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

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

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

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

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 T : ℝ) : ‖Complex.exp (-b * (↑T + ↑c * Complex.I) ^ 2)‖ = Real.exp (-(b.re * (T - b.im * c / b.re) ^ 2 - c ^ 2 * (b.im ^ 2 / b.re + b.re)))

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

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

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

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

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

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

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

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

@[deprecated fourier_gaussian_pi' (since := "2025-11-16")]

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

Alias of fourier_gaussian_pi'.

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

@[deprecated fourier_gaussian_pi (since := "2025-11-16")]

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

Alias of fourier_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) => Complex.exp (-∑ i : ι, b i * ↑(v i) ^ 2 + ∑ i : ι, c i * ↑(v i))) 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) => Complex.exp (-b * ∑ i : ι, ↑(v i) ^ 2 + ∑ i : ι, c i * ↑(v i))) 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) => Complex.exp (-b * ↑‖v‖ ^ 2 + c * ↑(inner ℝ w v))) 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) => Complex.exp (-b * ↑‖v‖ ^ 2 + c * ↑(inner ℝ w v))) 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), Complex.exp (-∑ i : ι, b i * ↑(v i) ^ 2 + ∑ i : ι, c i * ↑(v i)) = ∏ i : ι, (↑Real.pi / b i) ^ (1 / 2) * Complex.exp (c i ^ 2 / (4 * b i))

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), Complex.exp (-b * ∑ i : ι, ↑(v i) ^ 2 + ∑ i : ι, c i * ↑(v i)) = (↑Real.pi / b) ^ (↑(Fintype.card ι) / 2) * Complex.exp ((∑ i : ι, c i ^ 2) / (4 * b))

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), Complex.exp (-b * ↑‖v‖ ^ 2 + c * ↑(inner ℝ w v)) = (↑Real.pi / b) ^ (↑(Fintype.card ι) / 2) * Complex.exp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))

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), Complex.exp (-b * ↑‖v‖ ^ 2 + c * ↑(inner ℝ w v)) = (↑Real.pi / b) ^ (↑(Module.finrank ℝ V) / 2) * Complex.exp (c ^ 2 * ↑‖w‖ ^ 2 / (4 * b))

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), Complex.exp (-b * ↑‖v‖ ^ 2) = (↑Real.pi / b) ^ (↑(Module.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), Real.exp (-b * ‖v‖ ^ 2) = (Real.pi / b) ^ (↑(Module.finrank ℝ V) / 2)

theorem fourier_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 w : V) : FourierTransform.fourier (fun (v : V) => Complex.exp (-b * ↑‖v‖ ^ 2 + 2 * ↑Real.pi * Complex.I * ↑(inner ℝ x v))) w = (↑Real.pi / b) ^ (↑(Module.finrank ℝ V) / 2) * Complex.exp (-↑Real.pi ^ 2 * ↑‖x - w‖ ^ 2 / b)

@[deprecated fourier_gaussian_innerProductSpace' (since := "2025-11-16")]

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 w : V) : FourierTransform.fourier (fun (v : V) => Complex.exp (-b * ↑‖v‖ ^ 2 + 2 * ↑Real.pi * Complex.I * ↑(inner ℝ x v))) w = (↑Real.pi / b) ^ (↑(Module.finrank ℝ V) / 2) * Complex.exp (-↑Real.pi ^ 2 * ↑‖x - w‖ ^ 2 / b)

Alias of fourier_gaussian_innerProductSpace'.

theorem fourier_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) : FourierTransform.fourier (fun (v : V) => Complex.exp (-b * ↑‖v‖ ^ 2)) w = (↑Real.pi / b) ^ (↑(Module.finrank ℝ V) / 2) * Complex.exp (-↑Real.pi ^ 2 * ↑‖w‖ ^ 2 / b)

@[deprecated fourier_gaussian_innerProductSpace (since := "2025-11-16")]

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) : FourierTransform.fourier (fun (v : V) => Complex.exp (-b * ↑‖v‖ ^ 2)) w = (↑Real.pi / b) ^ (↑(Module.finrank ℝ V) / 2) * Complex.exp (-↑Real.pi ^ 2 * ↑‖w‖ ^ 2 / b)

Alias of fourier_gaussian_innerProductSpace.