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).
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Received January 27, 1997 / Revised version received April 1, 1998
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Hwang, J. Rational approximation to orthogonal bases and of the solutions of elliptic equations. Numer. Math. 81, 561–575 (1999). https://doi.org/10.1007/s002110050404
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DOI: https://doi.org/10.1007/s002110050404