Theorems · Definition · harmonic analysis
UnitAddTorus.mFourierBasis
{d : Type u_1} → [inst : Fintype d] → HilbertBasis (d → ℤ) ℂ ↥(MeasureTheory.Lp ℂ 2 MeasureTheory.volume)We define mFourierBasis to be a ℤᵈ-indexed Hilbert basis for the L² space of functions
on UnitAddTorus d, which by definition is an isometric isomorphism from L²(UnitAddTorus d)
to ℓ²(ℤᵈ, ℂ).
- Defined in
- Mathlib.Analysis.Fourier.AddCircleMulti
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 271 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Fintype
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- ENNRealstatement · cited by 9,879
- Fintypestatement and proof · cited by 7,736
- Complexstatement · cited by 5,565
- AddSubgroupstatement · cited by 3,232
- MeasureTheory.MeasureSpace.volumestatement · cited by 1,323
- MeasureTheory.AEEqFunstatement · cited by 856
- MeasureTheory.Lpstatement · cited by 715
- UnitAddCirclestatement · cited by 157
- UnitAddTorusstatement · cited by 29
- HilbertBasisstatement · cited by 28
- HilbertBasis.mkproof · cited by 1
- UnitAddTorus.orthonormal_mFourierproof · cited by 1
Cited by4
Results whose statement or proof uses this declaration.
- UnitAddTorus.hasSum_mFourier_series_L2proof · cited by 1
- UnitAddTorus.hasSum_prod_mFourierCoeffproof · cited by 1
- UnitAddTorus.mFourierBasis_reprstatement and proof · cited by 1
- UnitAddTorus.coe_mFourierBasisstatement · cited by 1