Asymptotic behaviour of the Hadamard determinants and the behaviour of the rows of the Padé and Chebyshev tables for a sum of exponentials

A. P. Starovoitov   ; N. A. Starovoitova

Sbornik. Mathematics, Tome 187 (1996) no. 2, pp. 297-313

Résumé

For function $f(z)=\sum _{j=1}^k e^{\lambda _j z}$ asymptotic equalities for the Hadamard determinants constructed from its Taylor coefficients are established. Using them, the asymptotics of the deviations from $f(z)$ of its Padé approximations $\Pi _{n,m}(z)$ and of the corresponding rational functions of best uniform approximation $r_{n,m}^*(z)=p_n^*(z)=q_m^*(z)$ is found for $m$ fixed as $n \to \infty$.

Bibliographie

[1] Padé H., “Memoire sur les développements en fractions continues de la fonction exponentielle”, Ann. Sci. Ecole Norm. Super, 16:3 (1899), 394–426 | MR

[2] Padé H., “Sur la représentation approchée d'une fonction par des fractions rationnelles”, Ann. Sci. Ecole Norm. Super, 9 (1892), 1–93 | MR

[3] Perron O., Die Lehre von den Kettenbrüchen, Chel. Publ. Comp., New York, 1948 | MR

[4] Luke Y. L., The special functions and their approximations, Mathematics in Science Engineering, 53, Acad. Press, New York–London, 1969 | Zbl

[5] Dzyadyk V. K., Filozof L. I., “O skorosti skhodimosti approksimatsii Pade dlya nekotorykh elementarnykh funktsii”, Matem. sb., 107(149) (1978), 347–363 | MR | Zbl

[6] Saff E. B., “The convergence of rational functions of best approximation to the exponential function”, Trans. Amer. Math. Soc., 153 (1971), 483–493 | DOI | MR | Zbl

[7] Saff E. B., “On the degree of best rational approximation to the exponential function”, J. Approx. Theory, 9:2 (1973), 97–101 | DOI | MR | Zbl

[8] Saff E. B., “The covergence of rational functions of best approximation to the exponential function”, Proc. Amer. Math. Soc., 32:1 (1972), 187–194 | DOI | MR | Zbl

[9] Beiker Dzh., Greivs-Morris P., Approksimatsii Pade, Mir, M., 1986 | MR

[10] Gonchar A. A., Rakhmanov E. A., “Ravnovesnye raspredeleniya i skorost ratsionalnoi approksimatsii analiticheskikh funktsii”, Matem. sb., 134(176) (1987), 306–352 | MR | Zbl

[11] Varga R. S., Topics in polynomial and rational interpolation and approximation, University of Montreal Press, 1982 | MR

[12] Varga R. S., “Scientific computation on some mathematical conjectures”, Approximation theory, eds. S. K. Chui, L. L. Schumaker, J. D. Ward, Academic Press, 1986, 191–209 | MR

[13] Trefethen L. N., Gutknecht M., “The Caratheodory–Fejér method for real rational approximation”, SIAM J. Numer. Anal., 20 (1983), 420–436 | DOI | MR | Zbl

[14] Braess D., “On the conjecture of Meinardus on rational approximation of $e^x$”, J. Approx. Theory, 40:4 (1984), 375–379 | DOI | MR | Zbl

[15] Trefethen L. N., “The asymptotic accuracy of rational best approximations to $e^z$ on a disk”, J. Approx. Theory, 40:4 (1984), 380–384 | DOI | MR

[16] Aptekarev A. I., “Asimptotika opredelitelei Adamara i skhodimost strok approksimatsii Pade dlya summy eksponent”, Matem. sb., 113(155) (1980), 520–537 | MR | Zbl

[17] Gonchar A. A., “O skhodimosti obobschennykh approksimatsii Pade meromorfnykh funktsii”, Matem. sb., 98(140) (1975), 564–577 | MR | Zbl

[18] Starovoitov A. P., Starovoitova N. A., “Povedenie strok tablits Pade i Chebysheva dlya summy eksponent”, Dokl. Akademii nauk Belarusi, 36:3–4 (1992), 202–204 | MR | Zbl

[19] Berezkina L. L., Rusak V. N., “O nailuchshei ratsionalnoi approksimatsii nekotorykh tselykh funktsii”, Vestsi AN BSSR. Ser. fiz.-matem. navuk, 1990, no. 4, 27–32 | MR | Zbl

[20] Rusak V. N., “Issledovanie strok ratsionalnoi tablitsy Chebyshëva dlya individualnykh analiticheskikh funktsii”, Vestsi AN BSSR. Ser. fiz.-matem. navuk, 1988, no. 6, 26–30 | MR | Zbl

[21] Dzyadyk V. K., “Ob asimptotike diagonalnykh approksimatsii Pade funktsii $\sin z$, $\cos z$, $\sh z$ i $\ch z$”, Matem. sb., 108(150):2 (1979), 247–267 | MR | Zbl

[22] Polia G., Sege G., Zadachi i teoremy iz analiza, ch. 2, Nauka, M., 1978

[23] Dzhrbashyan M. M., “K teorii ryadov Fure po ratsionalnym funktsiyam”, Izv. AN Arm. SSR. Ser. fiz.-matem. nauk, 9:7 (1957), 5–83 | MR

[24] Rusak V. N., Ratsionalnye funktsii kak apparat priblizheniya, Izd-vo BGU, Minsk, 1979 | MR