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Suppose we have a function $\rho(x,t)$ defined over $[-1,1]\times[0,1]$. Define the integral $ f_T(x)=\int_{[0,1]} (\rho(.,t)*K_{T-t})(x) dt $ for $x\in R$ and $T>1$, where $*$ is the convolution, and $K_a$ is the Gaussian kernel with variance $a$.

The question is given $f_T$ for some $T>1$, can we determine $\rho$ uniquely (noting that $\rho$ has a bounded support)?

$f_T$ has the following intuitive meaning: Suppose we have an ideal metal rod of infinite length . $\rho(x,t)$ is the rate of heat flow into $x$ at time $t$. So $f_T$ is the temperature at time $T$.

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  • $\begingroup$ sounds like recovering a 2d function from 1d measurement? $\endgroup$
    – Yimin
    Commented Feb 23, 2016 at 3:53
  • $\begingroup$ @yimin Sounds strange at first sight, that this may well be possible (note that $f_T(x)$ can be an analytic function in $x$ and $T$ and hence, determined if known on a set with some cluster point). More to the OP: I don't know the answer but I vaguely remember that the book The One-Dimensional Heat Equation by Cannon is a great source for such questions. $\endgroup$
    – Dirk
    Commented Feb 23, 2016 at 10:22
  • $\begingroup$ I second Dirk's suggestion -- if the heat equation is one-dimensional, Cannon's book has more about it than you've ever wanted to know. From my experience, determining space- and time-dependent $\rho$ could be tricky; you'd have much better chance if you have $\rho=\rho_1(t)\rho_2(x)$, where one of the two factors is known. $\endgroup$
    – Christian Clason
    Commented Feb 23, 2016 at 12:56
  • $\begingroup$ you can recover $\rho(x,t)$ iff you can reconstruct $(\rho \ast K)(x,t)$ for every $x,t$. so you'll need more than $f_T(.)$ for a single $T$, for example you'll need $\displaystyle\frac{\partial^k f}{\partial T^k}(x,T)$ for every $k,x$ $\endgroup$
    – reuns
    Commented Feb 25, 2016 at 0:35
  • $\begingroup$ @jian Uniqueness holds according to Terry Tao's comment mathoverflow.net/questions/72195/… $\endgroup$
    – Andrew
    Commented Feb 26, 2016 at 13:27

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