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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
##### Keywords:
simultaneous approximations
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