The Definition — Question 8

Question 8 – Solution

Strategy. Distinguish uniqueness of a few model parameters from uniqueness among all possible functions.

Step 1: Integrate and calibrate. For $s>0$, $$F(s)=A\int_a^\infty e^{-st}\,dt=\frac{Ae^{-as}}s.$$ For $s\le 0$, the positive tail integral diverges. Dividing the two measured equations gives $2F(2)/F(1)=e^{-a}=e^{-1}$, so $\boxed{a=1,\ A=2}$. These values reproduce both measurements.

Step 2: Characterize all consistent data. For positive $M_1,M_2$, the family requires $2M_2/M_1=e^{-a}\in(0,1]$. Thus consistency is exactly $\boxed{2M_2\le M_1}$. In that case the unique parameters are $$\boxed{a=\ln\frac{M_1}{2M_2},\qquad A=\frac{M_1^2}{2M_2}.}$$ They satisfy $a\ge 0$, $A>0$ and both equations, proving sufficiency as well as necessity. Equality corresponds to onset at zero.

Step 3: Construct an invisible perturbation. Set $b=s+1>0$. Integration by parts gives the three integrals of $1,t,t^2$ against $e^{-bt}$ as $1/b,1/b^2,2/b^3$. Therefore $$H(s)=\frac 2{(s+1)^3}-\frac 5{3(s+1)^2}+\frac 1{3(s+1)} =\boxed{\frac{(s-1)(s-2)}{3(s+1)^3}}.$$ The exponential boundary terms vanish, justifying those integrations. In particular, $H(1)=H(2)=0$, although $h$ is not the zero function.

Step 4: State the identification limit. For every real $\lambda$, the signal $f+\lambda h$ has a convergent transform for $s>0$. Splitting its defining integral gives $F(s)+\lambda H(s)$, so both recorded measurements are unchanged. Distinct $\lambda$ give distinct real-valued signals; for nonzero $\lambda$ they are not members of the original two-parameter family. The recovery formulas establish uniqueness only under that family assumption, not among arbitrary signals.