Asymptotic behavior of periodic, periodic biharmonic and periodic harmonic functions

The behavior of periodic functions defined on domains containing the upper half space, $\left \{ {\left ( {{x^1},{x^2},...,{x^n}} \right ):{x^n} > 0} \right \}$, is investigated as ${x^n}$ approaches infinity. Bounds on some of the first order derivatives of these functions are obtained which are directly proportional to bounds on derivatives of arbitrary orders in certain directions. It is shown that a periodic biharmonic and a periodic harmonic function can be approximated, respectively, by a third degree and a first degree polynomial in the variable ${x^n}$ and that, as ${x^n}$ approaches infinity, the error in using this approximation vanishes faster than the reciprocal of ${x^n}$ raised to any power.