| import OAI.Geometry.ProjectionBodies.SignCalculus |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped ENNReal RealInnerProductSpace |
| namespace PettyProjection.Spherical |
|
|
| lemma posPower_eq_max_pow {r : ℕ} (hr : 0 < r) (x : ℝ) : |
| posPower r x = max x 0 ^ r := by |
| by_cases hx : 0 < x |
| · simp [posPower, hx, max_eq_left hx.le] |
| · simp [posPower, hx, max_eq_right (not_lt.mp hx), hr.ne'] |
|
|
| / |
| def higherKernel {n : ℕ} (r : ℕ) (u : Sphere n) (f : Sphere n → ℝ) : ℝ := |
| mean (fun x => (max ⟪(u : Space n),(x : Space n)⟫ 0)^r*f x) |
|
|
| lemma higherKernel_const_pos {n : ℕ} [NeZero n] (r : ℕ) (u : Sphere n) : |
| 0 < higherKernel r u (fun _ => 1) := by |
| have hc : Continuous (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0 ^ r) := by |
| fun_prop |
| simp only [higherKernel, mul_one, mean] |
| exact integral_pos_of_integrable_nonneg_nonzero hc (continuous_integrable hc) |
| (fun x => pow_nonneg (le_max_right _ _) _) (x := u) |
| (by rw [real_inner_self_eq_norm_sq, norm_coe]; norm_num) |
|
|
| lemma higherKernel_recurrence {n r d : ℕ} [NeZero n] (hr : 3 ≤ r) |
| (u : Sphere n) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) |
| (heig : ∀ x : Sphere n, casimir f x = -eigenvalue n d*f x) : |
| (eigenvalue n r-eigenvalue n d)*higherKernel r u (fun x => f x) = |
| (r : ℝ)*((r : ℝ)-1)*higherKernel (r-2) u (fun x => f x) := by |
| let g : Space n → ℝ := fun y => posPower r ⟪(u : Space n),y⟫ |
| have hg : ContDiff ℝ 2 g := (posPower_contDiff_two hr).comp (innerSL ℝ (u : Space n)).contDiff |
| have h := mean_casimir_comm hf hg |
| have he (x : Sphere n) : f x*casimir g x = |
| (r : ℝ)*((r : ℝ)-1)*((max ⟪(u : Space n),(x : Space n)⟫ 0)^(r-2)*f x) - |
| eigenvalue n r*((max ⟪(u : Space n),(x : Space n)⟫ 0)^r*f x) := by |
| rw [show casimir g x = _ from casimir_posPower_inner hr u x, |
| posPower_eq_max_pow (show 0 < r-2 by omega), posPower_eq_max_pow (show 0 < r by omega)] |
| ring |
| simp_rw [he, heig, g, posPower_eq_max_pow (show 0 < r by omega)] at h |
| have hct (r : ℕ) : Continuous (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0 ^ r * f x) := by |
| exact ((continuous_const.inner continuous_subtype_val).max continuous_const |>.pow r).mul |
| (hf.continuous.comp continuous_subtype_val) |
| have hh : (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0 ^ r * |
| (-eigenvalue n d*f x)) = fun x : Sphere n => -eigenvalue n d* |
| (max ⟪(u : Space n),(x : Space n)⟫ 0 ^ r*f x) := by funext x; ring |
| rw [hh, mean, integral_sub |
| ((continuous_integrable (hct (r-2))).const_mul _) ((continuous_integrable (hct r)).const_mul _), |
| integral_const_mul, integral_const_mul, mean, integral_const_mul] at h |
| change (r : ℝ)*((r : ℝ)-1)*higherKernel (r-2) u (fun x => f x) - |
| eigenvalue n r*higherKernel r u (fun x => f x) = |
| -eigenvalue n d*higherKernel r u (fun x => f x) at h |
| linarith |
|
|
| lemma higherKernel_const_recurrence {n r : ℕ} [NeZero n] (hr : 3 ≤ r) (u : Sphere n) : |
| eigenvalue n r*higherKernel r u (fun _ => 1) = |
| (r : ℝ)*((r : ℝ)-1)*higherKernel (r-2) u (fun _ => 1) := by |
| have he (x : Sphere n) : casimir (fun _ : Space n => (1 : ℝ)) x = -eigenvalue n 0*1 := by |
| have hc (T : Space n →L[ℝ] Space n) : rotDeriv T (fun _ : Space n => (1 : ℝ)) = fun _ => 0 := |
| funext (rotDeriv_const T 1) |
| simp [casimir, hc, rotDeriv_const, eigenvalue] |
| simpa [eigenvalue] using higherKernel_recurrence hr u contDiff_const he |
|
|
| lemma higherKernel_peak {n : ℕ} [NeZero n] (u : Sphere n) {f : Sphere n → ℝ} |
| (hf : Continuous f) : |
| Tendsto (fun r : ℕ => higherKernel r u f / higherKernel r u (fun _ => 1)) |
| atTop (𝓝 (f u)) := by |
| have hc : Continuous (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0) := by fun_prop |
| have hmax (x : Sphere n) (_ : x ∈ (univ : Set (Sphere n))) (hne : x ≠ u) : |
| max ⟪(u : Space n),(x : Space n)⟫ 0 < max ⟪(u : Space n),(u : Space n)⟫ 0 := by |
| rw [real_inner_self_eq_norm_sq, norm_coe] |
| norm_num only [one_pow, max_eq_left zero_le_one] |
| apply max_lt |
| · exact (inner_lt_one_iff_real_of_norm_eq_one (norm_coe u) (norm_coe x)).mpr |
| (by intro h; exact hne (Subtype.ext h.symm)) |
| · norm_num |
| have h := tendsto_setIntegral_pow_smul_of_unique_maximum_of_isCompact_of_continuousOn |
| (μ := sigma n) isCompact_univ hc.continuousOn hmax |
| (fun x _ => le_max_right _ _) (by rw [real_inner_self_eq_norm_sq, norm_coe]; norm_num) |
| (by simp) hf.continuousOn |
| simpa [smul_eq_mul, div_eq_mul_inv, higherKernel, mean, mul_comm] using h |
|
|
| end PettyProjection.Spherical |
| end |
|
|
| noncomputable section |
| open Filter Topology |
| namespace PettyProjection.Spherical |
|
|
| / |
| def higherFactor (n j : ℕ) (s : ℝ) : ℝ := |
| ∏ a ∈ Finset.range j, (s-2*a)/(s+n+2*a) |
|
|
| lemma higherFactor_zero (n : ℕ) (s : ℝ) : higherFactor n 0 s = 1 := by |
| simp [higherFactor] |
|
|
| lemma higherFactor_succ (n j : ℕ) (s : ℝ) : |
| higherFactor n (j+1) s = higherFactor n j s*((s-2*j)/(s+n+2*j)) := by |
| simp only [higherFactor, Finset.prod_range_succ] |
|
|
| lemma higherFactor_odd_ne_zero {n : ℕ} (hn : 2 ≤ n) (j m : ℕ) : |
| higherFactor n j (2*m+1) ≠ 0 := by |
| apply Finset.prod_ne_zero_iff.mpr |
| intro a _ |
| apply div_ne_zero |
| · apply sub_ne_zero.mpr |
| have h : (2*m+1 : ℕ) ≠ 2*a := by omega |
| exact_mod_cast h |
| · have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn |
| have hm : (0 : ℝ) ≤ m := Nat.cast_nonneg _ |
| have ha : (0 : ℝ) ≤ a := Nat.cast_nonneg _ |
| linarith |
|
|
| lemma higherFactor_step {n : ℕ} (hn : 2 ≤ n) (j : ℕ) {s : ℝ} (hs : 3 ≤ s) : |
| (s-2*j)*(s+2*j+n-2)*higherFactor n j s = |
| s*(s+n-2)*higherFactor n j (s-2) := by |
| have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn |
| induction j with |
| | zero => simp [higherFactor] |
| | succ j ih => |
| rw [higherFactor_succ, higherFactor_succ] |
| have hj : (0 : ℝ) ≤ j := Nat.cast_nonneg _ |
| have h₁ : s+n+2*j ≠ 0 := by linarith |
| have h₂ : s-2+n+2*j ≠ 0 := by linarith |
| push_cast |
| field_simp |
| linear_combination (s-2*j-2)*(s+n+2*j)*ih |
|
|
| lemma higherFactor_tendsto (n j : ℕ) : |
| Tendsto (fun m : ℕ => higherFactor n j (2*m+1)) atTop (𝓝 1) := by |
| induction j with |
| | zero => simpa only [higherFactor_zero] using tendsto_const_nhds |
| | succ j ih => |
| have h : Tendsto (fun m : ℕ => ((2*(m : ℝ)+1)-2*j)/((2*(m : ℝ)+1)+n+2*j)) atTop (𝓝 1) := by |
| have h := tendsto_add_mul_div_add_mul_atTop_nhds (1-2*(j : ℝ)) |
| (1+n+2*(j : ℝ)) (2 : ℝ) (show (2 : ℝ) ≠ 0 by norm_num) |
| convert! h using 1 |
| · funext m; congr 1 <;> ring |
| · norm_num |
| simpa only [higherFactor_succ, mul_one] using ih.mul h |
|
|
| end PettyProjection.Spherical |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped ENNReal RealInnerProductSpace |
| namespace PettyProjection.Spherical |
|
|
| lemma eigenvalue_sub_even {n j : ℕ} (s : ℕ) : |
| eigenvalue n s-eigenvalue n (2*j) = |
| ((s : ℝ)-2*j)*((s : ℝ)+2*j+n-2) := by |
| simp only [eigenvalue, Nat.cast_mul, Nat.cast_ofNat] |
| ring |
|
|
| lemma eigenvalue_odd_sub_even_ne_zero {n : ℕ} (hn : 2 ≤ n) (m j : ℕ) : |
| eigenvalue n (2*m+1)-eigenvalue n (2*j) ≠ 0 := by |
| rw [eigenvalue_sub_even] |
| apply mul_ne_zero |
| · apply sub_ne_zero.mpr |
| have h : (2*m+1 : ℕ) ≠ 2*j := by omega |
| exact_mod_cast h |
| · have hn' : (2 : ℝ) ≤ n := by exact_mod_cast hn |
| have hm : (0 : ℝ) ≤ m := Nat.cast_nonneg _ |
| have hj : (0 : ℝ) ≤ j := Nat.cast_nonneg _ |
| push_cast |
| linarith |
|
|
| lemma higherKernel_normalized_recurrence {n r d : ℕ} [NeZero n] (hr : 3 ≤ r) |
| (u : Sphere n) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) |
| (heig : ∀ x : Sphere n, casimir f x = -eigenvalue n d*f x) : |
| (eigenvalue n r-eigenvalue n d)*(higherKernel r u (fun x => f x)/higherKernel r u (fun _ => 1)) = |
| eigenvalue n r*(higherKernel (r-2) u (fun x => f x)/higherKernel (r-2) u (fun _ => 1)) := by |
| have h := higherKernel_recurrence hr u hf heig |
| have hz := higherKernel_const_recurrence hr u |
| have h₁ := (higherKernel_const_pos r u).ne' |
| have h₂ := (higherKernel_const_pos (r-2) u).ne' |
| field_simp |
| linear_combination higherKernel (r-2) u (fun _ => 1)*h - |
| higherKernel (r-2) u (fun x => f x)*hz |
|
|
| lemma higherKernel_eigenfunction {n j : ℕ} (hn : 2 ≤ n) [NeZero n] |
| (u : Sphere n) {f : Space n → ℝ} (hf : ContDiff ℝ 2 f) |
| (heig : ∀ x : Sphere n, casimir f x = -eigenvalue n (2*j)*f x) : |
| higherKernel 1 u (fun x => f x)/higherKernel 1 u (fun _ => 1) = |
| higherFactor n j 1*f u := by |
| let A (m : ℕ) := higherKernel (2*m+1) u (fun x => f x)/higherKernel (2*m+1) u (fun _ => 1) |
| let F (m : ℕ) := higherFactor n j (2*m+1) |
| have hinv (m : ℕ) : A m*F 0 = A 0*F m := by |
| induction m with |
| | zero => rfl |
| | succ m ih => |
| have hc := eigenvalue_odd_sub_even_ne_zero hn (m+1) j |
| have hA := higherKernel_normalized_recurrence (show 3 ≤ 2*(m+1)+1 by omega) u hf heig |
| rw [show 2*(m+1)+1-2 = 2*m+1 by omega] at hA |
| change (eigenvalue n (2*(m+1)+1)-eigenvalue n (2*j))*A (m+1) = |
| eigenvalue n (2*(m+1)+1)*A m at hA |
| have hF : (eigenvalue n (2*(m+1)+1)-eigenvalue n (2*j))*F (m+1) = |
| eigenvalue n (2*(m+1)+1)*F m := by |
| have hm : (0 : ℝ) ≤ m := Nat.cast_nonneg _ |
| have h := higherFactor_step hn j (s := 2*(m+1)+1) (by linarith) |
| have he : (2*((m : ℝ)+1)+1)-2 = 2*m+1 := by ring |
| rw [he] at h |
| rw [eigenvalue_sub_even] |
| simpa only [F, eigenvalue, Nat.cast_add, |
| Nat.cast_mul, Nat.cast_one, Nat.cast_ofNat] using h |
| apply mul_left_cancel₀ hc |
| calc |
| _ = ((eigenvalue n (2*(m+1)+1)-eigenvalue n (2*j))*A (m+1))*F 0 := by ring |
| _ = eigenvalue n (2*(m+1)+1)*(A m*F 0) := by rw [hA]; ring |
| _ = eigenvalue n (2*(m+1)+1)*(A 0*F m) := by rw [ih] |
| _ = A 0*((eigenvalue n (2*(m+1)+1)-eigenvalue n (2*j))*F (m+1)) := by rw [hF]; ring |
| _ = _ := by ring |
| have ht : Tendsto (fun m : ℕ => 2*m+1) atTop atTop := by |
| apply Filter.tendsto_atTop.mpr |
| intro b |
| filter_upwards [eventually_ge_atTop b] with m hm |
| omega |
| have hA : Tendsto A atTop (𝓝 (f u)) := |
| (higherKernel_peak u (hf.continuous.comp continuous_subtype_val)).comp ht |
| have hF : Tendsto F atTop (𝓝 1) := higherFactor_tendsto n j |
| have hlim₁ := hA.mul (tendsto_const_nhds (x := F 0)) |
| have hlim₂ := (tendsto_const_nhds (x := A 0)).mul hF |
| have heq : (fun m => A m*F 0) = (fun m => A 0*F m) := funext hinv |
| rw [heq] at hlim₁ |
| have h := tendsto_nhds_unique hlim₁ hlim₂ |
| simpa [A, F, mul_comm] using h.symm |
|
|
| lemma higherKernel_harmonic {n j : ℕ} (hn : 2 ≤ n) [NeZero n] |
| (u : Sphere n) {p : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous (2*j)) |
| (hΔ : laplacePolynomial p = 0) : |
| higherKernel 1 u (fun x => polynomialFunction p x)/higherKernel 1 u (fun _ => 1) = |
| higherFactor n j 1*polynomialFunction p u := by |
| exact higherKernel_eigenfunction hn u ((polynomialFunction_contDiff p).of_le (by norm_num)) |
| (fun x => casimir_harmonic hp hΔ x) |
|
|
| end PettyProjection.Spherical |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped ENNReal RealInnerProductSpace |
| namespace PettyProjection.Spherical |
|
|
| def pole (n : ℕ) [NeZero n] : Sphere n := |
| ⟨EuclideanSpace.single (0 : Fin n) 1, by simp⟩ |
|
|
| lemma higherKernel_const_rotation {n : ℕ} (r : ℕ) (u : Sphere n) |
| (U : Space n ≃ₗᵢ[ℝ] Space n) : |
| higherKernel r (sphereMap U u) (fun _ => 1) = higherKernel r u (fun _ => 1) := by |
| have h := mean_rotation U (fun x : Sphere n => max ⟪U (u : Space n),(x : Space n)⟫ 0 ^ r) |
| change mean (fun x : Sphere n => max ⟪U (u : Space n),U (x : Space n)⟫ 0 ^ r) = _ at h |
| simp only [U.inner_map_map] at h |
| simp only [higherKernel, mul_one] |
| change mean (fun x : Sphere n => max ⟪U (u : Space n),(x : Space n)⟫ 0 ^ r) = _ |
| exact h.symm |
|
|
| lemma higherKernel_const_independent {n : ℕ} (r : ℕ) (u v : Sphere n) : |
| higherKernel r u (fun _ => 1) = higherKernel r v (fun _ => 1) := by |
| let U := (ℝ ∙ ((u : Space n)-(v : Space n)))ᗮ.reflection |
| have h : sphereMap U u = v := by |
| apply Subtype.ext |
| exact Submodule.reflection_sub ((norm_coe u).trans (norm_coe v).symm) |
| rw [← h, higherKernel_const_rotation] |
|
|
| / |
| def cosineConstant (n : ℕ) [NeZero n] : ℝ := |
| 2*higherKernel 1 (pole n) (fun _ => 1) |
|
|
| lemma cosineConstant_pos (n : ℕ) [NeZero n] : 0 < cosineConstant n := |
| mul_pos (by norm_num) (higherKernel_const_pos 1 (pole n)) |
|
|
| lemma mean_pos_inner_even {n : ℕ} [NeZero n] (u : Sphere n) {f : Space n → ℝ} |
| (heven : Function.Even f) (hf : Continuous f) : |
| mean (fun x : Sphere n => |⟪(u : Space n),(x : Space n)⟫| *f x) = |
| 2*higherKernel 1 u (fun x => f x) := by |
| have h := mean_rotation (LinearIsometryEquiv.neg ℝ : Space n ≃ₗᵢ[ℝ] Space n) |
| (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0*f x) |
| have he : (fun x : Sphere n => max ⟪(u : Space n),((sphereMap (LinearIsometryEquiv.neg ℝ)) x : Space n)⟫ 0 * |
| f (sphereMap (LinearIsometryEquiv.neg ℝ) x)) = |
| (fun x : Sphere n => max (-⟪(u : Space n),(x : Space n)⟫) 0*f x) := by |
| funext x |
| change max ⟪(u : Space n),-(x : Space n)⟫ 0*f (-(x : Space n)) = _ |
| rw [inner_neg_right, heven] |
| rw [he] at h |
| have hk (t : ℝ) : |t| = max t 0+max (-t) 0 := by |
| rcases le_total 0 t with ht | ht <;> simp [abs_of_nonneg, abs_of_nonpos, ht] |
| simp only [higherKernel, pow_one] |
| simp_rw [hk, add_mul] |
| rw [mean, integral_add] |
| · change mean (fun x : Sphere n => max ⟪(u : Space n),(x : Space n)⟫ 0*f x) + |
| mean (fun x : Sphere n => max (-⟪(u : Space n),(x : Space n)⟫) 0*f x) = _ |
| rw [h] |
| ring |
| · exact continuous_integrable (by fun_prop) |
| · exact continuous_integrable (by fun_prop) |
|
|
| lemma mean_abs_inner {n : ℕ} [NeZero n] (u : Sphere n) : |
| mean (fun x : Sphere n => |⟪(u : Space n),(x : Space n)⟫|) = cosineConstant n := by |
| have h := mean_pos_inner_even u (f := fun _ => 1) (fun _ => rfl) continuous_const |
| simpa only [mul_one, higherKernel_const_independent 1 u (pole n), cosineConstant] using h |
|
|
| / |
| def cosine {n : ℕ} [NeZero n] (f : Sphere n → ℝ) (u : Sphere n) : ℝ := |
| mean (fun x : Sphere n => |⟪(u : Space n),(x : Space n)⟫| *f x)/cosineConstant n |
|
|
| lemma cosine_continuous {n : ℕ} [NeZero n] {f : Sphere n → ℝ} (hf : Continuous f) : |
| Continuous (cosine f) := by |
| apply Continuous.div_const |
| have h := continuous_parametric_integral_of_continuous (μ := sigma n) |
| (f := fun u x : Sphere n => |⟪(u : Space n),(x : Space n)⟫| *f x) |
| (by fun_prop) isCompact_univ |
| simpa only [Measure.restrict_univ, mean] using h |
|
|
| lemma cosine_const {n : ℕ} [NeZero n] (c : ℝ) : cosine (fun _ : Sphere n => c) = fun _ => c := by |
| funext u |
| rw [cosine, mean, integral_mul_const] |
| change mean (fun x : Sphere n => |⟪(u : Space n),(x : Space n)⟫|)*c/cosineConstant n = c |
| rw [mean_abs_inner] |
| field_simp [(cosineConstant_pos n).ne'] |
|
|
| lemma polynomialFunction_even {n d : ℕ} {p : MvPolynomial (Fin n) ℝ} |
| (hp : p.IsHomogeneous d) (hd : Even d) : Function.Even (polynomialFunction p) := by |
| intro x |
| simpa only [neg_one_smul, hd.neg_one_pow, one_mul] using |
| polynomialFunction_smul_argument hp (-1) x |
|
|
| lemma cosine_harmonic {n j : ℕ} (hn : 2 ≤ n) [NeZero n] |
| {p : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous (2*j)) |
| (hΔ : laplacePolynomial p = 0) (u : Sphere n) : |
| cosine (fun x => polynomialFunction p x) u = higherFactor n j 1*polynomialFunction p u := by |
| rw [cosine, mean_pos_inner_even u (polynomialFunction_even hp (even_two_mul j)) |
| (polynomialFunction_contDiff p).continuous, cosineConstant, |
| ← higherKernel_const_independent 1 u (pole n), mul_div_mul_left _ _ (by norm_num : (2 : ℝ) ≠ 0)] |
| exact higherKernel_harmonic hn u hp hΔ |
|
|
| end PettyProjection.Spherical |
| end |
|
|
| noncomputable section |
| open Set MeasureTheory Metric Filter Topology Function |
| open scoped ENNReal RealInnerProductSpace |
| namespace PettyProjection.Spherical |
|
|
| def cosineMap {n : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) : C(Sphere n, ℝ) := |
| ⟨cosine f, cosine_continuous f.continuous⟩ |
|
|
| lemma cosine_selfadj {n : ℕ} [NeZero n] {f g : Sphere n → ℝ} |
| (hf : Continuous f) (hg : Continuous g) : |
| mean (fun u => f u*cosine g u) = mean (fun u => cosine f u*g u) := by |
| have hi : Integrable (fun z : Sphere n × Sphere n => |
| f z.1 * (|⟪(z.1 : Space n),(z.2 : Space n)⟫| * g z.2)) ((sigma n).prod (sigma n)) := |
| (by fun_prop : Continuous _).integrable_of_hasCompactSupport (HasCompactSupport.of_compactSpace _) |
| have h := integral_integral_swap (f := fun u v : Sphere n => f u*(|⟪(u : Space n),(v : Space n)⟫| * g v)) hi |
| simp only [cosine, mean, ← mul_div_assoc, div_mul_eq_mul_div, integral_div, ← integral_const_mul] |
| rw [h] |
| congr 1 |
| apply integral_congr_ae |
| filter_upwards with u |
| rw [← integral_mul_const] |
| apply integral_congr_ae |
| filter_upwards with v |
| rw [real_inner_comm (u : Space n) (v : Space n)] |
| ring |
|
|
| lemma mean_cosine {n : ℕ} [NeZero n] {f : Sphere n → ℝ} (hf : Continuous f) : |
| mean (cosine f) = mean f := by |
| have h := cosine_selfadj (f := fun _ => 1) continuous_const hf |
| simpa only [one_mul, cosine_const] using h |
|
|
| lemma cosine_neg {n : ℕ} [NeZero n] (f : Sphere n → ℝ) (u : Sphere n) : |
| cosine f (sphereMap (LinearIsometryEquiv.neg ℝ) u) = cosine f u := by |
| unfold cosine |
| congr 1 |
| congr 1 |
| funext x |
| change |⟪-(u : Space n),(x : Space n)⟫| * f x = _ |
| rw [inner_neg_left, abs_neg] |
|
|
| lemma mean_sphere_even_mul_odd {n d : ℕ} {f : Sphere n → ℝ} |
| (heven : ∀ u, f (sphereMap (LinearIsometryEquiv.neg ℝ) u) = f u) |
| {p : MvPolynomial (Fin n) ℝ} (hp : p.IsHomogeneous d) (hd : Odd d) : |
| mean (fun u : Sphere n => f u*polynomialFunction p u) = 0 := by |
| have h := mean_rotation (LinearIsometryEquiv.neg ℝ : Space n ≃ₗᵢ[ℝ] Space n) |
| (fun u : Sphere n => f u*polynomialFunction p u) |
| have he : (fun u : Sphere n => f (sphereMap (LinearIsometryEquiv.neg ℝ) u)* |
| polynomialFunction p (sphereMap (LinearIsometryEquiv.neg ℝ) u)) = |
| (fun u : Sphere n => -(f u*polynomialFunction p u)) := by |
| funext u |
| rw [heven] |
| change f u*polynomialFunction p (-(u : Space n)) = _ |
| rw [polynomialFunction_odd hp hd, mul_neg] |
| rw [he, mean, integral_neg] at h |
| change -mean (fun u : Sphere n => f u*polynomialFunction p u) = _ at h |
| linarith |
|
|
| lemma cosine_harmonic_odd {n d : ℕ} [NeZero n] {p : MvPolynomial (Fin n) ℝ} |
| (hp : p.IsHomogeneous d) (hd : Odd d) (u : Sphere n) : |
| cosine (fun x => polynomialFunction p x) u = 0 := by |
| rw [cosine, mean_even_mul_odd (f := fun x => |⟪(u : Space n),x⟫|) _ hp hd, zero_div] |
| intro x |
| simp only [inner_neg_right, abs_neg] |
|
|
| lemma cosine_projection_even {n j : ℕ} (hn : 2 ≤ n) [NeZero n] (f : C(Sphere n, ℝ)) : |
| (harmonicSpace n (2*j)).starProjection (toH n (cosineMap f)) = |
| higherFactor n j 1 • (harmonicSpace n (2*j)).starProjection (toH n f) := by |
| apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero |
| ((harmonicSpace n (2*j)).smul_mem _ ((harmonicSpace n (2*j)).starProjection_apply_mem _)) |
| rintro v ⟨p, hp, rfl⟩ |
| rw [inner_sub_left, real_inner_smul_left, Submodule.inner_starProjection_left_eq_right] |
| rw [Submodule.starProjection_eq_self_iff.mpr (show polynomialL2 n p ∈ harmonicSpace n (2*j) from ⟨p,hp,rfl⟩)] |
| change ⟪toH n (cosineMap f),toH n (polynomialRestriction n p)⟫ - |
| higherFactor n j 1 * ⟪toH n f,toH n (polynomialRestriction n p)⟫ = 0 |
| rw [inner_toH, inner_toH] |
| change mean (fun u : Sphere n => cosine f u*(polynomialRestriction n p) u) - _ = 0 |
| rw [← cosine_selfadj f.continuous (polynomialRestriction n p).continuous] |
| have he : (fun u : Sphere n => f u*cosine (polynomialRestriction n p) u) = |
| (fun u : Sphere n => higherFactor n j 1*(f u*polynomialFunction p u)) := by |
| funext u |
| rw [show cosine (polynomialRestriction n p) u = _ from cosine_harmonic hn hp.1 hp.2 u] |
| ring |
| rw [he, mean, integral_const_mul] |
| change _ - higherFactor n j 1 * (∫ u : Sphere n, f u * polynomialFunction p u ∂sigma n) = 0 |
| exact sub_self _ |
|
|
| lemma cosine_projection_odd {n d : ℕ} [NeZero n] (f : C(Sphere n, ℝ)) (hd : Odd d) : |
| (harmonicSpace n d).starProjection (toH n (cosineMap f)) = 0 := by |
| apply Submodule.eq_starProjection_of_mem_of_inner_eq_zero (Submodule.zero_mem _) |
| rintro v ⟨p, hp, rfl⟩ |
| rw [sub_zero] |
| change ⟪toH n (cosineMap f),toH n (polynomialRestriction n p)⟫ = 0 |
| rw [inner_toH] |
| exact mean_sphere_even_mul_odd (cosine_neg f) hp.1 hd |
|
|
| end PettyProjection.Spherical |
| end |
|
|
| end OAI |
|
|
