Extraction of harmonics from trigonometric polynomials by phase-amplitude operators

D. G. Vasilchenkova ; V. I. Danchenko

Algebra i analiz, Tome 32 (2020) no. 2, pp. 21-44.

Résumé

The method of phase-amplitude transformations is used for extraction of harmonics $ \tau _{\mu }$ of a given order $ \mu $ from trigonometric polynomials $\displaystyle T_n(t)=\sum _{k=1}^n\tau _k(t), \tau _k(t):= a_k\cos kt+b_k\sin kt.$     Such transformations take polynomials $ T_n(t)$ to similar polynomials by using two simplest operations: multiplication by a real constant $ X$ and shift by a real phase $ \lambda $, i.e., $ T_n(t)\to X\cdot T_n(t-\lambda )$. The harmonic $ \tau _{\mu }$ is extracted by addition of similar polynomials: $\displaystyle \tau _{\mu }(t)=\sum _{k=1}^{m}X_k\cdot T_n(t-\lambda _k), m\le n,$     where the $ X_k$ and $ \lambda _k$ are defined by explicit formulas. Similar formulas for harmonics are obtained on a fairly large class of convergent trigonometric series. This representation yields sharp estimates of Fejér type for harmonics and coefficients of the polynomial $ T_n$.

Détail

Keywords: discrete moment problem, Prony method, regularization.

Bibliographie

[1] Grenander U., Szego G., Toeplitz forms and their applications, 2nd ed., Chelsea Publ. Co., New York, 1984 | MR | Zbl

[2] Pisarenko V. F., “The retrieval of harmonics from a covariance function”, Geophys. J. R. Astr. Soc., 33 (1973), 347–366 | DOI | Zbl

[3] Beylkin G., Monzón L., “On generalized gaussian quadratures for exponentials and their applications”, Appl. and Comput. Harmon. Anal., 12:3 (2002), 332–373 | DOI | MR | Zbl

[4] Suetin P. K., Klassicheskie ortogonalnye mnogochleny, 3-e izd., pererab. i dop., Fizmatlit, M., 2005

[5] Prony R., “Sur les lois de la Dilatabilité des fluides élastiques et sur celles de la Force expansive de la vapeur de l'eau et de la vapeur de l'alkool, à différentes températures”, J. de l'Ecole Polytech., 2:4 (1795), 28–35

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

[7] Gashkov S. B., “Neravenstvo Feiera–Egervari–Sassa dlya neotritsatelnykh trigonometricheskikh mnogochlenov”, Mat. prosveschenie, 9 (2005), 69–75 | Zbl

[8] Stechkin S. B., “O nekotorykh ekstremalnykh svoistvakh polozhitelnykh trigonometricheskikh polinomov”, Mat. zametki, 7:4 (1970), 411–422

[9] Arestov V. V., “Ob ekstremalnykh svoistvakh neotritsatelnykh trigonometricheskikh polinomov”, Tr. In-ta mat. i mekh. UrO RAN, 1, 1992, 50–70

[10] Belov A. C., “Nekotorye svoistva i otsenki dlya neotritsatelnykh trigonometricheskikh polinomov”, Izvestiya RAN. Ser. mat., 67:4 (2003), 3–20 | DOI

[11] Belov A. S., “Ob otsenkakh sverkhu chastnykh summ trigonometricheskogo ryada cherez otsenki snizu”, Mat. sb., 183:11 (1992), 55–74 | Zbl

[12] Danchenko V. I., Danchenko D. Ya., “Extraction of pairs of harmonics from trigonometric polynomials by phase-amplitude operators”, J. Math. Sci. (N. Y.), 232:3 (2018), 322–337 | DOI | MR | Zbl

[13] Chunaev P., Danchenko V., “Approximation by amplitude and frequency operators”, J. Approx. Theory, 207 (2016), 1–31 | DOI | MR | Zbl

[14] Danchenko V. I., Dodonov A. E., “Otsenki eksponentsialnykh summ. Prilozheniya”, Probl. mat. anal., 67 (2012), 23–30 | Zbl

[15] Sylvester J. J., “On a remarkable discovery in the theory of canonical forms and of hyperdeterminants”, Phil. Magazine, 2 (1851), 391–410

[16] Kung J. P. S., “Canonical forms of binary forms: variations on a theme of Sylvester”, Invariant Theory and Tableaux (Minneapolis, MN, 1988), IMA Vol. Math. Appl., 19, Springer, New York, 1990, 46–58 | MR

[17] Lyubich Y. I., “The Sylvester–Ramanujan system of equations and the complex power moment problem”, Ramanujan J., 8:1 (2004), 23–45 | DOI | MR | Zbl

[18] Boley D., Luk F., Vandevoorde D., “Vandermonde factorization of a Hankel matrix”, Scientific Computing (Hong Kong, 1997), Springer, Singapore, 1997, 27–39 | MR

[19] Prasolov V. V., Zadachi i teoremy lineinoi algebry, Nauka, M., 1996

[20] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady. Elementarnye funktsii, v. 1, 2-e izd., isprav., Fizmatlit, M., 2002