The Main triangle truncation Sample Clauses

The Main triangle truncation. Using the approach illustrated in (2.1.1), we can also define a continuous analogue of the transformator SΨN in the following way.‌ Let 1τ denote the characteristic function of the interval (0, τ) and let us consider ∆τ (t, s) = 1τ (t + s), t, s > 0. Define S∆τ to be the transformator defined formally by: S∆τ (Op(k)) = ∆τ ٨ Op(k). Let us also define the operator Jτ : L2(R+) → L2(0, τ) to be the operator such that for any f ∈ L2(R+) one has: Jτ f(t) = 1τ (t)f(τ − t), 0 < t < τ. It is immediate to see that for any fixed τ > 0, Jτ is a partial isometry onto L2(0, τ) and so Jτ = 1. Let Jτ be the transformator defined as Jτ (A) = Jτ A, A ∈ B(L2(R+)), then we see that Jτ is a partial isometry onto B(L2(0, τ)). The importance of such a transformator can be seen in the following: