Theorems · Theorem · harmonic analysis

MeasureTheory.Lp.fourierInv_toTemperedDistribution_eq

∀ {E : Type u_1} {F : Type u_2} [inst : NormedAddCommGroup E] [inst_1 : MeasurableSpace E] [inst_2 : BorelSpace E]
  [inst_3 : NormedAddCommGroup F] [inst_4 : InnerProductSpace ℂ F] [inst_5 : CompleteSpace F]
  [inst_6 : InnerProductSpace ℝ E] [inst_7 : FiniteDimensional ℝ E] (f : ↥(MeasureTheory.Lp F 2 MeasureTheory.volume)),
  FourierTransformInv.fourierInv (MeasureTheory.Lp.toTemperedDistribution f) =
    MeasureTheory.Lp.toTemperedDistribution (FourierTransformInv.fourierInv f)

The 𝓢'-inverse Fourier transform and the L2-inverse Fourier transform coincide on L2.

Defined in: Mathlib.Analysis.Fourier.LpSpace
Cited by: 1 results in Mathlib
Foundations: Depth 312 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NormedAddCommGroup MeasurableSpace BorelSpace NormedAddCommGroup InnerProductSpace CompleteSpace InnerProductSpace FiniteDimensional

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Cites25

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
MeasurableSpacestatement and proof · cited by 13,106
NormedSpaceproof · cited by 12,499
MeasureTheory.Measureproof · cited by 10,939
ENNRealstatement and proof · cited by 9,879
Complexstatement and proof · cited by 5,565
InnerProductSpacestatement and proof · cited by 3,523
AddSubgroupstatement · cited by 3,232
Factproof · cited by 2,726
CompleteSpacestatement and proof · cited by 2,532
FiniteDimensionalstatement and proof · cited by 1,854

Cited by1

Results whose statement or proof uses this declaration.

TemperedDistribution.memSobolev_iff_exists_smulLeftCLM_fourierproof · cited by 2