Let$A$and$B$be positive numbers and$m$and$n$positive integers,$m<n$. Then there is for complex valued functions φ on$R$with sufficient differentiability and boundedness properties a representationwhere$v$_{$1$}and$v$_{$2$}are bounded Borel measures with$v$_{$1$}absolutely continuous, such that there exists a function φ with ∣φ^{(n)}∣ ⩽$A$and ∣φ∣ ⩽$A$on$R$and satisfying $$\varphi ^{(m)} (0) = A\int_R {\left| {d\nu _1 } \right|} + B\int_R {\left| {d\nu _2 } \right|} .$$ This result is formulated and proved in a general setting also applicable to derivatives of fractional order. Necessary and sufficient conditions are given in order that the measures and the optimal functions have the same essential properties as those which occur in the particular case stated above.

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The paper deals with the possibility to solve the heat equation backwards in time. More specifically, we treat the following problem. Given the temperature at a finite number of points of a homogeneous bar, how old can the heat distribution be? In the case that the temperature is given at equidistant points x_{1}, the problem is completely solved. In the case of nonequidistant x_{i}we find an upper bound for the age. Such a bound is also obtained when the information about the heat distribution is given by the value of a finite number of linear functionals.