Rational approximation to orthogonal bases and of the solutions of elliptic equations (Q1282360)

One of the main results is Theorem 4: Let \(N\geq 2\) be given and let \(\{\Phi_j\}\) be an orthogonal basis in \(\mathbb{R}^n\). Then there exists an orthogonal basis \(\{b_j\}\) as rational approximation to \(\{\Phi_j\}\) and satisfying \[ F(n)\equiv \max (|b_1|,\ldots,|b_n|)\leq \begin{cases} 2N,&\text{ if } n=2,\\ 10n^3N&\text{ if }n\geq 3. \end{cases} \] The theorem is used to difference schemes for second-order elliptic equations of the form \(u(x)\) - \(\text{trace}(AD^2u(x))=0\), where \(A\) is a symmetric \(n\times n\) matrix, \(\lambda I\leq A\leq \Lambda I,\) \(0<\lambda <\Lambda ,\) \(D^2u(x)=(\frac{\partial ^2u(x)}{\partial x_i\partial x_j})_{i,j=1,\dots,n}\). Let the set of permissible numbers be \(\omega h\) and define \[ N_A=\inf_{F_h}\{\lim_{h\to 0}\sup \omega h\} , \] where the infimum is taken over the family of all constant and monotone difference operators \(F_h\) on the grid \(G(h)\) with step \(h\), that are an approximation of the operator \(F(n)\). By means of theorem 4 the following Theorem 2 is proved: Let \(E=\Lambda /\lambda ,\) then we have \[ N_A\leq \begin{cases} 64E,&\text{ if }n=2,\\ 40n^6E&\text{ if }n\geq 3. \end{cases} \] Theorem 2 is the improvement of appropriate \textit{M. Kocan}'s result [ibid. 72, No. 1, 73-92 (1995; Zbl 0846.65053)].