The authors consider the problem of determining the temperature distribution u ( x , t ) on the half-line x = 0 , 0$"> t > 0 , from measurements at an interior point, for all 0$"> t > 0 . As is well-known, this is an ill-posed problem Using the Tikhonov method, the authors give a regularized solution, and assuming the (unknown) exact solution is in H α ( ℝ ) , 0$"> α > 0 . They give an error estimate of the order 1 / ( 1 n 1 /  ϵ  ) α for ϵ  → 0 , where 0$"> ϵ  > 0 is a bound on the measurement error.