Theorems · Theorem · harmonic analysis
fourierCoeffOn_of_hasDerivAt
∀ {a b : ℝ} (hab : a < b) {f f' : ℝ → ℂ} {n : ℤ},
n ≠ 0 →
(∀ x ∈ Set.uIcc a b, HasDerivAt f (f' x) x) →
IntervalIntegrable f' MeasureTheory.volume a b →
fourierCoeffOn hab f n =
1 / (-2 * ↑Real.pi * Complex.I * ↑n) * ((fourier (-n)) ↑a * (f b - f a) - (↑b - ↑a) * fourierCoeffOn hab f' n)Express Fourier coefficients of f on an interval in terms of those of its derivative.
- Defined in
- Mathlib.Analysis.Fourier.AddCircle
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 271 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites22
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement · cited by 62,936
- Setstatement · cited by 53,352
- Realstatement and proof · cited by 25,697
- Complexstatement and proof · cited by 5,565
- ContinuousMapstatement · cited by 2,491
- Real.pistatement · cited by 1,774
- Complex.ofRealstatement · cited by 1,654
- MeasureTheory.MeasureSpace.volumestatement and proof · cited by 1,323
- Set.Iooproof · cited by 1,214
- Complex.Istatement · cited by 866
- HasDerivAtstatement and proof · cited by 493
- AddSubgroup.zmultiplesstatement · cited by 493
Cited by1
Results whose statement or proof uses this declaration.
- bernoulliFourierCoeff_recurrenceproof · cited by 2