charFun_eq_fourierIntegral — Mathlib

English

The characteristic function can be expressed as a Fourier-type integral against the inner product character: χ_μ(t) = VectorFourier.fourierIntegral probChar μ bilinFormOfRealInner 1 (−t).

Русский

Характеристическая функция выражается через четыреier-интал, как χ_μ(t) = VectorFourier.fourierIntegral probChar μ bilinFormOfRealInner 1 (−t).

LaTeX

$$$$\\chi_{\\mu}(t) = \\mathrm{VectorFourier.fourierIntegral}(probChar, \\mu, bilinFormOfRealInner, 1, -t).$$$$

Lean4

/-- `charFun` is a Fourier integral for the inner product and the character `probChar`. -/
theorem charFun_eq_fourierIntegral (t : E) :
    charFun μ t = VectorFourier.fourierIntegral probChar μ bilinFormOfRealInner 1 (-t) := by
  simp [charFun_apply, VectorFourier.fourierIntegral_probChar]

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