Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Dirichlet Kernel eq

dirichletKernel_eq

harmonic analysis Fourier analysis measure theory dirichlet kernel

Plain-language statement

At every real $x$ for which $e^{ix}\ne1$ , the finite-sum definition of the $N$ th Dirichlet kernel agrees with the project’s closed-form expression:

$D_N(x)=D'_N(x).$

Exact Lean statement

lemma dirichletKernel_eq {x : ℝ} (h : cexp (I * x) ≠ 1) :
    dirichletKernel N x = dirichletKernel' N x

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma dirichletKernel_eq {x : ℝ} (h : cexp (I * x) ≠ 1) :    dirichletKernel N x = dirichletKernel' N x := by  have : (cexp (1 / 2 * I * x) - cexp (-1 / 2 * I * x)) * dirichletKernel N x      = cexp ((N + 1 / 2) * I * x) - cexp (-(N + 1 / 2) * I * x) := by    calc (cexp (1 / 2 * I * x) - cexp (-1 / 2 * I * x)) * dirichletKernel N x      _ = ∑ n ∈ Icc (-(N : ℤ)) N, (cexp ((n + 1 / 2) * I * ↑x) - cexp ((n - 1 / 2) * I * ↑x)) := by        rw [dirichletKernel, mul_sum]        congr 1 with n        simp [sub_mul, ← exp_add, ← exp_add]        congr 2 <;>        · ring_nf          congr 1          rw [mul_assoc, mul_assoc]          congr          rw_mod_cast [← mul_assoc, mul_comm, ← mul_assoc, inv_mul_cancel₀, one_mul]          exact Real.pi_pos.ne.symm      _ = ∑ n ∈ Icc (-(N : ℤ)) N, cexp ((n + 1 / 2) * I * ↑x)          - ∑ n ∈ Icc (-(N : ℤ)) N, cexp ((n - 1 / 2) * I * ↑x) := by        rw [sum_sub_distrib]      _ = cexp ((N + 1 / 2) * I * x) - cexp (-(N + 1 / 2) * I * x) := by        rw [← sum_Ico_add_eq_sum_Icc, ← sum_Ioc_add_eq_sum_Icc, add_sub_add_comm,          ← zero_add (cexp ((N + 1 / 2) * I * ↑x) - cexp (-(N + 1 / 2) * I * ↑x))]        · congr 2          · rw [sub_eq_zero]            conv => lhs; rw [← Int.add_sub_cancel (-(N : ℤ)) 1, sub_eq_add_neg,              ← Int.add_sub_cancel (Nat.cast N) 1, sub_eq_add_neg, ← sum_Ico_add']            congr 2 with n            · rw [mem_Ico, mem_Ioc, Int.lt_iff_add_one_le, add_le_add_iff_right,              ← mem_Icc, Int.lt_iff_add_one_le, ← mem_Icc]              simp            · simp only [Int.reduceNeg, Int.cast_add, Int.cast_neg, Int.cast_one, one_div, add_assoc,                sub_eq_add_neg]              norm_num          · rw [neg_add_rev, add_comm, Int.cast_neg, sub_eq_add_neg]            norm_cast        all_goals simp  have h' : (cexp (1 / 2 * I * x) - cexp (-1 / 2 * I * x)) ≠ 0 := by    contrapose! h    rw [sub_eq_zero] at h    calc cexp (I * ↑x)      _ = cexp (1 / 2 * I * ↑x) * cexp (1 / 2 * I * ↑x) := by        rw [← exp_add, mul_assoc, ← mul_add]        ring_nf      _ = cexp (1 / 2 * I * ↑x) * cexp (-1 / 2 * I * ↑x) := by congr 2      _ = 1 := by        rw [← exp_add]        ring_nf        exact exp_zero  rw [dirichletKernel']  apply mul_left_cancel₀ h'  rw [this, mul_add, sub_eq_add_neg]  congr  · rw [mul_div]    apply eq_div_of_mul_eq    · contrapose! h      rwa [sub_eq_zero, neg_mul, exp_neg, eq_comm, inv_eq_one] at h    ring_nf    rw [← exp_add, ← exp_add, ← exp_add]    congr 2 <;> ring  · rw [mul_div]    apply eq_div_of_mul_eq (sub_ne_zero_of_ne h.symm)    ring_nf    rw [← exp_add, ← exp_add, ← exp_add, neg_add_eq_sub]    congr 2 <;> ring

Project: Carleson formalization
License: Apache-2.0
Commit: 74ef907d6bdb
Source: Carleson/Classical/DirichletKernel.lean:69-132

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.32.0

Adjoint Carleson adjoint

adjointCarleson_adjoint

Plain-language statement

adjointCarleson is the adjoint of carlesonOn.

harmonic analysisFourier analysismeasure theory

Source project: Carleson formalization

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Ae tendsto zero of distribution le

ae_tendsto_zero_of_distribution_le

Plain-language statement

Suppose that, for every error threshold $\delta>0$ and every measure tolerance $\varepsilon>0$ , one can choose $N_0$ so that the set where $\sup_{N>N_0}\lVert f(x)-F_N(x)\rVert$ exceeds $\delta$ has measure at most $\varepsilon$ . Then $F_N(x)$ converges to $f(x)$ for almost every $x$ .

harmonic analysisFourier analysismeasure theory

Source project: Carleson formalization

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Antichain operator

antichain_operator

Plain-language statement

For an antichain $\mathfrak{A}$ of pairwise incomparable tiles, and measurable functions $f$ and $g$ bounded by the indicators of $F$ and $G$ , the pairing of $g$ with the Carleson sum over $\mathfrak{A}$ is controlled by the $L^2$ norms of $f$ and $g$ and by positive powers of the two tile-density parameters. Concretely, the bound is

$C(a,q)\,\mathrm{dens}_1(\mathfrak{A})^{(q-1)/(8a^4)}\,\mathrm{dens}_2(\mathfrak{A})^{1/q-1/2}\,\lVert f\rVert_2\lVert g\rVert_2.$

harmonic analysisFourier analysismeasure theory

Source project: Carleson formalization

Person-level attribution pending.

View proof record