File: Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean

full_name: _aux_Mathlib_Analysis_Fourier_RiemannLebesgueLemma___macroRules_termI_1
code: local notation3 "i" => fun (w : V) => (1 / (2 * ‖w‖ ^ 2) : ℝ) • w
goal: Lean.Macro

full_name: fourierIntegral_half_period_translate
code: /-- Shifting `f` by `(1 / (2 * ‖w‖ ^ 2)) • w` negates the integral in the Riemann-Lebesgue lemma. -/
theorem fourierIntegral_half_period_translate {w : V} (hw : w ≠ 0) :
    (∫ v : V, 𝐞 (-⟪v, w⟫) • f (v + i w)) = -∫ v : V, 𝐞 (-⟪v, w⟫) • f v 
goal: ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f (v + (fun w => (1 / (2 * ‖w‖ ^ 2)) • w) w) = -∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v

full_name: fourierIntegral_eq_half_sub_half_period_translate
code: /-- Rewrite the Fourier integral in a form that allows us to use uniform continuity. -/
theorem fourierIntegral_eq_half_sub_half_period_translate {w : V} (hw : w ≠ 0)
    (hf : Integrable f) :
    ∫ v : V, 𝐞 (-⟪v, w⟫) • f v = (1 / (2 : ℂ)) • ∫ v : V, 𝐞 (-⟪v, w⟫) • (f v - f (v + i w)) 
goal: ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v =
  (1 / 2) • ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • (f v - f (v + (fun w => (1 / (2 * ‖w‖ ^ 2)) • w) w))

full_name: tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support
code: /-- Riemann-Lebesgue Lemma for continuous and compactly-supported functions: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 wrt `cocompact V`. Note that this is primarily
of interest as a preparatory step for the more general result
`tendsto_integral_exp_inner_smul_cocompact` in which `f` can be arbitrary. -/
theorem tendsto_integral_exp_inner_smul_cocompact_of_continuous_compact_support (hf1 : Continuous f)
    (hf2 : HasCompactSupport f) :
    Tendsto (fun w : V => ∫ v : V, 𝐞 (-⟪v, w⟫) • f v) (cocompact V) (𝓝 0) 
goal: Filter.Tendsto (fun w => ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v) (Filter.cocompact V) (𝓝 0)

full_name: tendsto_integral_exp_inner_smul_cocompact
code: /-- Riemann-Lebesgue lemma for functions on a real inner-product space: the integral
`∫ v, exp (-2 * π * ⟪w, v⟫ * I) • f v` tends to 0 as `w → ∞`. -/
theorem tendsto_integral_exp_inner_smul_cocompact :
    Tendsto (fun w : V => ∫ v, 𝐞 (-⟪v, w⟫) • f v) (cocompact V) (𝓝 0) 
goal: Filter.Tendsto (fun w => ∫ (v : V), 𝐞 (-⟪v, w⟫_ℝ) • f v) (Filter.cocompact V) (𝓝 0)

full_name: Real.tendsto_integral_exp_smul_cocompact
code: /-- The Riemann-Lebesgue lemma for functions on `ℝ`. -/
theorem Real.tendsto_integral_exp_smul_cocompact (f : ℝ → E) :
    Tendsto (fun w : ℝ => ∫ v : ℝ, 𝐞 (-(v * w)) • f v) (cocompact ℝ) (𝓝 0) 
goal: Filter.Tendsto (fun w => ∫ (v : ℝ), 𝐞 (-(v * w)) • f v) (Filter.cocompact ℝ) (𝓝 0)

full_name: Real.zero_at_infty_fourierIntegral
code: /-- The Riemann-Lebesgue lemma for functions on `ℝ`, formulated via `Real.fourierIntegral`. -/
theorem Real.zero_at_infty_fourierIntegral (f : ℝ → E) : Tendsto (𝓕 f) (cocompact ℝ) (𝓝 0) 
goal: Filter.Tendsto (𝓕 f) (Filter.cocompact ℝ) (𝓝 0)

full_name: tendsto_integral_exp_smul_cocompact_of_inner_product
code: /-- Riemann-Lebesgue lemma for functions on a finite-dimensional inner-product space, formulated
via dual space. **Do not use** -- it is only a stepping stone to
`tendsto_integral_exp_smul_cocompact` where the inner-product-space structure isn't required. -/
theorem tendsto_integral_exp_smul_cocompact_of_inner_product (μ : Measure V) [μ.IsAddHaarMeasure] :
    Tendsto (fun w : V →L[ℝ] ℝ => ∫ v, 𝐞 (-w v) • f v ∂μ) (cocompact (V →L[ℝ] ℝ)) (𝓝 0) 
goal: Filter.Tendsto (fun w => ∫ (v : V), 𝐞 (-w v) • f v ∂μ) (Filter.cocompact (V →L[ℝ] ℝ)) (𝓝 0)

full_name: tendsto_integral_exp_smul_cocompact
code: /-- Riemann-Lebesgue lemma for functions on a finite-dimensional real vector space, formulated via
dual space. -/
theorem tendsto_integral_exp_smul_cocompact (μ : Measure V) [μ.IsAddHaarMeasure] :
    Tendsto (fun w : V →L[ℝ] ℝ => ∫ v, 𝐞 (-w v) • f v ∂μ) (cocompact (V →L[ℝ] ℝ)) (𝓝 0) 
goal: Filter.Tendsto (fun w => ∫ (v : V), 𝐞 (-w v) • f v ∂μ) (Filter.cocompact (V →L[ℝ] ℝ)) (𝓝 0)

full_name: Real.zero_at_infty_vector_fourierIntegral
code: /-- The Riemann-Lebesgue lemma, formulated in terms of `VectorFourier.fourierIntegral` (with the
pairing in the definition of `fourier_integral` taken to be the canonical pairing between `V` and
its dual space). -/
theorem Real.zero_at_infty_vector_fourierIntegral (μ : Measure V) [μ.IsAddHaarMeasure] :
    Tendsto (VectorFourier.fourierIntegral 𝐞 μ (topDualPairing ℝ V).flip f) (cocompact (V →L[ℝ] ℝ))
      (𝓝 0) 
goal: Filter.Tendsto (VectorFourier.fourierIntegral 𝐞 μ (topDualPairing ℝ V).flip f) (Filter.cocompact (V →L[ℝ] ℝ)) (𝓝 0)