The infinitely differentiable extension of systems of functions

The “best” extension of systems of functions of real variables from an $(n-1)$-dimensional hyperplane $E_{n?1}$ to the whole of $E_n$ is investigated. It is shown that extension can be realized to a function, infinitely differentiable outside $E_{n?1}$, whose derivatives have in a certain sense the best possible rate of growth close to $E_{n?1}$ functions (the $B$-class).