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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

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