Fritz John pp 461-465 | Cite as

# Commentary on [39] and [47]

Chapter

## Abstract

Paper [39] is concerned with the approximate solution of the backward heat equation: and | where h real and m an integer are parameters to be chosen, and f̄ is an approximation to f, obtained, e. g., by measurement. The measurement error is denoted by

*u*_{ t }=*u*_{ xx },*x*in*R*, -*T*<*t*≤ 0.*u*(*x*, 0)*f*(*x*), under the assumption that*u*(*x, t*) ≥ 0. John shows that under this condition$$\left| {f(x + iy)} \right| \leqslant {e^{{v^{2/4l}}}}\mu ,\mu = Maxf(x)$$

*u*(x, - a)| ≤ (1 -*a T*)^{-1 2}*μ*. John considers approximations*ū*to*u*of the following kind:$$\bar u(x, - t) = \sum\limits_m^m {c_j^m} (t)\bar f(x + jh)$$

*ɛ*:*ɛ*= Max |*f*(*x*) -*f̄*(*x*)|.## Preview

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