Existence and uniqueness of consistent Hermite – Fourier approximations

A. P. Starovoitov ; E. P. Kechko ; T. M. Osnath

Problemy fiziki, matematiki i tehniki, no. 2 (2023), pp. 68-73.

Résumé

The trigonometric analogues of algebraic Hermite – Padé approximations were defined, these are Hermite – Fourier approximations. In particular, the theorem of existence of Hermite – Fourier approximations was proved, the sufficient condition of their uniqueness was obtained, and the criterion of the existence and uniqueness of Hermite – Fourier polynomials, which are the numerator and denominator of Hermite – Fourier approximations associated with an arbitrary set of trigonometric series k. When the conditions of the criterion were met, the explicit type of the specified polynomials was established.

Détail

Keywords: trigonometric series, Fourier sums, trigonometric Padé approximants, Hermite – Padé polynomials, Hermite – Padé approximations.

Bibliographie

[1] C. Hermite, “Sur la fonction exponentielle”, C.R. Akad. Sci., 77 (1873), 18–293

[2] K. Mahler, “Zur Approximation der Exponentialfunktion und des Logarithmus, I”, Journal fur die reine und angewandte Mathematik, 166:2 (1932), 118–136 | DOI | MR

[3] K. Mahler, “Zur Approximation der Exponentialfunktion und des Logarithmus, II”, Journal fur die reine und angewandte Mathematik, 166:3 (1932), 137–150 | DOI | MR | Zbl

[4] K. Mahler, “Perfect systems”, Compositio Mathematica, 19:2 (1968), 95–166 | MR | Zbl

[5] Dzh. Beiker ml., P. Greivs-Morris, Approksimatsii Pade. 1. Osnovy teorii. 2. Obobscheniya i prilozheniya, Mir, M., 1986, 502 pp.

[6] E.M. Nikishin, V.N. Sorokin, Ratsionalnye approksimatsii i ortogonalnost, Nauka, M., 1988, 256 pp.

[7] A.I. Aptekarev, V.I. Buslaev, A. Martines-Finkelshtein, S.P. Suetin, “Approksimatsii Pade, nepreryvnye drobi i ortogonalnye mnogochleny”, Uspekhi matematicheskikh nauk, 66:6 (402) (2011), 37–122 | DOI | Zbl

[8] A.I. Aptekarev, P.M. Bleher, A.B.J. Kuijlaars, “Large n limit of Gaussian random matrices with external source, II”, Communications in Mathematical Physics, 259:2 (2005), 367–389 | DOI | MR | Zbl

[9] B. Beckermann, V. Kalyagin, A.C. Matos, F. Wielonsky, How well does the Hermite - Pade approximation smooth the Gibbs phenomenon?, Mathematics of Computation, 80:274 (2011), 931–958 | DOI | MR | Zbl

[10] A.I. Aptekarev, V.G. Lysov, D.N. Tulyakov, “Sluchainye matritsy s vneshnim istochnikom i asimptotika sovmestno ortogonalnykh mnogochlenov”, Matematicheskii sbornik, 202:2 (2011), 3–56 | DOI | Zbl

[11] S.P. Suetin, “Approksimatsii Pade i effektivnoe analiticheskoe prodolzhenie stepennogo ryada”, Uspekhi matematicheskikh nauk, 57:1 (343) (2002), 45–142 | DOI | Zbl

[12] S.P. Suetin, “Raspredelenie nulei polinomov Pade i analiticheskoe prodolzhenie”, Uspekhi matematicheskikh nauk, 70:5 (425) (2015), 121–174 | DOI | Zbl

[13] G. Chudnovsky, “On the method of ThueSiegel”, Ann. of Math., 117:2 (1983), 325–382 | DOI | MR | Zbl

[14] A.P. Starovoitov, N.V. Ryabchenko, D.A. Volkov, “O suschestvovanii i edinstvennosti mnogochlenov Ermita - Pade vtorogo roda”, Problemy fiziki, matematiki i tekhniki, 2019, no. 2 (39), 92–96

[15] A.P. Starovoitov, N.V. Ryabchenko, “O determinantnykh predstavleniyakh mnogochlenov Ermita - Pade”, Trudy Moskovskogo matematicheskogo obschestva, 83, no. 1, 2022, 17–36

[16] Yu.A. Labych, A.P. Starovoitov, “Trigonometricheskie approksimatsii Pade funktsii s regulyarno ubyvayuschimi koeffitsientami Fure”, Matematicheskii sbornik, 200:7 (2009), 107–130 | DOI | Zbl