Theorems · Theorem · measure theory
Real.circleAverage_eq_integral_add
∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] {f : ℂ → E} {c : ℂ} {R : ℝ} (η : ℝ),
Real.circleAverage f c R = (2 * Real.pi)⁻¹ • ∫ (θ : ℝ) in 0..2 * Real.pi, f (circleMap c R (θ + η))Circle averages do not change when shifting the angle.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 261 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Realstatement and proof · cited by 25,697
- NormedAddCommGroupstatement and proof · cited by 15,752
- NormedSpacestatement and proof · cited by 12,499
- Complexstatement and proof · cited by 5,565
- zero_addproof · cited by 2,366
- Real.pistatement and proof · cited by 1,774
- add_commproof · cited by 1,535
- MeasureTheory.MeasureSpace.volumestatement and proof · cited by 1,323
- intervalIntegralstatement and proof · cited by 546
- mul_inv_revproof · cited by 270
- Function.Periodicproof · cited by 154
- circleMapstatement and proof · cited by 117
Cited by1
Results whose statement or proof uses this declaration.
- Real.circleAverage_neg_radiusproof · cited by 2