Approximation of analytic functions by polynomials with integer coefficients (Q1358462)

In this paper, the author investigates the degree (or rate) of approximation by polynomials with integer coefficients for complex valued functions \(f\) defined on the unit square \(G: =[0,1] \times [0,1]\). Let \(A(G)\) denote the class of functions continuous on \(G\) and analytic in its interior. For \(r\geq 1\), let \(A^r(G)\) denote the class of functions \(f\), such that \(f^{(j)} \in A(G)\), \(0\leq j\leq r\). Let \(\Gamma\) denote the boundary of \(G\) and \(\{z_j\}^4_{j=1}\) denote its four corners. Let \(f\in A^r(G)\) and let \(q\) be the interpolating polynomial determined by the conditions. \[ q^{(s)}(z_j)= f^{(s)}(z_j),\;1\leq j\leq 4,\;0\leq s\leq r. \] Assume that the coefficients of \(q\) are integers (this is the major integer assumption). Then there exists an algebraic polynomial \(Q\) of degree \(\leq n\) with coefficients in \(\mathbb{Z}[i]\) and such that of \(z\in\Gamma\) and \(0\leq s \leq r\), \[ |f^{(s)}- Q^{(s)} |(z) \leq M \rho^{r-s}_{1+1/n} (z) \omega \bigl(f^{(s)}; \rho_{1+1/n} (z) \bigr). \] Here \(M\) is independent of \(z,n\); \(\rho_{1+1/n} (z)\) is the distance from \(z\) to the level curve \(\Gamma_{1+1/n}\), and \(\omega (f^{(s)}; \cdot)\) denotes the modulus of continuity of \(f^{(s)}\) on \(G\). The level curve of course refers to a level curve of the conformal mapping of the exterior of \(G\) onto the exterior of the unit ball. The author relates this to results of Belyi on the degree of approximation of functions on a domain whose boundary is quasiconformal, and to work of Trigub and Alper on polynomials with integer coefficients, especially on the real line.