theorem Frourio.meyers_serrin_L1_L2_density (f : ℝ → ℂ) (hf_L1 : MeasureTheory.Integrable f MeasureTheory.volume) (hf_L2 : MeasureTheory.MemLp f 2 MeasureTheory.volume) (ε : ℝ) (hε : 0 < ε) : ∃ (g : ℝ → ℂ), HasCompactSupport g ∧ ContDiff ℝ (↑⊤) g ∧ MeasureTheory.eLpNorm (fun (t : ℝ) => f t - g t) 1 MeasureTheory.volume < ENNReal.ofReal ε ∧ MeasureTheory.eLpNorm (fun (t : ℝ) => f t - g t) 2 MeasureTheory.volume < ENNReal.ofReal ε

Meyers–Serrin density theorem (real line version): smooth compactly supported functions are dense in Integrable ∩ MemLp 2.

theorem Frourio.exists_smooth_compact_support_L1_L2_close (f : ℝ → ℂ) (hf_L1 : MeasureTheory.Integrable f MeasureTheory.volume) (hf_L2 : MeasureTheory.MemLp f 2 MeasureTheory.volume) (ε : ℝ) (hε : 0 < ε) : ∃ (g : ℝ → ℂ), HasCompactSupport g ∧ ContDiff ℝ (↑⊤) g ∧ MeasureTheory.eLpNorm (fun (t : ℝ) => f t - g t) 1 MeasureTheory.volume < ENNReal.ofReal ε ∧ MeasureTheory.eLpNorm (fun (t : ℝ) => f t - g t) 2 MeasureTheory.volume < ENNReal.ofReal ε

Approximating an L¹ ∩ L² function by a smooth compactly supported function in both norms.

theorem Frourio.exists_schwartz_L1_L2_close (f : ℝ → ℂ) (hf_L1 : MeasureTheory.Integrable f MeasureTheory.volume) (hf_L2 : MeasureTheory.MemLp f 2 MeasureTheory.volume) (ε : ℝ) (hε : 0 < ε) : ∃ (φ : SchwartzMap ℝ ℂ), MeasureTheory.eLpNorm (fun (t : ℝ) => f t - φ t) 1 MeasureTheory.volume < ENNReal.ofReal ε ∧ MeasureTheory.eLpNorm (fun (t : ℝ) => f t - φ t) 2 MeasureTheory.volume < ENNReal.ofReal ε

Helper lemma for simultaneously approximating an L¹ ∩ L² function by a Schwartz function with small error in both norms.

theorem Frourio.exists_schwartz_L1_L2_approx (f : ℝ → ℂ) (hf_L1 : MeasureTheory.Integrable f MeasureTheory.volume) (hf_L2 : MeasureTheory.MemLp f 2 MeasureTheory.volume) : ∃ (φ : ℕ → SchwartzMap ℝ ℂ), (∀ (n : ℕ), MeasureTheory.Integrable (fun (t : ℝ) => (φ n) t) MeasureTheory.volume) ∧ (∀ (n : ℕ), MeasureTheory.MemLp (fun (t : ℝ) => (φ n) t) 2 MeasureTheory.volume) ∧ Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (fun (t : ℝ) => f t - (φ n) t) 1 MeasureTheory.volume) Filter.atTop (nhds 0) ∧ Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (fun (t : ℝ) => f t - (φ n) t) 2 MeasureTheory.volume) Filter.atTop (nhds 0)

Placeholder: simultaneously approximate an L¹ ∩ L² function by Schwartz functions that converge in both norms.

theorem Frourio.continuous_integral_norm_sq_of_L2_tendsto {φ : ℕ → ℝ → ℂ} {g : ℝ → ℂ} (hg_L2 : MeasureTheory.MemLp g 2 MeasureTheory.volume) (hφ_L2 : ∀ (n : ℕ), MeasureTheory.MemLp (φ n) 2 MeasureTheory.volume) (hφ_tendsto : Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (fun (t : ℝ) => g t - φ n t) 2 MeasureTheory.volume) Filter.atTop (nhds 0)) : Filter.Tendsto (fun (n : ℕ) => ∫ (t : ℝ), ‖φ n t‖ ^ 2) Filter.atTop (nhds (∫ (t : ℝ), ‖g t‖ ^ 2))

Placeholder: convergence of squared norms under L² convergence.

Once proved, this should assert that if φ n tends to g in L² and all the functions lie in L², then the squared norms of φ n converge to the squared norm of g.

theorem Frourio.fourierIntegral_L2_convergence {φ : ℕ → SchwartzMap ℝ ℂ} {g : ℝ → ℂ} (hg_L1 : MeasureTheory.Integrable g MeasureTheory.volume) (hg_L2 : MeasureTheory.MemLp g 2 MeasureTheory.volume) (hφ_L1 : ∀ (n : ℕ), MeasureTheory.Integrable (fun (t : ℝ) => (φ n) t) MeasureTheory.volume) (hφ_L2 : ∀ (n : ℕ), MeasureTheory.MemLp (fun (t : ℝ) => (φ n) t) 2 MeasureTheory.volume) (hφ_tendsto_L1 : Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (fun (t : ℝ) => g t - (φ n) t) 1 MeasureTheory.volume) Filter.atTop (nhds 0)) (hφ_tendsto_L2 : Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (fun (t : ℝ) => g t - (φ n) t) 2 MeasureTheory.volume) Filter.atTop (nhds 0)) : Filter.Tendsto (fun (n : ℕ) => MeasureTheory.eLpNorm (fun (ξ : ℝ) => fourierIntegral g ξ - fourierIntegral (fun (t : ℝ) => (φ n) t) ξ) 2 MeasureTheory.volume) Filter.atTop (nhds 0)

Placeholder: convergence of Fourier transforms in L² when the original functions converge in both L¹ and L².

The eventual goal is to show that if φ n → g in L¹ ∩ L², then the Fourier transforms also converge in L².

theorem Frourio.fourierIntegral_memLp_L1_L2 {g : ℝ → ℂ} (hg_L1 : MeasureTheory.Integrable g MeasureTheory.volume) (hg_L2 : MeasureTheory.MemLp g 2 MeasureTheory.volume) : MeasureTheory.MemLp (fun (ξ : ℝ) => fourierIntegral g ξ) 2 MeasureTheory.volume

Placeholder: the Fourier transform of an L¹ ∩ L² function lies in L².

Ultimately this should follow from the Plancherel theorem once the preceding lemmas are established.

theorem Frourio.fourier_plancherel_L1_L2 (g : ℝ → ℂ) (hg_L1 : MeasureTheory.Integrable g MeasureTheory.volume) (hg_L2 : MeasureTheory.MemLp g 2 MeasureTheory.volume) : ∫ (t : ℝ), ‖g t‖ ^ 2 = 1 / (2 * Real.pi) * ∫ (ξ : ℝ), ‖fourierIntegral g ξ‖ ^ 2

Fourier-Plancherel theorem for L¹ ∩ L² functions.

This is the CORRECT version of the Plancherel identity for functions in both L¹ and L². Unlike the invalid fourierIntegral_l2_norm_INVALID, this version has both:

• L¹ assumption (Integrable g): ensures fourierIntegral g is well-defined pointwise
• L² assumption (MemLp g 2): ensures the L² norms on both sides are finite

With both assumptions, we can prove:

1. fourierIntegral g ∈ L² (by Plancherel)
2. ∫ ‖g‖² = (1/(2π)) * ∫ ‖fourierIntegral g‖²

This is the standard textbook version of Plancherel for L¹ ∩ L².