Approximation and Interpolation by Entire Functions of Several Variables

Let f: R(n) -> R be C(infinity) and let h: R(n) -> R be positive and continuous. For any unbounded nondecreasing sequence {c(k)} of nonnegative real numbers and for any sequence without accumulation points {x(m)} in R(n), there exists an entire function g: C(n) -> C taking real values on R(n) such that g((alpha))(x) - f((alpha))(x)vertical bar < h(x), vertical bar x vertical bar >= c(k), vertical bar alpha vertical bar <= k, k = 0, 1, 2,..., g((alpha))(x(m)) - f((alpha))(x(m)), vertical bar x(m)vertical bar >= c(k), vertical bar alpha vertical bar <= k, m, k = 0, 1, 2,..., This is a version for functions of several variables of the case n = 1 due to L. Hoischcn.