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Convergence of simultaneous Padé approximants for a class of functions. (Russian) Zbl 0633.41013
Let \(\sigma\) be a finite Borel measure on [0,1] and let \(f(x)=\int^{1}_{0}d\sigma (t)/(x-t)\). The nth diagonal Padé approximation to f is the unique rational function \(\pi_ n=P_ n/Q_ n\) of order n such that \(f-\pi_ n\) has a zero of order \(2n+1\) at infinity. \(Q_ n\) is the nth orthogonal polynomial relative to \(\sigma\). Of interest in number theory is the construction of simultaneous approximations to the functions \(f(x\xi^ m)\), \(m=0,...,r-1\), where \(\xi\) is an rth root of unity. Subject to certain conditions on \(\sigma\), approximations can be found of the form \(\pi_ n=P_ n/Q_ n\), where \(Q_ n\) is a polynomial of degree nr in \(x^ r\) and orthogonal to \(x^ k\) for \(k<n\). The functions \(\pi_ n(x\xi^ m)\) simultaneously approximate \(f(x\xi^ m)\), \(0\leq m<r.\)
It follows from an extension of a result of A. A. Gonchar and E. A. Rakhmanov [Tr. Mat. Inst. Steklova 157, 31-48 (1981; Zbl 0492.41027)] that there exists a measure \(\mu\), independent of \(\sigma\), such that on \(D_*={\mathbb{C}}-\{t\xi^ m|\) \(t\in [0,1]\), \(0\leq m<r\}\), one has \(| Q_ n(x)|^{1/n}\to \exp (-V_*(x))\) and \(| f(x)-\pi_ n(x)|^{1/n}\to \exp (W(x)-w),\) where \(V(x)=- \int^{1}_{0}\log | x-t| d\mu (t)\) is the logarithmic potential of \(\mu\), \(V_*(x)=\sum^{r-1}_{m=0}V(x\xi^ m)\), \(W=V+V_*\) and w is some constant with \(W(x)<w\) on \(D_*\). This paper derives explicit expressions for \(\mu\), V and w.
Reviewer: C.Series

MSC:
41A21 Padé approximation
41A28 Simultaneous approximation
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