convolution_lsmul — Mathlib

English

Convolution with the zero function yields the zero function: f ⋆[L, μ] 0 = 0.

Русский

Свёртка нулевой функции давала нулёвую функцию: f ⋆ 0 = 0.

LaTeX

$$$f ⋆[L, μ] 0 = 0$$$

Lean4

/-- The definition of convolution where the bilinear operator is scalar multiplication.
Note: it often helps the elaborator to give the type of the convolution explicitly. -/
theorem convolution_lsmul [Sub G] {f : G → 𝕜} {g : G → F} : (f ⋆[lsmul 𝕜 𝕜, μ] g : G → F) x = ∫ t, f t • g (x - t) ∂μ :=
  rfl

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