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A variational approach to the theory of multipoint Padé approximants
1.S. T. Epstein, J. Chem. Phys. 48, 4716 (1968).
1.See also S. T. Epstein and J. H. Epstein, Wisconsin Theoretical Chemistry Institute Report, WIS‐TCI‐297 (1968).
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3.J. Nuttal, Phys. Rev. 157, 1312 (1967).
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5.J. Nuttal, “The Connection of Padé Approximants with Stationary Variational Principles,” in The Padé Approximant in Theoretical Physics, edited by G. A. Baker, Jr. and J. L. Gammel (Academic, New York, 1970).
6.J. O. Hirschfelder, W. Byers Brown, and S. T. Epstein, “Recent Developments in Perturbations Theory,” in Advances in Quantum Chemistry (Academic, New York, 1964), Vol. I.
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8.P. W. Langhoff, Chem. Phys. Lett. 12, 223 (1971).
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14.L. Wuytack, to appear in Rocky Mountain J. Math.
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17.M. Karplus and H. J. Kolker, J. Chem. Phys. 41, 3955 (1964).
18.The trial function (in obvious notation) is such that it could be accurate through the term in The corresponding optimization condition reads This set of equations is clearly equivalent to the set oneobtains on projecting the exact equation onto the space spanned by Since this projection cannot alter the solution to (*) through the term in we conclude that must indeed be accurate through the term in
19.The set of Eqs. (18) can be linearized to read Together with the normalization condition these equations constitute linear equations in unknowns.
20.It is also possible to use a lower bound to The bounding properties of the resulting approximant are much the same as those described later in this section, except that now one has only an upper bound for rather than the bounds for and for
21.G. A. Baker, Jr., J. Math. Phys. 10, 814 (1969).
22.P. W. Langhoff and M. Karplus, Phys. Rev. Lett. 19, 1461 (1967). See also the references given by Langhoff in Ref. 8.
23.For example, a nonoptimal Padé approximant would be better than an optimal Padé approximant if the function to be approximated is of the form where is a polynomial of finite degree and is representable by a series of Stieltjes.
24.Here one could make many other choices for the trial function such that only the given information would be utilized in the resulting approximants. For example, one could take the range of summation to be rather than In this way one can generate whole families of rational approximations to each member of which matches at the data points and which imposes rigorous bounds on elsewhere for
25.It is clearly possible that as Correspondingly we then have and the formal use of such a in the variational formulation given in the body of the paper might then be considered unrigorous. However, this objection can be circumvented as follows. Instead of starting with the representation (C3), we could have proceeded directly from (C1), writing If we then choose an H whose spectrum is the points of increase of in together with the number 0, so that then we can define a self‐adjoint operator W such that We then have and avariational lower bound is, for example, The operator W can clearly be chosen so that it is bounded, and the derivation can be carried through in a manner similar to that used in the paper.
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