Rational approximation to orthogonal bases and of the solutions of elliptic equations

Abstract.

It is shown that given $N \ge 2$, an orthogonal basis $\phi_1, ..., \phi_n$ of $\mathbb{R}^n$ can be approximated by an orthogonal basis $b_1, ..., b_n$, where $b_1$ has integral and $b_2, \, ... \, , \, b_n$ have rational components, such that the angle between $\phi_i$ and $b_i$ is at most $\frac 1N$ and the length $|b_i| \le 10n^3N$, $i = 1, ..., n$. This improves the length of the integral approximation due to Schmidt (1995). As an application, we improve a theorem of Kocan (1995) about the minimal size of grids in the solutions of elliptic equations. Our result fits the need in Kuo and Trudinger (1990).