We study the average of sum, in the sense of integral.

Left shift operator

Throughout we consider the Hilbert space \(L^2=L^2(\mathbb{R})\), the space of all complex-valued functions with real variable such that \(f \in L^2\) if and only if \[ \lVert f \rVert_2^2=\int_{-\infty}^{\infty}|f(t)|^2dm(t)<\infty \] where \(m\) denotes the ordinary Lebesgue measure (in fact it's legitimate to consider Riemann integral in this context).

For each \(t \geq 0\), we assign an bounded linear operator \(Q(t)\) such that \[ (Q(t)f)(s)=f(s+t). \] This is indeed bounded since we have \(\lVert Q(t)f \rVert_2 = \lVert f \rVert_2\) as the Lebesgue measure is translate-invariant. This is a left translation operator with a single step \(t\).