Knowlpedia

Tame Function (Segal)

A function on an infinite-dimensional Hilbert space depending on finitely many coordinates

A function ff on a real Hilbert space MM is tame if f(x)=fˉ(Px)f(x)=\bar f(Px) for some finite-dimensional subspace M′M' and projection P:M→M′P:M\to M'.

Remarks

Key properties (paper use):

  • Tame functions generate the σ-algebra R\mathfrak R of the Gaussian probability space.
  • Integrals are defined first on tame functions via finite-dimensional Gaussian integrals.
Examples
  • f(x)=exp⁡(i(x,e))f(x)=\exp(i(x,e)) depends only on the 1D coordinate (x,e)(x,e).