeomorph.neg G).trans (Homeomorph.addLeft... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.convolutionExistsAt | null |
_root_.HasCompactSupport.convolutionExists_right (hcg : HasCompactSupport g)
(hf : LocallyIntegrable f μ) (hg : Continuous g) : ConvolutionExists f g L μ := by
intro x₀
refine HasCompactSupport.convolutionExistsAt L ?_ hf hg
refine (hcg.comp_homeomorph (Homeomorph.subLeft x₀)).mono ?_
refine fun t => mt fun... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.convolutionExists_right | null |
_root_.HasCompactSupport.convolutionExists_left_of_continuous_right
(hcf : HasCompactSupport f) (hf : LocallyIntegrable f μ) (hg : Continuous g) :
ConvolutionExists f g L μ := by
intro x₀
refine HasCompactSupport.convolutionExistsAt L ?_ hf hg
refine hcf.mono ?_
refine fun t => mt fun ht : f t = 0 => ?_... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.convolutionExists_left_of_continuous_right | null |
_root_.BddAbove.convolutionExistsAt [MeasurableAdd₂ G] [SFinite μ] {x₀ : G} {s : Set G}
(hbg : BddAbove ((fun i => ‖g i‖) '' ((fun t => x₀ - t) ⁻¹' s))) (hs : MeasurableSet s)
(h2s : (support fun t => L (f t) (g (x₀ - t))) ⊆ s) (hf : IntegrableOn f s μ)
(hmg : AEStronglyMeasurable g μ) : ConvolutionExistsAt... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.BddAbove.convolutionExistsAt | A sufficient condition to prove that `f ⋆[L, μ] g` exists.
We assume that the integrand has compact support and `g` is bounded on this support (note that
both properties hold if `g` is continuous with compact support). We also require that `f` is
integrable on the support of the integrand, and that both functions are s... |
convolutionExistsAt_flip :
ConvolutionExistsAt g f x L.flip μ ↔ ConvolutionExistsAt f g x L μ := by
simp_rw [ConvolutionExistsAt, ← integrable_comp_sub_left (fun t => L (f t) (g (x - t))) x,
sub_sub_cancel, flip_apply] | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolutionExistsAt_flip | null |
ConvolutionExistsAt.integrable_swap (h : ConvolutionExistsAt f g x L μ) :
Integrable (fun t => L (f (x - t)) (g t)) μ := by
convert h.comp_sub_left x
simp_rw [sub_sub_self] | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | ConvolutionExistsAt.integrable_swap | null |
convolutionExistsAt_iff_integrable_swap :
ConvolutionExistsAt f g x L μ ↔ Integrable (fun t => L (f (x - t)) (g t)) μ :=
convolutionExistsAt_flip.symm | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolutionExistsAt_iff_integrable_swap | null |
_root_.HasCompactSupport.convolutionExists_left
(hcf : HasCompactSupport f) (hf : Continuous f)
(hg : LocallyIntegrable g μ) : ConvolutionExists f g L μ := fun x₀ =>
convolutionExistsAt_flip.mp <| hcf.convolutionExists_right L.flip hg hf x₀ | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.convolutionExists_left | null |
_root_.HasCompactSupport.convolutionExists_right_of_continuous_left
(hcg : HasCompactSupport g) (hf : Continuous f) (hg : LocallyIntegrable g μ) :
ConvolutionExists f g L μ := fun x₀ =>
convolutionExistsAt_flip.mp <| hcg.convolutionExists_left_of_continuous_right L.flip hg hf x₀ | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.convolutionExists_right_of_continuous_left | null |
noncomputable convolution [Sub G] (f : G → E) (g : G → E') (L : E →L[𝕜] E' →L[𝕜] F)
(μ : Measure G := by volume_tac) : G → F := fun x =>
∫ t, L (f t) (g (x - t)) ∂μ | def | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution | The convolution of two functions `f` and `g` with respect to a continuous bilinear map `L` and
measure `μ`. It is defined to be `(f ⋆[L, μ] g) x = ∫ t, L (f t) (g (x - t)) ∂μ`. |
convolution_lsmul [Sub G] {f : G → 𝕜} {g : G → F} :
(f ⋆[lsmul 𝕜 𝕜, μ] g : G → F) x = ∫ t, f t • g (x - t) ∂μ :=
rfl | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_lsmul | The convolution of two functions with respect to a bilinear operation `L` and a measure `μ`. -/
scoped[Convolution] notation:67 f " ⋆[" L:67 ", " μ:67 "] " g:66 => convolution f g L μ
/-- The convolution of two functions with respect to a bilinear operation `L` and the volume. -/
scoped[Convolution]
notation:67 f " ... |
convolution_mul [Sub G] [NormedSpace ℝ 𝕜] {f : G → 𝕜} {g : G → 𝕜} :
(f ⋆[mul 𝕜 𝕜, μ] g) x = ∫ t, f t * g (x - t) ∂μ :=
rfl | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_mul | The definition of convolution where the bilinear operator is multiplication. |
smul_convolution [SMulCommClass ℝ 𝕜 F] {y : 𝕜} : y • f ⋆[L, μ] g = y • (f ⋆[L, μ] g) := by
ext; simp only [Pi.smul_apply, convolution_def, ← integral_smul, L.map_smul₂] | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | smul_convolution | null |
convolution_smul [SMulCommClass ℝ 𝕜 F] {y : 𝕜} : f ⋆[L, μ] y • g = y • (f ⋆[L, μ] g) := by
ext; simp only [Pi.smul_apply, convolution_def, ← integral_smul, (L _).map_smul]
@[simp] | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_smul | null |
zero_convolution : 0 ⋆[L, μ] g = 0 := by
ext
simp_rw [convolution_def, Pi.zero_apply, L.map_zero₂, integral_zero]
@[simp] | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | zero_convolution | null |
convolution_zero : f ⋆[L, μ] 0 = 0 := by
ext
simp_rw [convolution_def, Pi.zero_apply, (L _).map_zero, integral_zero] | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_zero | null |
ConvolutionExistsAt.distrib_add {x : G} (hfg : ConvolutionExistsAt f g x L μ)
(hfg' : ConvolutionExistsAt f g' x L μ) :
(f ⋆[L, μ] (g + g')) x = (f ⋆[L, μ] g) x + (f ⋆[L, μ] g') x := by
simp only [convolution_def, (L _).map_add, Pi.add_apply, integral_add hfg hfg'] | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | ConvolutionExistsAt.distrib_add | null |
ConvolutionExists.distrib_add (hfg : ConvolutionExists f g L μ)
(hfg' : ConvolutionExists f g' L μ) : f ⋆[L, μ] (g + g') = f ⋆[L, μ] g + f ⋆[L, μ] g' := by
ext x
exact (hfg x).distrib_add (hfg' x) | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | ConvolutionExists.distrib_add | null |
ConvolutionExistsAt.add_distrib {x : G} (hfg : ConvolutionExistsAt f g x L μ)
(hfg' : ConvolutionExistsAt f' g x L μ) :
((f + f') ⋆[L, μ] g) x = (f ⋆[L, μ] g) x + (f' ⋆[L, μ] g) x := by
simp only [convolution_def, L.map_add₂, Pi.add_apply, integral_add hfg hfg'] | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | ConvolutionExistsAt.add_distrib | null |
ConvolutionExists.add_distrib (hfg : ConvolutionExists f g L μ)
(hfg' : ConvolutionExists f' g L μ) : (f + f') ⋆[L, μ] g = f ⋆[L, μ] g + f' ⋆[L, μ] g := by
ext x
exact (hfg x).add_distrib (hfg' x) | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | ConvolutionExists.add_distrib | null |
convolution_mono_right {f g g' : G → ℝ} (hfg : ConvolutionExistsAt f g x (lsmul ℝ ℝ) μ)
(hfg' : ConvolutionExistsAt f g' x (lsmul ℝ ℝ) μ) (hf : ∀ x, 0 ≤ f x) (hg : ∀ x, g x ≤ g' x) :
(f ⋆[lsmul ℝ ℝ, μ] g) x ≤ (f ⋆[lsmul ℝ ℝ, μ] g') x := by
apply integral_mono hfg hfg'
simp only [lsmul_apply, Algebra.id.smul... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_mono_right | null |
convolution_mono_right_of_nonneg {f g g' : G → ℝ}
(hfg' : ConvolutionExistsAt f g' x (lsmul ℝ ℝ) μ) (hf : ∀ x, 0 ≤ f x) (hg : ∀ x, g x ≤ g' x)
(hg' : ∀ x, 0 ≤ g' x) : (f ⋆[lsmul ℝ ℝ, μ] g) x ≤ (f ⋆[lsmul ℝ ℝ, μ] g') x := by
by_cases H : ConvolutionExistsAt f g x (lsmul ℝ ℝ) μ
· exact convolution_mono_right ... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_mono_right_of_nonneg | null |
convolution_congr [MeasurableAdd₂ G] [MeasurableNeg G] [SFinite μ]
[IsAddRightInvariant μ] (h1 : f =ᵐ[μ] f') (h2 : g =ᵐ[μ] g') : f ⋆[L, μ] g = f' ⋆[L, μ] g' := by
ext x
apply integral_congr_ae
exact (h1.prodMk <| h2.comp_tendsto
(quasiMeasurePreserving_sub_left_of_right_invariant μ x).tendsto_ae).fun_comp... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_congr | null |
support_convolution_subset_swap : support (f ⋆[L, μ] g) ⊆ support g + support f := by
intro x h2x
by_contra hx
apply h2x
simp_rw [Set.mem_add, ← exists_and_left, not_exists, not_and_or, notMem_support] at hx
rw [convolution_def]
convert integral_zero G F using 2
ext t
rcases hx (x - t) t with (h | h | h... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | support_convolution_subset_swap | null |
Integrable.integrable_convolution (hf : Integrable f μ)
(hg : Integrable g μ) : Integrable (f ⋆[L, μ] g) μ :=
(hf.convolution_integrand L hg).integral_prod_left | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | Integrable.integrable_convolution | null |
protected _root_.HasCompactSupport.convolution [T2Space G] (hcf : HasCompactSupport f)
(hcg : HasCompactSupport g) : HasCompactSupport (f ⋆[L, μ] g) :=
(hcg.isCompact.add hcf).of_isClosed_subset isClosed_closure <|
closure_minimal
((support_convolution_subset_swap L).trans <| add_subset_add subset_closu... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.convolution | null |
continuousOn_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
(hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : LocallyIntegrable f μ) (hg : ContinuousOn ↿g (s ×ˢ univ)) :
ContinuousOn (fun q : P × G => (f ⋆[L, μ] g q.1) q.2) (s ×ˢ univ) := by
/- First get rid of th... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | continuousOn_convolution_right_with_param | The convolution `f * g` is continuous if `f` is locally integrable and `g` is continuous and
compactly supported. Version where `g` depends on an additional parameter in a subset `s` of
a parameter space `P` (and the compact support `k` is independent of the parameter in `s`). |
continuousOn_convolution_right_with_param_comp {s : Set P} {v : P → G}
(hv : ContinuousOn v s) {g : P → G → E'} {k : Set G} (hk : IsCompact k)
(hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIntegrable f μ)
(hg : ContinuousOn ↿g (s ×ˢ univ)) : ContinuousOn (fun x => (f ⋆[L, μ] g x) (v x)) s := by
... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | continuousOn_convolution_right_with_param_comp | The convolution `f * g` is continuous if `f` is locally integrable and `g` is continuous and
compactly supported. Version where `g` depends on an additional parameter in an open subset `s` of
a parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
given in terms of compositions with ... |
_root_.HasCompactSupport.continuous_convolution_right (hcg : HasCompactSupport g)
(hf : LocallyIntegrable f μ) (hg : Continuous g) : Continuous (f ⋆[L, μ] g) := by
rw [← continuousOn_univ]
let g' : G → G → E' := fun _ q => g q
have : ContinuousOn ↿g' (univ ×ˢ univ) := (hg.comp continuous_snd).continuousOn
e... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.continuous_convolution_right | The convolution is continuous if one function is locally integrable and the other has compact
support and is continuous. |
_root_.BddAbove.continuous_convolution_right_of_integrable
[FirstCountableTopology G] [SecondCountableTopologyEither G E']
(hbg : BddAbove (range fun x => ‖g x‖)) (hf : Integrable f μ) (hg : Continuous g) :
Continuous (f ⋆[L, μ] g) := by
refine continuous_iff_continuousAt.mpr fun x₀ => ?_
have : ∀ᶠ x in... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.BddAbove.continuous_convolution_right_of_integrable | The convolution is continuous if one function is integrable and the other is bounded and
continuous. |
support_convolution_subset : support (f ⋆[L, μ] g) ⊆ support f + support g :=
(support_convolution_subset_swap L).trans (add_comm _ _).subset
variable [IsAddLeftInvariant μ] [IsNegInvariant μ] | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | support_convolution_subset | null |
convolution_flip : g ⋆[L.flip, μ] f = f ⋆[L, μ] g := by
ext1 x
simp_rw [convolution_def]
rw [← integral_sub_left_eq_self _ μ x]
simp_rw [sub_sub_self, flip_apply] | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_flip | Commutativity of convolution |
convolution_eq_swap : (f ⋆[L, μ] g) x = ∫ t, L (f (x - t)) (g t) ∂μ := by
rw [← convolution_flip]; rfl | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_eq_swap | The symmetric definition of convolution. |
convolution_lsmul_swap {f : G → 𝕜} {g : G → F} :
(f ⋆[lsmul 𝕜 𝕜, μ] g : G → F) x = ∫ t, f (x - t) • g t ∂μ :=
convolution_eq_swap _ | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_lsmul_swap | The symmetric definition of convolution where the bilinear operator is scalar multiplication. |
convolution_mul_swap [NormedSpace ℝ 𝕜] {f : G → 𝕜} {g : G → 𝕜} :
(f ⋆[mul 𝕜 𝕜, μ] g) x = ∫ t, f (x - t) * g t ∂μ :=
convolution_eq_swap _ | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_mul_swap | The symmetric definition of convolution where the bilinear operator is multiplication. |
convolution_neg_of_neg_eq (h1 : ∀ᵐ x ∂μ, f (-x) = f x) (h2 : ∀ᵐ x ∂μ, g (-x) = g x) :
(f ⋆[L, μ] g) (-x) = (f ⋆[L, μ] g) x :=
calc
∫ t : G, (L (f t)) (g (-x - t)) ∂μ = ∫ t : G, (L (f (-t))) (g (x + t)) ∂μ := by
apply integral_congr_ae
filter_upwards [h1, (eventually_add_left_iff μ x).2 h2] with t ... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_neg_of_neg_eq | The convolution of two even functions is also even. |
_root_.HasCompactSupport.continuous_convolution_left
(hcf : HasCompactSupport f) (hf : Continuous f) (hg : LocallyIntegrable g μ) :
Continuous (f ⋆[L, μ] g) := by
rw [← convolution_flip]
exact hcf.continuous_convolution_right L.flip hg hf | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.continuous_convolution_left | null |
_root_.BddAbove.continuous_convolution_left_of_integrable
[FirstCountableTopology G] [SecondCountableTopologyEither G E]
(hbf : BddAbove (range fun x => ‖f x‖)) (hf : Continuous f) (hg : Integrable g μ) :
Continuous (f ⋆[L, μ] g) := by
rw [← convolution_flip]
exact hbf.continuous_convolution_right_of_in... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.BddAbove.continuous_convolution_left_of_integrable | null |
convolution_eq_right' {x₀ : G} {R : ℝ} (hf : support f ⊆ ball (0 : G) R)
(hg : ∀ x ∈ ball x₀ R, g x = g x₀) : (f ⋆[L, μ] g) x₀ = ∫ t, L (f t) (g x₀) ∂μ := by
have h2 : ∀ t, L (f t) (g (x₀ - t)) = L (f t) (g x₀) := fun t ↦ by
by_cases ht : t ∈ support f
· have h2t := hf ht
rw [mem_ball_zero_iff] at h... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_eq_right' | Compute `(f ⋆ g) x₀` if the support of the `f` is within `Metric.ball 0 R`, and `g` is constant
on `Metric.ball x₀ R`.
We can simplify the RHS further if we assume `f` is integrable, but also if `L = (•)` or more
generally if `L` has an `AntilipschitzWith`-condition. |
dist_convolution_le' {x₀ : G} {R ε : ℝ} {z₀ : E'} (hε : 0 ≤ ε) (hif : Integrable f μ)
(hf : support f ⊆ ball (0 : G) R) (hmg : AEStronglyMeasurable g μ)
(hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε) :
dist ((f ⋆[L, μ] g : G → F) x₀) (∫ t, L (f t) z₀ ∂μ) ≤ (‖L‖ * ∫ x, ‖f x‖ ∂μ) * ε := by
have hfg : Convolution... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | dist_convolution_le' | Approximate `(f ⋆ g) x₀` if the support of the `f` is bounded within a ball, and `g` is near
`g x₀` on a ball with the same radius around `x₀`. See `dist_convolution_le` for a special case.
We can simplify the second argument of `dist` further if we add some extra type-classes on `E`
and `𝕜` or if `L` is scalar multi... |
dist_convolution_le {f : G → ℝ} {x₀ : G} {R ε : ℝ} {z₀ : E'} (hε : 0 ≤ ε)
(hf : support f ⊆ ball (0 : G) R) (hnf : ∀ x, 0 ≤ f x) (hintf : ∫ x, f x ∂μ = 1)
(hmg : AEStronglyMeasurable g μ) (hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε) :
dist ((f ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀) z₀ ≤ ε := by
have hif : Integrabl... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | dist_convolution_le | Approximate `f ⋆ g` if the support of the `f` is bounded within a ball, and `g` is near `g x₀`
on a ball with the same radius around `x₀`.
This is a special case of `dist_convolution_le'` where `L` is `(•)`, `f` has integral 1 and `f` is
nonnegative. |
convolution_tendsto_right {ι} {g : ι → G → E'} {l : Filter ι} {x₀ : G} {z₀ : E'}
{φ : ι → G → ℝ} {k : ι → G} (hnφ : ∀ᶠ i in l, ∀ x, 0 ≤ φ i x)
(hiφ : ∀ᶠ i in l, ∫ x, φ i x ∂μ = 1)
(hφ : Tendsto (fun n => support (φ n)) l (𝓝 0).smallSets)
(hmg : ∀ᶠ i in l, AEStronglyMeasurable (g i) μ) (hcg : Tendsto (u... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_tendsto_right | `(φ i ⋆ g i) (k i)` tends to `z₀` as `i` tends to some filter `l` if
* `φ` is a sequence of nonnegative functions with integral `1` as `i` tends to `l`;
* The support of `φ` tends to small neighborhoods around `(0 : G)` as `i` tends to `l`;
* `g i` is `mu`-a.e. strongly measurable as `i` tends to `l`;
* `g i x` tends t... |
integral_convolution [MeasurableAdd₂ G] [MeasurableNeg G] [NormedSpace ℝ E]
[NormedSpace ℝ E'] [CompleteSpace E] [CompleteSpace E'] (hf : Integrable f ν)
(hg : Integrable g μ) : ∫ x, (f ⋆[L, ν] g) x ∂μ = L (∫ x, f x ∂ν) (∫ x, g x ∂μ) := by
refine (integral_integral_swap (by apply hf.convolution_integrand L hg... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | integral_convolution | null |
convolution_assoc' (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z))
{x₀ : G} (hfg : ∀ᵐ y ∂μ, ConvolutionExistsAt f g y L ν)
(hgk : ∀ᵐ x ∂ν, ConvolutionExistsAt g k x L₄ μ)
(hi : Integrable (uncurry fun x y => (L₃ (f y)) ((L₄ (g (x - y))) (k (x₀ - x)))) (μ.prod ν)) :
((f ⋆[L, ν] g) ⋆[L₂,... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_assoc' | Convolution is associative. This has a weak but inconvenient integrability condition.
See also `MeasureTheory.convolution_assoc`. |
convolution_assoc (hL : ∀ (x : E) (y : E') (z : E''), L₂ (L x y) z = L₃ x (L₄ y z)) {x₀ : G}
(hf : AEStronglyMeasurable f ν) (hg : AEStronglyMeasurable g μ) (hk : AEStronglyMeasurable k μ)
(hfg : ∀ᵐ y ∂μ, ConvolutionExistsAt f g y L ν)
(hgk : ∀ᵐ x ∂ν, ConvolutionExistsAt (fun x => ‖g x‖) (fun x => ‖k x‖) x ... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_assoc | Convolution is associative. This requires that
* all maps are a.e. strongly measurable w.r.t. one of the measures
* `f ⋆[L, ν] g` exists almost everywhere
* `‖g‖ ⋆[μ] ‖k‖` exists almost everywhere
* `‖f‖ ⋆[ν] (‖g‖ ⋆[μ] ‖k‖)` exists at `x₀` |
convolution_precompR_apply {g : G → E'' →L[𝕜] E'} (hf : LocallyIntegrable f μ)
(hcg : HasCompactSupport g) (hg : Continuous g) (x₀ : G) (x : E'') :
(f ⋆[L.precompR E'', μ] g) x₀ x = (f ⋆[L, μ] fun a => g a x) x₀ := by
have := hcg.convolutionExists_right (L.precompR E'' :) hf hg x₀
simp_rw [convolution_def,... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | convolution_precompR_apply | null |
_root_.HasCompactSupport.hasFDerivAt_convolution_right (hcg : HasCompactSupport g)
(hf : LocallyIntegrable f μ) (hg : ContDiff 𝕜 1 g) (x₀ : G) :
HasFDerivAt (f ⋆[L, μ] g) ((f ⋆[L.precompR G, μ] fderiv 𝕜 g) x₀) x₀ := by
rcases hcg.eq_zero_or_finiteDimensional 𝕜 hg.continuous with (rfl | fin_dim)
· have : ... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.hasFDerivAt_convolution_right | Compute the total derivative of `f ⋆ g` if `g` is `C^1` with compact support and `f` is locally
integrable. To write down the total derivative as a convolution, we use
`ContinuousLinearMap.precompR`. |
_root_.HasCompactSupport.hasFDerivAt_convolution_left [IsNegInvariant μ]
(hcf : HasCompactSupport f) (hf : ContDiff 𝕜 1 f) (hg : LocallyIntegrable g μ) (x₀ : G) :
HasFDerivAt (f ⋆[L, μ] g) ((fderiv 𝕜 f ⋆[L.precompL G, μ] g) x₀) x₀ := by
simp +singlePass only [← convolution_flip]
exact hcf.hasFDerivAt_conv... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.hasFDerivAt_convolution_left | null |
_root_.HasCompactSupport.hasDerivAt_convolution_right (hf : LocallyIntegrable f₀ μ)
(hcg : HasCompactSupport g₀) (hg : ContDiff 𝕜 1 g₀) (x₀ : 𝕜) :
HasDerivAt (f₀ ⋆[L, μ] g₀) ((f₀ ⋆[L, μ] deriv g₀) x₀) x₀ := by
convert (hcg.hasFDerivAt_convolution_right L hf hg x₀).hasDerivAt using 1
rw [convolution_precom... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.hasDerivAt_convolution_right | null |
_root_.HasCompactSupport.hasDerivAt_convolution_left [IsNegInvariant μ]
(hcf : HasCompactSupport f₀) (hf : ContDiff 𝕜 1 f₀) (hg : LocallyIntegrable g₀ μ) (x₀ : 𝕜) :
HasDerivAt (f₀ ⋆[L, μ] g₀) ((deriv f₀ ⋆[L, μ] g₀) x₀) x₀ := by
simp +singlePass only [← convolution_flip]
exact hcf.hasDerivAt_convolution_ri... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.hasDerivAt_convolution_left | null |
hasFDerivAt_convolution_right_with_param {g : P → G → E'} {s : Set P} {k : Set G}
(hs : IsOpen s) (hk : IsCompact k) (hgs : ∀ p, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 1 ↿g (s ×ˢ univ)) (q₀ : P × G)
(hq₀ : q₀.1 ∈ s) :
HasFDerivAt (fun q : P × G => (f ⋆[L, μ] g q... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | hasFDerivAt_convolution_right_with_param | The derivative of the convolution `f * g` is given by `f * Dg`, when `f` is locally integrable
and `g` is `C^1` and compactly supported. Version where `g` depends on an additional parameter in an
open subset `s` of a parameter space `P` (and the compact support `k` is independent of the
parameter in `s`). |
contDiffOn_convolution_right_with_param_aux {G : Type uP} {E' : Type uP} {F : Type uP}
{P : Type uP} [NormedAddCommGroup E'] [NormedAddCommGroup F] [NormedSpace 𝕜 E']
[NormedSpace ℝ F] [NormedSpace 𝕜 F] [MeasurableSpace G]
{μ : Measure G}
[NormedAddCommGroup G] [BorelSpace G] [NormedSpace 𝕜 G] [Norme... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | contDiffOn_convolution_right_with_param_aux | The convolution `f * g` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`).
In this version, all the types belong to the... |
contDiffOn_convolution_right_with_param {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, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIntegrable f μ)
(hg : ContDiffOn 𝕜 n ↿g (s ×ˢ univ)) :
ContDiffOn 𝕜 n (fun q : P ... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | contDiffOn_convolution_right_with_param | The convolution `f * g` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`). |
contDiffOn_convolution_right_with_param_comp {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, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIntegrable f μ)
(hg : ContDiffOn 𝕜 n ↿g (s... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | contDiffOn_convolution_right_with_param_comp | The convolution `f * g` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
given in terms of composition with an additi... |
contDiffOn_convolution_left_with_param [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, ∀ x, p ∈ s → x ∉ k → g p x = 0)
(hf : LocallyIntegrable f μ) (hg : ContDiffOn 𝕜 n ↿g (s ×ˢ... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | contDiffOn_convolution_left_with_param | The convolution `g * f` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`). |
contDiffOn_convolution_left_with_param_comp [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, ∀ x, p ∈ s → x ∉ k → g p x = 0) (hf : LocallyIn... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | contDiffOn_convolution_left_with_param_comp | The convolution `g * f` is `C^n` when `f` is locally integrable and `g` is `C^n` and compactly
supported. Version where `g` depends on an additional parameter in an open subset `s` of a
parameter space `P` (and the compact support `k` is independent of the parameter in `s`),
given in terms of composition with additiona... |
_root_.HasCompactSupport.contDiff_convolution_right {n : ℕ∞} (hcg : HasCompactSupport g)
(hf : LocallyIntegrable f μ) (hg : ContDiff 𝕜 n g) : ContDiff 𝕜 n (f ⋆[L, μ] g) := by
rcases exists_compact_iff_hasCompactSupport.2 hcg with ⟨k, hk, h'k⟩
rw [← contDiffOn_univ]
exact contDiffOn_convolution_right_with_pa... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.contDiff_convolution_right | null |
_root_.HasCompactSupport.contDiff_convolution_left [μ.IsAddLeftInvariant] [μ.IsNegInvariant]
{n : ℕ∞} (hcf : HasCompactSupport f) (hf : ContDiff 𝕜 n f) (hg : LocallyIntegrable g μ) :
ContDiff 𝕜 n (f ⋆[L, μ] g) := by
rw [← convolution_flip]
exact hcf.contDiff_convolution_right L.flip hg hf | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | _root_.HasCompactSupport.contDiff_convolution_left | null |
noncomputable posConvolution (f : ℝ → E) (g : ℝ → E') (L : E →L[ℝ] E' →L[ℝ] F)
(ν : Measure ℝ := by volume_tac) : ℝ → F :=
indicator (Ioi (0 : ℝ)) fun x => ∫ t in 0..x, L (f t) (g (x - t)) ∂ν | def | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | posConvolution | The forward convolution of two functions `f` and `g` on `ℝ`, with respect to a continuous
bilinear map `L` and measure `ν`. It is defined to be the function mapping `x` to
`∫ t in 0..x, L (f t) (g (x - t)) ∂ν` if `0 < x`, and 0 otherwise. |
posConvolution_eq_convolution_indicator (f : ℝ → E) (g : ℝ → E') (L : E →L[ℝ] E' →L[ℝ] F)
(ν : Measure ℝ := by volume_tac) [NoAtoms ν] :
posConvolution f g L ν = convolution (indicator (Ioi 0) f) (indicator (Ioi 0) g) L ν := by
ext1 x
rw [convolution, posConvolution, indicator]
split_ifs with h
· rw [in... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | posConvolution_eq_convolution_indicator | null |
integrable_posConvolution {f : ℝ → E} {g : ℝ → E'} {μ ν : Measure ℝ} [SFinite μ]
[SFinite ν] [IsAddRightInvariant μ] [NoAtoms ν] (hf : IntegrableOn f (Ioi 0) ν)
(hg : IntegrableOn g (Ioi 0) μ) (L : E →L[ℝ] E' →L[ℝ] F) :
Integrable (posConvolution f g L ν) μ := by
rw [← integrable_indicator_iff (measurable... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | integrable_posConvolution | null |
integral_posConvolution [CompleteSpace E] [CompleteSpace E'] [CompleteSpace F]
{μ ν : Measure ℝ}
[SFinite μ] [SFinite ν] [IsAddRightInvariant μ] [NoAtoms ν] {f : ℝ → E} {g : ℝ → E'}
(hf : IntegrableOn f (Ioi 0) ν) (hg : IntegrableOn g (Ioi 0) μ) (L : E →L[ℝ] E' →L[ℝ] F) :
∫ x : ℝ in Ioi 0, ∫ t : ℝ in 0.... | theorem | Analysis | [
"Mathlib.Analysis.Calculus.ContDiff.Basic",
"Mathlib.Analysis.Calculus.ParametricIntegral",
"Mathlib.MeasureTheory.Integral.Prod",
"Mathlib.MeasureTheory.Function.LocallyIntegrable",
"Mathlib.MeasureTheory.Group.Integral",
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Integral.IntervalInteg... | Mathlib/Analysis/Convolution.lean | integral_posConvolution | The integral over `Ioi 0` of a forward convolution of two functions is equal to the product
of their integrals over this set. (Compare `integral_convolution` for the two-sided convolution.) |
of a proof needing to construct a sequence by induction in the middle of the proof. | example | Analysis | [
"Mathlib.Analysis.SpecificLimits.Basic"
] | Mathlib/Analysis/Hofer.lean | of | null |
hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
{ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' := by
by_contra H
have reformulation : ∀ (x') (k : ℕ), ε * ϕ x... | theorem | Analysis | [
"Mathlib.Analysis.SpecificLimits.Basic"
] | Mathlib/Analysis/Hofer.lean | hofer | null |
@[to_additive /-- Additive convolution of functions -/]
noncomputable mlconvolution (f g : G → ℝ≥0∞) (μ : Measure G) :
G → ℝ≥0∞ := fun x ↦ ∫⁻ y, (f y) * (g (y⁻¹ * x)) ∂μ | def | Analysis | [
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Group.LIntegral"
] | Mathlib/Analysis/LConvolution.lean | mlconvolution | Multiplicative convolution of functions. |
@[to_additive /-- The definition of additive convolution of functions. -/]
mlconvolution_def {f g : G → ℝ≥0∞} {μ : Measure G} {x : G} :
(f ⋆ₘₗ[μ] g) x = ∫⁻ y, (f y) * (g (y⁻¹ * x)) ∂μ := rfl | theorem | Analysis | [
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Group.LIntegral"
] | Mathlib/Analysis/LConvolution.lean | mlconvolution_def | Scoped notation for the multiplicative convolution of functions with respect to a measure `μ`.
-/
scoped[MeasureTheory] notation:67 f " ⋆ₘₗ["μ:67"] " g:66 => MeasureTheory.mlconvolution f g μ
/-- Scoped notation for the multiplicative convolution of functions with respect to `volume`. -/
scoped[MeasureTheory] notation... |
@[to_additive (attr := simp)
/-- Convolution of the zero function with a function returns the zero function. -/]
zero_mlconvolution (f : G → ℝ≥0∞) (μ : Measure G) : 0 ⋆ₘₗ[μ] f = 0 := by
ext; simp [mlconvolution] | theorem | Analysis | [
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Group.LIntegral"
] | Mathlib/Analysis/LConvolution.lean | zero_mlconvolution | Convolution of the zero function with a function returns the zero function. |
@[to_additive (attr := simp)
/-- Convolution of a function with the zero function returns the zero function. -/]
mlconvolution_zero (f : G → ℝ≥0∞) (μ : Measure G) : f ⋆ₘₗ[μ] 0 = 0 := by
ext; simp [mlconvolution] | theorem | Analysis | [
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Group.LIntegral"
] | Mathlib/Analysis/LConvolution.lean | mlconvolution_zero | Convolution of a function with the zero function returns the zero function. |
@[to_additive (attr := measurability, fun_prop)
/-- The convolution of measurable functions is measurable. -/]
measurable_mlconvolution {f g : G → ℝ≥0∞} (μ : Measure G) [SFinite μ]
(hf : Measurable f) (hg : Measurable g) : Measurable (f ⋆ₘₗ[μ] g) := by
unfold mlconvolution
fun_prop | theorem | Analysis | [
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Group.LIntegral"
] | Mathlib/Analysis/LConvolution.lean | measurable_mlconvolution | The convolution of measurable functions is measurable. |
@[to_additive (attr := measurability, fun_prop)
/-- The convolution of `AEMeasurable` functions is `AEMeasurable`. -/]
aemeasurable_mlconvolution {f g : G → ℝ≥0∞}
(hf : AEMeasurable f μ) (hg : AEMeasurable g μ) :
AEMeasurable (f ⋆ₘₗ[μ] g) μ := by
unfold mlconvolution
fun_prop
@[to_additive] | theorem | Analysis | [
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Group.LIntegral"
] | Mathlib/Analysis/LConvolution.lean | aemeasurable_mlconvolution | The convolution of `AEMeasurable` functions is `AEMeasurable`. |
mlconvolution_assoc₀ {f g k : G → ℝ≥0∞}
(hf : AEMeasurable f μ) (hg : AEMeasurable g μ) (hk : AEMeasurable k μ) :
f ⋆ₘₗ[μ] g ⋆ₘₗ[μ] k = (f ⋆ₘₗ[μ] g) ⋆ₘₗ[μ] k := by
ext x
simp only [mlconvolution_def]
conv in f _ * (∫⁻ _ , _ ∂μ) =>
rw [← lintegral_const_mul'' _ (by fun_prop), ← lintegral_mul_left_eq_se... | theorem | Analysis | [
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Group.LIntegral"
] | Mathlib/Analysis/LConvolution.lean | mlconvolution_assoc₀ | null |
mlconvolution_assoc {f g k : G → ℝ≥0∞}
(hf : Measurable f) (hg : Measurable g) (hk : Measurable k) :
f ⋆ₘₗ[μ] g ⋆ₘₗ[μ] k = (f ⋆ₘₗ[μ] g) ⋆ₘₗ[μ] k :=
mlconvolution_assoc₀ hf.aemeasurable hg.aemeasurable hk.aemeasurable | theorem | Analysis | [
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Group.LIntegral"
] | Mathlib/Analysis/LConvolution.lean | mlconvolution_assoc | null |
@[to_additive /-- Convolution is commutative when the group is commutative. -/]
mlconvolution_comm [IsMulLeftInvariant μ] [IsInvInvariant μ] {f g : G → ℝ≥0∞} :
(f ⋆ₘₗ[μ] g) = (g ⋆ₘₗ[μ] f) := by
ext x
simp only [mlconvolution_def]
rw [← lintegral_mul_left_eq_self _ x, ← lintegral_inv_eq_self]
simp [mul_comm] | theorem | Analysis | [
"Mathlib.MeasureTheory.Group.Prod",
"Mathlib.MeasureTheory.Group.LIntegral"
] | Mathlib/Analysis/LConvolution.lean | mlconvolution_comm | Convolution is commutative when the group is commutative. |
protected seminormedAddCommGroup : SeminormedAddCommGroup (Matrix m n α) :=
Pi.seminormedAddCommGroup
attribute [local instance] Matrix.seminormedAddCommGroup | def | Analysis | [
"Mathlib.Analysis.InnerProductSpace.PiL2"
] | Mathlib/Analysis/Matrix.lean | seminormedAddCommGroup | Seminormed group instance (using sup norm of sup norm) for matrices over a seminormed group. Not
declared as an instance because there are several natural choices for defining the norm of a
matrix. |
norm_def (A : Matrix m n α) : ‖A‖ = ‖fun i j => A i j‖ := rfl | theorem | Analysis | [
"Mathlib.Analysis.InnerProductSpace.PiL2"
] | Mathlib/Analysis/Matrix.lean | norm_def | null |
norm_eq_sup_sup_nnnorm (A : Matrix m n α) :
‖A‖ = Finset.sup Finset.univ fun i ↦ Finset.sup Finset.univ fun j ↦ ‖A i j‖₊ := by
simp_rw [Matrix.norm_def, Pi.norm_def, Pi.nnnorm_def] | lemma | Analysis | [
"Mathlib.Analysis.InnerProductSpace.PiL2"
] | Mathlib/Analysis/Matrix.lean | norm_eq_sup_sup_nnnorm | The norm of a matrix is the sup of the sup of the nnnorm of the entries |
nnnorm_def (A : Matrix m n α) : ‖A‖₊ = ‖fun i j => A i j‖₊ := rfl | theorem | Analysis | [
"Mathlib.Analysis.InnerProductSpace.PiL2"
] | Mathlib/Analysis/Matrix.lean | nnnorm_def | null |
norm_le_iff {r : ℝ} (hr : 0 ≤ r) {A : Matrix m n α} : ‖A‖ ≤ r ↔ ∀ i j, ‖A i j‖ ≤ r := by
simp_rw [norm_def, pi_norm_le_iff_of_nonneg hr] | theorem | Analysis | [
"Mathlib.Analysis.InnerProductSpace.PiL2"
] | Mathlib/Analysis/Matrix.lean | norm_le_iff | null |
nnnorm_le_iff {r : ℝ≥0} {A : Matrix m n α} : ‖A‖₊ ≤ r ↔ ∀ i j, ‖A i j‖₊ ≤ r := by
simp_rw [nnnorm_def, pi_nnnorm_le_iff] | theorem | Analysis | [
"Mathlib.Analysis.InnerProductSpace.PiL2"
] | Mathlib/Analysis/Matrix.lean | nnnorm_le_iff | null |
norm_lt_iff {r : ℝ} (hr : 0 < r) {A : Matrix m n α} : ‖A‖ < r ↔ ∀ i j, ‖A i j‖ < r := by
simp_rw [norm_def, pi_norm_lt_iff hr] | theorem | Analysis | [
"Mathlib.Analysis.InnerProductSpace.PiL2"
] | Mathlib/Analysis/Matrix.lean | norm_lt_iff | null |
nnnorm_lt_iff {r : ℝ≥0} (hr : 0 < r) {A : Matrix m n α} :
‖A‖₊ < r ↔ ∀ i j, ‖A i j‖₊ < r := by
simp_rw [nnnorm_def, pi_nnnorm_lt_iff hr] | theorem | Analysis | [
"Mathlib.Analysis.InnerProductSpace.PiL2"
] | Mathlib/Analysis/Matrix.lean | nnnorm_lt_iff | null |
norm_entry_le_entrywise_sup_norm (A : Matrix m n α) {i : m} {j : n} : ‖A i j‖ ≤ ‖A‖ :=
(norm_le_pi_norm (A i) j).trans (norm_le_pi_norm A i) | theorem | Analysis | [
"Mathlib.Analysis.InnerProductSpace.PiL2"
] | Mathlib/Analysis/Matrix.lean | norm_entry_le_entrywise_sup_norm | null |
nnnorm_entry_le_entrywise_sup_nnnorm (A : Matrix m n α) {i : m} {j : n} : ‖A i j‖₊ ≤ ‖A‖₊ :=
(nnnorm_le_pi_nnnorm (A i) j).trans (nnnorm_le_pi_nnnorm A i)
@[simp] | theorem | Analysis | [
"Mathlib.Analysis.InnerProductSpace.PiL2"
] | Mathlib/Analysis/Matrix.lean | nnnorm_entry_le_entrywise_sup_nnnorm | null |
nnnorm_map_eq (A : Matrix m n α) (f : α → β) (hf : ∀ a, ‖f a‖₊ = ‖a‖₊) :
‖A.map f‖₊ = ‖A‖₊ := by
simp only [nnnorm_def, Pi.nnnorm_def, Matrix.map_apply, hf]
@[simp] | theorem | Analysis | [
"Mathlib.Analysis.InnerProductSpace.PiL2"
] | Mathlib/Analysis/Matrix.lean | nnnorm_map_eq | null |
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