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On Padé approximants associated with Hamburger series. (English) Zbl 0571.41013
The author considers Hamburger series $$f(z)=\sum^{\infty}_{k=0}\mu_ kz^ k$$ whose coefficients are the moments $$\mu_ k=\int^{\infty}_{-\infty}t^ kd\lambda (t)$$ of an increasing function $$\lambda$$ having infinitely many points of increase. For $$j\geq 0$$ even, $$\pi_{n,j}$$ is the monic orthogonal polynomial of degree n with respect to the weight $$t^ jd\lambda$$, the numbers $$\tau^{(n)}_{\nu,j}$$ are the zeros of $$\pi_{n,j}$$, $$\lambda^{(n)}_{\nu,j}$$ the associated Christoffel numbers and $$f[n- 1+j,n]$$ the Padé approximant to f(z). An entry of a Padé table is said to be normal if it does not occur in any other location of the table; it is known that every entry of the Padé table of a Stieltjes series (i.e., when $$\lambda (t)=const$$. for $$t<0)$$ is normal. On the other hand, if $$d\lambda$$ is symmetric with respect to the origin, then no entry of the Padé table is normal. Set $\sigma_{n,j}(z)=\int^{\infty}_{-\infty}((\pi_{n,j}(z)- \pi_{n,j}(t))/(z-t))t^ jd\lambda (t)$ and $$\rho_{n,j}(z)=\int^{\infty}_{-\infty}(\pi_{n,j}(t)/(z-t))t^ jd\lambda (t)$$. The author proves that for $$n\geq 1$$ the entry f[n-1,n] is normal if and only if $$\pi_{n,0}(0)\sigma_{n,0}(0)\neq 0$$, and for $$j>0$$ even $$f[n-1+j,n]$$ is normal if and only if $$\pi_{n,j}(0)\rho_{n,j}(0)\neq 0$$. As a corollary he obtains that all entries $$f[n-1+j,n]$$, $$n\geq 1$$, $$j\geq 0$$ (even or odd) will be normal if and only if $$\sigma_{n,0}(0)\neq 0$$ for all $$n\geq 1$$ and $$\pi_{n,j}(0)\neq 0$$ for all $$n\geq 1$$ and all even $$j\geq 0$$. Next assume that $$d\lambda (t)=w(t)dt$$, where w(t)$$\geq 0$$ on a symmetric interval $$I=[-a,a]$$ $$(0<a\leq +\infty)$$, continuous in (-a,a). Using a result of D. B. Hunter [Math. Comput. 29, 559-565 (1975; Zbl 0304.42016)], for which the author gives a simplified proof, and writing $$\mu_ k^{(n)}$$ for $$\mu^{(n)}_{k,0}$$, he proves: (a) if w(t)/w(-t) is strictly increasing on I, then (*) $$0<\mu_ k^{(1)}<\mu_ k^{(2)}<...<\mu_ k^{([k/2]+1)}=\mu_ k^{([k/2]+2)}=...=\mu_ k$$; (b) if $$w(t)=w(-t)$$ on I, then $$0=\mu_ k^{(1)}<\mu_ k^{(2)}<...<\mu_ k^{((k/2)+1)}=\mu_ k^{((k/2)+2)}=...=\mu_ k$$ if k is even, and (trivially) $$0=\mu_ k^{(1)}=\mu_ k^{(2)}=...=\mu_ k$$ if k is odd; (c) if w(t)/w(-t) is strictly decreasing on I, then (*) holds for even k, and $$0>\mu_ k^{(1)}>\mu_ k^{(2)}>...>\mu_ k^{([k/2])}>\mu_ k^{([k/2]+1)}=\mu_ k^{([k/2]+2)}=...=\mu_ k$$ if k is odd. The last section discusses three numerical methods for computing the Padé approximants.
Reviewer: J.Horváth

##### MSC:
 41A21 Padé approximation 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 44A60 Moment problems
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