Fourier transform of the Gaussian

We prove that the Fourier transform of the Gaussian function is another Gaussian:

• integral_cexp_quadratic: general formula for ∫ (x : ℝ), exp (b * x ^ 2 + c * x + d)
• fourierIntegral_gaussian: for all complex b and t with 0 < re b, we have ∫ x:ℝ, exp (I * t * x) * exp (-b * x^2) = (π / b) ^ (1 / 2) * exp (-t ^ 2 / (4 * b)).
• fourierIntegral_gaussian_pi: a variant with b and t scaled to give a more symmetric statement, and formulated in terms of the Fourier transform operator 𝓕.

We also give versions of these formulas in finite-dimensional inner product spaces, see integral_cexp_neg_mul_sq_norm_add and fourierIntegral_gaussian_innerProductSpace.

Fourier integral of Gaussian functions

def GaussianFourier.verticalIntegral (b : ℂ) (c : ℝ) (T : ℝ) : ℂ

The integral of the Gaussian function over the vertical edges of a rectangle with vertices at (±T, 0) and (±T, c).

Equations

• GaussianFourier.verticalIntegral b c T = ∫ (y : ℝ) in 0 ..c, Complex.I * ((-b * (↑T + ↑y * Complex.I) ^ 2).exp - (-b * (↑T - ↑y * Complex.I) ^ 2).exp)

theorem GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I (b : ℂ) (c : ℝ) (T : ℝ) : ‖(-b * (↑T + ↑c * Complex.I) ^ 2).exp‖ = (-(b.re * T ^ 2 - 2 * b.im * c * T - b.re * c ^ 2)).exp

Explicit formula for the norm of the Gaussian function along the vertical edges.

theorem GaussianFourier.norm_cexp_neg_mul_sq_add_mul_I' {b : ℂ} (hb : b.re ≠ 0) (c : ℝ) (T : ℝ) : ‖(-b * (↑T + ↑c * Complex.I) ^ 2).exp‖ = (-(b.re * (T - b.im * c / b.re) ^ 2 - c ^ 2 * (b.im ^ 2 / b.re + b.re))).exp

theorem GaussianFourier.verticalIntegral_norm_le {b : ℂ} (hb : 0 < b.re) (c : ℝ) {T : ℝ} (hT : 0 ≤ T) : ‖GaussianFourier.verticalIntegral b c T‖ ≤ 2 * |c| * (-(b.re * T ^ 2 - 2 * |b.im| * |c| * T - b.re * c ^ 2)).exp

theorem GaussianFourier.tendsto_verticalIntegral {b : ℂ} (hb : 0 < b.re) (c : ℝ) : Filter.Tendsto (GaussianFourier.verticalIntegral b c) Filter.atTop (nhds 0)

theorem GaussianFourier.integrable_cexp_neg_mul_sq_add_real_mul_I {b : ℂ} (hb : 0 < b.re) (c : ℝ) : MeasureTheory.Integrable (fun (x : ℝ) => (-b * (↑x + ↑c * Complex.I) ^ 2).exp) MeasureTheory.volume

theorem GaussianFourier.integral_cexp_neg_mul_sq_add_real_mul_I {b : ℂ} (hb : 0 < b.re) (c : ℝ) : ∫ (x : ℝ), (-b * (↑x + ↑c * Complex.I) ^ 2).exp = (↑Real.pi / b) ^ (1 / 2)

theorem integral_cexp_quadratic {b : ℂ} (hb : b.re < 0) (c : ℂ) (d : ℂ) : ∫ (x : ℝ), (b * ↑x ^ 2 + c * ↑x + d).exp = (↑Real.pi / -b) ^ (1 / 2) * (d - c ^ 2 / (4 * b)).exp

theorem integrable_cexp_quadratic' {b : ℂ} (hb : b.re < 0) (c : ℂ) (d : ℂ) : MeasureTheory.Integrable (fun (x : ℝ) => (b * ↑x ^ 2 + c * ↑x + d).exp) MeasureTheory.volume

theorem integrable_cexp_quadratic {b : ℂ} (hb : 0 < b.re) (c : ℂ) (d : ℂ) : MeasureTheory.Integrable (fun (x : ℝ) => (-b * ↑x ^ 2 + c * ↑x + d).exp) MeasureTheory.volume

theorem fourierIntegral_gaussian {b : ℂ} (hb : 0 < b.re) (t : ℂ) : ∫ (x : ℝ), (Complex.I * t * ↑x).exp * (-b * ↑x ^ 2).exp = (↑Real.pi / b) ^ (1 / 2) * (-t ^ 2 / (4 * b)).exp

@[deprecated fourierIntegral_gaussian]

theorem fourier_transform_gaussian {b : ℂ} (hb : 0 < b.re) (t : ℂ) : ∫ (x : ℝ), (Complex.I * t * ↑x).exp * (-b * ↑x ^ 2).exp = (↑Real.pi / b) ^ (1 / 2) * (-t ^ 2 / (4 * b)).exp

Alias of fourierIntegral_gaussian.

theorem fourierIntegral_gaussian_pi' {b : ℂ} (hb : 0 < b.re) (c : ℂ) : (Real.fourierIntegral fun (x : ℝ) => (-↑Real.pi * b * ↑x ^ 2 + 2 * ↑Real.pi * c * ↑x).exp) = fun (t : ℝ) => 1 / b ^ (1 / 2) * (-↑Real.pi / b * (↑t + Complex.I * c) ^ 2).exp

@[deprecated fourierIntegral_gaussian_pi']

theorem fourier_transform_gaussian_pi' {b : ℂ} (hb : 0 < b.re) (c : ℂ) : (Real.fourierIntegral fun (x : ℝ) => (-↑Real.pi * b * ↑x ^ 2 + 2 * ↑Real.pi * c * ↑x).exp) = fun (t : ℝ) => 1 / b ^ (1 / 2) * (-↑Real.pi / b * (↑t + Complex.I * c) ^ 2).exp

Alias of fourierIntegral_gaussian_pi'.

theorem fourierIntegral_gaussian_pi {b : ℂ} (hb : 0 < b.re) : (Real.fourierIntegral fun (x : ℝ) => (-↑Real.pi * b * ↑x ^ 2).exp) = fun (t : ℝ) => 1 / b ^ (1 / 2) * (-↑Real.pi / b * ↑t ^ 2).exp

@[deprecated fourierIntegral_gaussian_pi]

theorem GaussianFourier.root_.fourier_transform_gaussian_pi {b : ℂ} (hb : 0 < b.re) : (Real.fourierIntegral fun (x : ℝ) => (-↑Real.pi * b * ↑x ^ 2).exp) = fun (t : ℝ) => 1 / b ^ (1 / 2) * (-↑Real.pi / b * ↑t ^ 2).exp

Alias of fourierIntegral_gaussian_pi.

theorem GaussianFourier.integrable_cexp_neg_sum_mul_add {ι : Type u_2} [Fintype ι] {b : ι → ℂ} (hb : ∀ (i : ι), 0 < (b i).re) (c : ι → ℂ) : MeasureTheory.Integrable (fun (v : ι → ℝ) => ((-Finset.univ.sum fun (i : ι) => b i * ↑(v i) ^ 2) + Finset.univ.sum fun (i : ι) => c i * ↑(v i)).exp) MeasureTheory.volume

theorem GaussianFourier.integrable_cexp_neg_mul_sum_add {b : ℂ} {ι : Type u_2} [Fintype ι] (hb : 0 < b.re) (c : ι → ℂ) : MeasureTheory.Integrable (fun (v : ι → ℝ) => ((-b * Finset.univ.sum fun (i : ι) => ↑(v i) ^ 2) + Finset.univ.sum fun (i : ι) => c i * ↑(v i)).exp) MeasureTheory.volume

theorem GaussianFourier.integrable_cexp_neg_mul_sq_norm_add_of_euclideanSpace {b : ℂ} {ι : Type u_2} [Fintype ι] (hb : 0 < b.re) (c : ℂ) (w : EuclideanSpace ℝ ι) : MeasureTheory.Integrable (fun (v : EuclideanSpace ℝ ι) => (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫_ℝ).exp) MeasureTheory.volume

theorem GaussianFourier.integrable_cexp_neg_mul_sq_norm_add {b : ℂ} {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (hb : 0 < b.re) (c : ℂ) (w : V) : MeasureTheory.Integrable (fun (v : V) => (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫_ℝ).exp) MeasureTheory.volume

In a real inner product space, the complex exponential of minus the square of the norm plus a scalar product is integrable. Useful when discussing the Fourier transform of a Gaussian.

theorem GaussianFourier.integral_cexp_neg_sum_mul_add {ι : Type u_2} [Fintype ι] {b : ι → ℂ} (hb : ∀ (i : ι), 0 < (b i).re) (c : ι → ℂ) : ∫ (v : ι → ℝ), ((-Finset.univ.sum fun (i : ι) => b i * ↑(v i) ^ 2) + Finset.univ.sum fun (i : ι) => c i * ↑(v i)).exp = Finset.univ.prod fun (i : ι) => (↑Real.pi / b i) ^ (1 / 2) * (c i ^ 2 / (4 * b i)).exp

theorem GaussianFourier.integral_cexp_neg_mul_sum_add {b : ℂ} {ι : Type u_2} [Fintype ι] (hb : 0 < b.re) (c : ι → ℂ) : ∫ (v : ι → ℝ), ((-b * Finset.univ.sum fun (i : ι) => ↑(v i) ^ 2) + Finset.univ.sum fun (i : ι) => c i * ↑(v i)).exp = (↑Real.pi / b) ^ (↑(Fintype.card ι) / 2) * ((Finset.univ.sum fun (i : ι) => c i ^ 2) / (4 * b)).exp

theorem GaussianFourier.integral_cexp_neg_mul_sq_norm_add_of_euclideanSpace {b : ℂ} {ι : Type u_2} [Fintype ι] (hb : 0 < b.re) (c : ℂ) (w : EuclideanSpace ℝ ι) : ∫ (v : EuclideanSpace ℝ ι), (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫_ℝ).exp = (↑Real.pi / b) ^ (↑(Fintype.card ι) / 2) * (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b)).exp

theorem GaussianFourier.integral_cexp_neg_mul_sq_norm_add {b : ℂ} {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (hb : 0 < b.re) (c : ℂ) (w : V) : ∫ (v : V), (-b * ↑‖v‖ ^ 2 + c * ↑⟪w, v⟫_ℝ).exp = (↑Real.pi / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2) * (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b)).exp

theorem GaussianFourier.integral_cexp_neg_mul_sq_norm {b : ℂ} {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (hb : 0 < b.re) : ∫ (v : V), (-b * ↑‖v‖ ^ 2).exp = (↑Real.pi / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2)

theorem GaussianFourier.integral_rexp_neg_mul_sq_norm {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] {b : ℝ} (hb : 0 < b) : ∫ (v : V), (-b * ‖v‖ ^ 2).exp = (Real.pi / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2)

theorem fourierIntegral_gaussian_innerProductSpace' {b : ℂ} {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (hb : 0 < b.re) (x : V) (w : V) : Real.fourierIntegral (fun (v : V) => (-b * ↑‖v‖ ^ 2 + 2 * ↑Real.pi * Complex.I * ↑⟪x, v⟫_ℝ).exp) w = (↑Real.pi / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2) * (-↑Real.pi ^ 2 * ↑‖x - w‖ ^ 2 / b).exp

theorem fourierIntegral_gaussian_innerProductSpace {b : ℂ} {V : Type u_1} [NormedAddCommGroup V] [InnerProductSpace ℝ V] [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V] (hb : 0 < b.re) (w : V) : Real.fourierIntegral (fun (v : V) => (-b * ↑‖v‖ ^ 2).exp) w = (↑Real.pi / b) ^ (↑(FiniteDimensional.finrank ℝ V) / 2) * (-↑Real.pi ^ 2 * ↑‖w‖ ^ 2 / b).exp