Substituting (4.15) into (4.24b) and using the sampling property of the Dirac delta (Zemanian 1987, Section 1.7) leads to
By using the convolution theorem for the Fourier transform (Champeney 1990, Chapter 6) one obtains (4.24d).
Starting from (5.1) with t1 = t + τ and t2 = t substituted into, one has
where, in the fourth equality the Fourier transform pair ↔ δ(f − f1) is used. Equation (4.27b) immediately follows from (5.2) by using the sampling property of Dirac delta (Zemanian 1987, Section 1.7).
Let ψ(·) be invertible and its inverse ϕ(·) differentiable. It results that (Zemanian 1987, Section 1.7)
(5.3)
Therefore
By using (4.91), (4.89), and (5.4), one has
from which (4.92) follows by using the sampling property of the Dirac delta (Zemanian 1987, Section 1.7).
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