The Fourier transform

We set up the Fourier transform for complex-valued functions on finite-dimensional spaces.

Design choices

In namespace VectorFourier, we define the Fourier integral in the following context:

• 𝕜 is a commutative ring.
• V and W are 𝕜-modules.
• e is a unitary additive character of 𝕜, i.e. an AddChar 𝕜 Circle.
• μ is a measure on V.
• L is a 𝕜-bilinear form V × W → 𝕜.
• E is a complete normed ℂ-vector space.

With these definitions, we define fourierIntegral to be the map from functions V → E to functions W → E that sends f to

fun w ↦ ∫ v in V, e (-L v w) • f v ∂μ,

This includes the cases W is the dual of V and L is the canonical pairing, or W = V and L is a bilinear form (e.g. an inner product).

In namespace Fourier, we consider the more familiar special case when V = W = 𝕜 and L is the multiplication map (but still allowing 𝕜 to be an arbitrary ring equipped with a measure).

The most familiar case of all is when V = W = 𝕜 = ℝ, L is multiplication, μ is volume, and e is Real.fourierChar, i.e. the character fun x ↦ exp ((2 * π * x) * I) (for which we introduce the notation 𝐞 in the locale FourierTransform).

Another familiar case (which generalizes the previous one) is when V = W is an inner product space over ℝ and L is the scalar product. We introduce two notations 𝓕 for the Fourier transform in this case and 𝓕⁻ f (v) = 𝓕 f (-v) for the inverse Fourier transform. These notations make in particular sense for V = W = ℝ.

Main results

At present the only nontrivial lemma we prove is fourierIntegral_continuous, stating that the Fourier transform of an integrable function is continuous (under mild assumptions).

Fourier theory for functions on general vector spaces

def VectorFourier.fourierIntegral {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 Circle) (μ : MeasureTheory.Measure V) (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (w : W) : E

The Fourier transform integral for f : V → E, with respect to a bilinear form L : V × W → 𝕜 and an additive character e.

Equations

• VectorFourier.fourierIntegral e μ L f w = ∫ (v : V), e (-(L v) w) • f v ∂μ

theorem VectorFourier.fourierIntegral_const_smul {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 Circle) (μ : MeasureTheory.Measure V) (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (r : ℂ) : fourierIntegral e μ L (r • f) = r • fourierIntegral e μ L f

theorem VectorFourier.norm_fourierIntegral_le_integral_norm {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 Circle) (μ : MeasureTheory.Measure V) (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (w : W) : ‖fourierIntegral e μ L f w‖ ≤ ∫ (v : V), ‖f v‖ ∂μ

The uniform norm of the Fourier integral of f is bounded by the L¹ norm of f.

theorem VectorFourier.fourierIntegral_comp_add_right {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] [MeasurableAdd V] (e : AddChar 𝕜 Circle) (μ : MeasureTheory.Measure V) [μ.IsAddRightInvariant] (L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜) (f : V → E) (v₀ : V) : fourierIntegral e μ L (f ∘ fun (v : V) => v + v₀) = fun (w : W) => e ((L v₀) w) • fourierIntegral e μ L f w

The Fourier integral converts right-translation into scalar multiplication by a phase factor.

In this section we assume 𝕜, V, W have topologies, and L, e are continuous (but f needn't be). This is used to ensure that e (-L v w) is (a.e. strongly) measurable. We could get away with imposing only a measurable-space structure on 𝕜 (it doesn't have to be the Borel sigma-algebra of a topology); but it seems hard to imagine cases where this extra generality would be useful, and allowing it would complicate matters in the most important use cases.

theorem VectorFourier.fourierIntegral_convergent_iff {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V] [TopologicalSpace W] {e : AddChar 𝕜 Circle} {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜} (he : Continuous ⇑e) (hL : Continuous fun (p : V × W) => (L p.1) p.2) {f : V → E} (w : W) : MeasureTheory.Integrable (fun (v : V) => e (-(L v) w) • f v) μ ↔ MeasureTheory.Integrable f μ

For any w, the Fourier integral is convergent iff f is integrable.

theorem VectorFourier.fourierIntegral_add {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V] [TopologicalSpace W] {e : AddChar 𝕜 Circle} {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜} (he : Continuous ⇑e) (hL : Continuous fun (p : V × W) => (L p.1) p.2) {f g : V → E} (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g μ) : fourierIntegral e μ L (f + g) = fourierIntegral e μ L f + fourierIntegral e μ L g

theorem VectorFourier.fourierIntegral_continuous {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace ℂ E] [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V] [TopologicalSpace W] {e : AddChar 𝕜 Circle} {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜} [FirstCountableTopology W] (he : Continuous ⇑e) (hL : Continuous fun (p : V × W) => (L p.1) p.2) {f : V → E} (hf : MeasureTheory.Integrable f μ) : Continuous (fourierIntegral e μ L f)

The Fourier integral of an L^1 function is a continuous function.

theorem VectorFourier.integral_bilin_fourierIntegral_eq_flip {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {E : Type u_4} {F : Type u_5} {G : Type u_6} [NormedAddCommGroup E] [NormedSpace ℂ E] [NormedAddCommGroup F] [NormedSpace ℂ F] [NormedAddCommGroup G] [NormedSpace ℂ G] [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V] [TopologicalSpace W] [MeasurableSpace W] [BorelSpace W] {e : AddChar 𝕜 Circle} {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜} {ν : MeasureTheory.Measure W} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] [SecondCountableTopology V] [CompleteSpace E] [CompleteSpace F] {f : V → E} {g : W → F} (M : E →L[ℂ] F →L[ℂ] G) (he : Continuous ⇑e) (hL : Continuous fun (p : V × W) => (L p.1) p.2) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g ν) : ∫ (ξ : W), (M (fourierIntegral e μ L f ξ)) (g ξ) ∂ν = ∫ (x : V), (M (f x)) (fourierIntegral e ν L.flip g x) ∂μ

The Fourier transform satisfies ∫ 𝓕 f * g = ∫ f * 𝓕 g, i.e., it is self-adjoint. Version where the multiplication is replaced by a general bilinear form M.

theorem VectorFourier.integral_fourierIntegral_smul_eq_flip {𝕜 : Type u_1} [CommRing 𝕜] {V : Type u_2} [AddCommGroup V] [Module 𝕜 V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module 𝕜 W] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace ℂ F] [TopologicalSpace 𝕜] [IsTopologicalRing 𝕜] [TopologicalSpace V] [BorelSpace V] [TopologicalSpace W] [MeasurableSpace W] [BorelSpace W] {e : AddChar 𝕜 Circle} {μ : MeasureTheory.Measure V} {L : V →ₗ[𝕜] W →ₗ[𝕜] 𝕜} {ν : MeasureTheory.Measure W} [MeasureTheory.SigmaFinite μ] [MeasureTheory.SigmaFinite ν] [SecondCountableTopology V] [CompleteSpace F] {f : V → ℂ} {g : W → F} (he : Continuous ⇑e) (hL : Continuous fun (p : V × W) => (L p.1) p.2) (hf : MeasureTheory.Integrable f μ) (hg : MeasureTheory.Integrable g ν) : ∫ (ξ : W), fourierIntegral e μ L f ξ • g ξ ∂ν = ∫ (x : V), f x • fourierIntegral e ν L.flip g x ∂μ

The Fourier transform satisfies ∫ 𝓕 f * g = ∫ f * 𝓕 g, i.e., it is self-adjoint.

theorem VectorFourier.fourierIntegral_continuousLinearMap_apply {𝕜 : Type u_1} {E : Type u_3} {F : Type u_4} {V : Type u_5} {W : Type u_6} [NontriviallyNormedField 𝕜] [NormedAddCommGroup V] [NormedSpace 𝕜 V] [MeasurableSpace V] [BorelSpace V] [NormedAddCommGroup W] [NormedSpace 𝕜 W] {e : AddChar 𝕜 Circle} {μ : MeasureTheory.Measure V} {L : V →L[𝕜] W →L[𝕜] 𝕜} [NormedAddCommGroup F] [NormedSpace ℝ F] [NormedAddCommGroup E] [NormedSpace ℂ E] {f : V → F →L[ℝ] E} {a : F} {w : W} (he : Continuous ⇑e) (hf : MeasureTheory.Integrable f μ) : (fourierIntegral e μ L.toLinearMap₂ f w) a = fourierIntegral e μ L.toLinearMap₂ (fun (x : V) => (f x) a) w

theorem VectorFourier.fourierIntegral_continuousMultilinearMap_apply {𝕜 : Type u_1} {ι : Type u_2} {E : Type u_3} {V : Type u_5} {W : Type u_6} [Fintype ι] [NontriviallyNormedField 𝕜] [NormedAddCommGroup V] [NormedSpace 𝕜 V] [MeasurableSpace V] [BorelSpace V] [NormedAddCommGroup W] [NormedSpace 𝕜 W] {e : AddChar 𝕜 Circle} {μ : MeasureTheory.Measure V} {L : V →L[𝕜] W →L[𝕜] 𝕜} [NormedAddCommGroup E] [NormedSpace ℂ E] {M : ι → Type u_7} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace ℝ (M i)] {f : V → ContinuousMultilinearMap ℝ M E} {m : (i : ι) → M i} {w : W} (he : Continuous ⇑e) (hf : MeasureTheory.Integrable f μ) : (fourierIntegral e μ L.toLinearMap₂ f w) m = fourierIntegral e μ L.toLinearMap₂ (fun (x : V) => (f x) m) w

Fourier theory for functions on 𝕜

def Fourier.fourierIntegral {𝕜 : Type u_1} [CommRing 𝕜] [MeasurableSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 Circle) (μ : MeasureTheory.Measure 𝕜) (f : 𝕜 → E) (w : 𝕜) : E

The Fourier transform integral for f : 𝕜 → E, with respect to the measure μ and additive character e.

Equations

• Fourier.fourierIntegral e μ f w = VectorFourier.fourierIntegral e μ (LinearMap.mul 𝕜 𝕜) f w

theorem Fourier.fourierIntegral_def {𝕜 : Type u_1} [CommRing 𝕜] [MeasurableSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 Circle) (μ : MeasureTheory.Measure 𝕜) (f : 𝕜 → E) (w : 𝕜) : fourierIntegral e μ f w = ∫ (v : 𝕜), e (-(v * w)) • f v ∂μ

theorem Fourier.fourierIntegral_const_smul {𝕜 : Type u_1} [CommRing 𝕜] [MeasurableSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 Circle) (μ : MeasureTheory.Measure 𝕜) (f : 𝕜 → E) (r : ℂ) : fourierIntegral e μ (r • f) = r • fourierIntegral e μ f

theorem Fourier.norm_fourierIntegral_le_integral_norm {𝕜 : Type u_1} [CommRing 𝕜] [MeasurableSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] (e : AddChar 𝕜 Circle) (μ : MeasureTheory.Measure 𝕜) (f : 𝕜 → E) (w : 𝕜) : ‖fourierIntegral e μ f w‖ ≤ ∫ (x : 𝕜), ‖f x‖ ∂μ

The uniform norm of the Fourier transform of f is bounded by the L¹ norm of f.

theorem Fourier.fourierIntegral_comp_add_right {𝕜 : Type u_1} [CommRing 𝕜] [MeasurableSpace 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace ℂ E] [MeasurableAdd 𝕜] (e : AddChar 𝕜 Circle) (μ : MeasureTheory.Measure 𝕜) [μ.IsAddRightInvariant] (f : 𝕜 → E) (v₀ : 𝕜) : fourierIntegral e μ (f ∘ fun (v : 𝕜) => v + v₀) = fun (w : 𝕜) => e (v₀ * w) • fourierIntegral e μ f w

The Fourier transform converts right-translation into scalar multiplication by a phase factor.

def Real.fourierChar : AddChar ℝ Circle

The standard additive character of ℝ, given by fun x ↦ exp (2 * π * x * I). Denoted as 𝐞 within the Real.FourierTransform namespace.

Equations

• Real.fourierChar = { toFun := fun (z : ℝ) => Circle.exp (2 * Real.pi * z), map_zero_eq_one' := Real.fourierChar.proof_2, map_add_eq_mul' := Real.fourierChar.proof_3 }

def FourierTransform.«term𝐞» : Lean.ParserDescr

The standard additive character of ℝ, given by fun x ↦ exp (2 * π * x * I). Denoted as 𝐞 within the Real.FourierTransform namespace.

Equations

• FourierTransform.«term𝐞» = Lean.ParserDescr.node `FourierTransform.«term𝐞» 1024 (Lean.ParserDescr.symbol "𝐞")

theorem Real.fourierChar_apply (x : ℝ) : ↑(fourierChar x) = Complex.exp (↑(2 * Real.pi * x) * Complex.I)

theorem Real.continuous_fourierChar : Continuous ⇑fourierChar

theorem Real.vector_fourierIntegral_eq_integral_exp_smul {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [AddCommGroup V] [Module ℝ V] [MeasurableSpace V] {W : Type u_3} [AddCommGroup W] [Module ℝ W] (L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ) (μ : MeasureTheory.Measure V) (f : V → E) (w : W) : VectorFourier.fourierIntegral fourierChar μ L f w = ∫ (v : V), Complex.exp (↑(-2 * Real.pi * (L v) w) * Complex.I) • f v ∂μ

@[simp]

theorem Real.fourierIntegral_convergent_iff' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} {W : Type u_3} [NormedAddCommGroup V] [NormedSpace ℝ V] [NormedAddCommGroup W] [NormedSpace ℝ W] [MeasurableSpace V] [BorelSpace V] {μ : MeasureTheory.Measure V} {f : V → E} (L : V →L[ℝ] W →L[ℝ] ℝ) (w : W) : MeasureTheory.Integrable (fun (v : V) => fourierChar (-(L v) w) • f v) μ ↔ MeasureTheory.Integrable f μ

The Fourier integral is well defined iff the function is integrable. Version with a general continuous bilinear function L. For the specialization to the inner product in an inner product space, see Real.fourierIntegral_convergent_iff.

theorem Real.fourierIntegral_continuousLinearMap_apply' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {F : Type u_3} {V : Type u_4} {W : Type u_5} [NormedAddCommGroup V] [NormedSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [NormedAddCommGroup W] [NormedSpace ℝ W] {μ : MeasureTheory.Measure V} {L : V →L[ℝ] W →L[ℝ] ℝ} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : V → F →L[ℝ] E} {a : F} {w : W} (hf : MeasureTheory.Integrable f μ) : (VectorFourier.fourierIntegral fourierChar μ L.toLinearMap₂ f w) a = VectorFourier.fourierIntegral fourierChar μ L.toLinearMap₂ (fun (x : V) => (f x) a) w

theorem Real.fourierIntegral_continuousMultilinearMap_apply' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {ι : Type u_2} {V : Type u_4} {W : Type u_5} [Fintype ι] [NormedAddCommGroup V] [NormedSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [NormedAddCommGroup W] [NormedSpace ℝ W] {μ : MeasureTheory.Measure V} {L : V →L[ℝ] W →L[ℝ] ℝ} {M : ι → Type u_6} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace ℝ (M i)] {f : V → ContinuousMultilinearMap ℝ M E} {m : (i : ι) → M i} {w : W} (hf : MeasureTheory.Integrable f μ) : (VectorFourier.fourierIntegral fourierChar μ L.toLinearMap₂ f w) m = VectorFourier.fourierIntegral fourierChar μ L.toLinearMap₂ (fun (x : V) => (f x) m) w

@[simp]

theorem Real.fourierIntegral_convergent_iff {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] {μ : MeasureTheory.Measure V} {f : V → E} (w : V) : MeasureTheory.Integrable (fun (v : V) => fourierChar (-inner v w) • f v) μ ↔ MeasureTheory.Integrable f μ

def Real.fourierIntegral {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E) (w : V) : E

The Fourier transform of a function on an inner product space, with respect to the standard additive character ω ↦ exp (2 i π ω). Denoted as 𝓕 within the Real.FourierTransform namespace.

Equations

• Real.fourierIntegral f w = VectorFourier.fourierIntegral Real.fourierChar MeasureTheory.volume (innerₗ V) f w

def Real.fourierIntegralInv {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E) (w : V) : E

The inverse Fourier transform of a function on an inner product space, defined as the Fourier transform but with opposite sign in the exponential. Denoted as 𝓕⁻¹ within the Real.FourierTransform namespace.

Equations

• Real.fourierIntegralInv f w = VectorFourier.fourierIntegral Real.fourierChar MeasureTheory.volume (-innerₗ V) f w

def FourierTransform.term𝓕 : Lean.ParserDescr

The Fourier transform of a function on an inner product space, with respect to the standard additive character ω ↦ exp (2 i π ω). Denoted as 𝓕 within the Real.FourierTransform namespace.

Equations

• FourierTransform.term𝓕 = Lean.ParserDescr.node `FourierTransform.term𝓕 1024 (Lean.ParserDescr.symbol "𝓕")

def FourierTransform.«term𝓕⁻» : Lean.ParserDescr

The inverse Fourier transform of a function on an inner product space, defined as the Fourier transform but with opposite sign in the exponential. Denoted as 𝓕⁻¹ within the Real.FourierTransform namespace.

Equations

• FourierTransform.«term𝓕⁻» = Lean.ParserDescr.node `FourierTransform.«term𝓕⁻» 1024 (Lean.ParserDescr.symbol "𝓕⁻")

theorem Real.fourierIntegral_eq {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E) (w : V) : fourierIntegral f w = ∫ (v : V), fourierChar (-inner v w) • f v

theorem Real.fourierIntegral_eq' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E) (w : V) : fourierIntegral f w = ∫ (v : V), Complex.exp (↑(-2 * Real.pi * inner v w) * Complex.I) • f v

theorem Real.fourierIntegralInv_eq {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E) (w : V) : fourierIntegralInv f w = ∫ (v : V), fourierChar (inner v w) • f v

theorem Real.fourierIntegralInv_eq' {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E) (w : V) : fourierIntegralInv f w = ∫ (v : V), Complex.exp (↑(2 * Real.pi * inner v w) * Complex.I) • f v

theorem Real.fourierIntegral_comp_linearIsometry {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] {W : Type u_3} [NormedAddCommGroup W] [InnerProductSpace ℝ W] [MeasurableSpace W] [BorelSpace W] [FiniteDimensional ℝ W] [FiniteDimensional ℝ V] (A : W ≃ₗᵢ[ℝ] V) (f : V → E) (w : W) : fourierIntegral (f ∘ ⇑A) w = fourierIntegral f (A w)

theorem Real.fourierIntegralInv_eq_fourierIntegral_neg {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E) (w : V) : fourierIntegralInv f w = fourierIntegral f (-w)

theorem Real.fourierIntegralInv_eq_fourierIntegral_comp_neg {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E) : fourierIntegralInv f = fourierIntegral fun (x : V) => f (-x)

theorem Real.fourierIntegralInv_comm {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] (f : V → E) : fourierIntegral (fourierIntegralInv f) = fourierIntegralInv (fourierIntegral f)

theorem Real.fourierIntegralInv_comp_linearIsometry {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] {W : Type u_3} [NormedAddCommGroup W] [InnerProductSpace ℝ W] [MeasurableSpace W] [BorelSpace W] [FiniteDimensional ℝ W] [FiniteDimensional ℝ V] (A : W ≃ₗᵢ[ℝ] V) (f : V → E) (w : W) : fourierIntegralInv (f ∘ ⇑A) w = fourierIntegralInv f (A w)

theorem Real.fourierIntegral_real_eq {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E) (w : ℝ) : fourierIntegral f w = ∫ (v : ℝ), fourierChar (-(v * w)) • f v

theorem Real.fourierIntegral_real_eq_integral_exp_smul {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] (f : ℝ → E) (w : ℝ) : fourierIntegral f w = ∫ (v : ℝ), Complex.exp (↑(-2 * Real.pi * v * w) * Complex.I) • f v

theorem Real.fourierIntegral_continuousLinearMap_apply {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {F : Type u_4} [NormedAddCommGroup F] [NormedSpace ℝ F] {f : V → F →L[ℝ] E} {a : F} {v : V} (hf : MeasureTheory.Integrable f MeasureTheory.volume) : (fourierIntegral f v) a = fourierIntegral (fun (x : V) => (f x) a) v

theorem Real.fourierIntegral_continuousMultilinearMap_apply {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {V : Type u_2} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [MeasurableSpace V] [BorelSpace V] [FiniteDimensional ℝ V] {ι : Type u_4} [Fintype ι] {M : ι → Type u_5} [(i : ι) → NormedAddCommGroup (M i)] [(i : ι) → NormedSpace ℝ (M i)] {f : V → ContinuousMultilinearMap ℝ M E} {m : (i : ι) → M i} {v : V} (hf : MeasureTheory.Integrable f MeasureTheory.volume) : (fourierIntegral f v) m = fourierIntegral (fun (x : V) => (f x) m) v