Knowlpedia

Wiener Transform W

A unitary transform on Gaussian L₂ intertwining T with T^{*-1}

The Wiener transform WW is the on the Gaussian space L2(M,n)L_2(M,n) whose action on the polynomial subspace is

(Wf)(x)=∫f(2 y+ix) dn(y).(Wf)(x)=\int f(\sqrt2\,y+i x)\,dn(y).

and whose action on all of L2(M,n)L_2(M,n) is obtained by continuous extension.

Intertwining properties

For TT in the , the satisfies

WU(T)W−1=U(T∗−1).W\mathfrak U(T)W^{-1}=\mathfrak U(T^{*-1}).

For the Fock–Cook field operators used in the paper,

P(x)=WQ(x)W−1.P(x)=WQ(x)W^{-1}.

In one dimension, under the standard unitary identification, WW becomes the Fourier transform.