Geodesic
On the existence of trigonometric Pad\'e approximations
Problemy fiziki, matematiki i tehniki, no. 3 (2024), pp. 71-76.

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In this work, based on the well-known results on classical Padé approximants of power series, the conditions are found under which trigonometric Padé – Jacobi approximants exist for a given Fourier series. This made it possible to describe the class of Fourier series in Chebyshev polynomials of the first and second kind, for which there are nonlinear Padé – Chebyshev approximants. In particular, another proof of the well-known theorem of S.P. Suetin is given.
Mots-clés : Padé approximants
Keywords: Padé – Chebyshev approximations, power series, Fourier series, series in Chebyshev polynomials. 
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A. P. Starovoitov; T. M. Osnach; N. V. Ryabchenko. On the existence of trigonometric Pad\'e approximations. Problemy fiziki, matematiki i tehniki, no. 3 (2024), pp. 71-76. http://geodesic.mathdoc.fr/item/PFMT_2024_3_a11/

[1] Dzh. Beiker ml., P. Greivs-Morris, Approksimatsii Pade, v. 1, Osnovy teorii, Mir, M., 1986; т. 2, Обобщения и приложения

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

[3] A.P. Starovoitov, E.P. Kechko, T.M. Osnach, “Suschestvovanie i edinstvennost sovmestnykh approksimatsii Ermita - Fure”, Problemy fiziki, matematiki i tekhniki, 2023, no. 2 (55), 68-73 | Zbl

[4] A.P. Starovoitov, E.P. Kechko, T.M. Osnach, “O suschestvovanii trigonometricheskikh approksimatsii Ermita-Yakobi i nelineinykh approksimatsii Ermita-Chebysheva”, Zhurnal Belorusskogo gosudarstvennogo universiteta. Matematika. Informatika, 2023, no. 2, 6-17

[5] S.P. Suetin, “O suschestvovanii nelineinykh approksimatsii Pade-Chebysheva dlya analiticheskikh funktsii”, Matematicheskie zametki, 86:2 (2009), 290-303 | DOI | Zbl

[6] H. Pade, “Sur la representation approchee d'une fonction par des fractions rationnelles”, Annales scientifiques de l'E.N.S. Ser. 3, 9 (1892), 3-93 | MR

[7] G. Frobenious, “Ueber Relationen zwischen den Naherungsbruchen von Potenzreihen”, Journal fur die reine und angewandte Mathematik, 90 (1881), 1-17 | DOI | MR

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

[9] C. Jacobi, “Uber die Darstellung einer Reihe gegebner Werthe durch eine gebrochne rationale Function”, J. Reine Angew. Math., 30 (1846), 127-156 | MR

[10] A.P. Starovoitov, N.V. Ryabchenko, “O determinantnykh predstavleniyakh mnogochlenov Ermita-Pade”, Trudy Moskovskogo matematicheskogo obschestva, 83, no. 1, 2022, 17-35

[11] A.A. Gonchar, E.A. Rakhmanov, S.P. Suetin, “Approksimatsii Pade-Chebysheva dlya mnogoznachnykh analiticheskikh funktsii, variatsiya ravnovesnoi energii i S-svoistvo statsionarnykh kompaktov”, UMN, 66:6 (2011), 3-36 | DOI