Geodesic
Extraction of Several Harmonics from Trigonometric Polynomials. Fejér-Type Inequalities
Informatics and Automation, Differential equations and dynamical systems, Tome 308 (2020), pp. 101-115 
D. G. Vasilchenkova; V. I. Danchenko. Extraction of Several Harmonics from Trigonometric Polynomials. Fejér-Type Inequalities. Informatics and Automation, Differential equations and dynamical systems, Tome 308 (2020), pp. 101-115. http://geodesic.mathdoc.fr/item/TRSPY_2020_308_a7/
@article{TRSPY_2020_308_a7,
     author = {D. G. Vasilchenkova and V. I. Danchenko},
     title = {Extraction of {Several} {Harmonics} from {Trigonometric} {Polynomials.} {Fej\'er-Type} {Inequalities}},
     journal = {Informatics and Automation},
     pages = {101--115},
     year = {2020},
     volume = {308},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2020_308_a7/}
}
TY  - JOUR
AU  - D. G. Vasilchenkova
AU  - V. I. Danchenko
TI  - Extraction of Several Harmonics from Trigonometric Polynomials. Fejér-Type Inequalities
JO  - Informatics and Automation
PY  - 2020
SP  - 101
EP  - 115
VL  - 308
UR  - http://geodesic.mathdoc.fr/item/TRSPY_2020_308_a7/
LA  - ru
ID  - TRSPY_2020_308_a7
ER  - 
%0 Journal Article
%A D. G. Vasilchenkova
%A V. I. Danchenko
%T Extraction of Several Harmonics from Trigonometric Polynomials. Fejér-Type Inequalities
%J Informatics and Automation
%D 2020
%P 101-115
%V 308
%U http://geodesic.mathdoc.fr/item/TRSPY_2020_308_a7/
%G ru
%F TRSPY_2020_308_a7

Voir la notice de l'article provenant de la source Math-Net.Ru

Given a trigonometric polynomial $T_n(t)=\sum _{k=1}^n\tau _k(t)$, $\tau _k(t):=a_k\cos kt+b_k\sin kt$, we consider the problem of extracting the sum of harmonics $\sum \tau _{\mu _s}(t)$ of prescribed orders $\mu _s$ by the method of amplitude and phase transformations. Such transformations map the polynomials $T_n(t)$ into similar ones using two simple operations: the multiplication by a real constant $X$ and the shift by a real phase $\lambda $, i.e., $T_n(t)\mapsto XT_n(t-\lambda )$. We represent the sum of harmonics as a sum of such polynomials and then use this representation to obtain sharp Fejér-type estimates.

[1] V. V. Arestov, “On extremal properties of the nonnegative trigonometric polynomials”, Tr. Inst. Mat. Mekh. (Ekaterinburg), 1 (1992), 50–70 | MR | Zbl

[2] A. S. Belov, “Some estimates for non-negative trigonometric polynomials and properties of these polynomials”, Izv. Math., 67:4 (2003), 637–653 | DOI | DOI | MR | Zbl

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

[4] Boley D.L., Luk F.T., Vandevoorde D., “Vandermonde factorization of a Hankel matrix”, Scientific computing, WSC'97, Proc. Workshop (Hong Kong, 1997), Springer, Berlin, 1998, 27–39 | MR | Zbl

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

[6] V. I. Danchenko and D. Ya. Danchenko, “Refinement of Fejér's inequality on a subclass of nonnegative trigonometric polynomials”, Modern Methods of Function Theory and Related Problems: Mater. Int. Conf. Voronezh Winter Math. Sch. (2019), Izd. Dom VGU, Voronezh, 2019, 114–115 (in Russian)

[7] Danchenko V.I., Dodonov A.E., “Estimates for exponential sums. Applications”, J. Math. Sci., 188:3 (2013), 197–206 | DOI | MR | Zbl

[8] S. B. Gashkov, “Fejér–Egerváry–Szász inequality for nonnegative trigonometric polynomials”, Mat. Prosveshch., Ser. 3, 9 (2005), 69–75

[9] Grenander U., Szegö G., Toeplitz forms and their applications, Chelsea Publ., New York, 1984 | MR | Zbl

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

[11] Pisarenko V.F., “The retrieval of harmonics from a covariance function”, Geophys. J. R. Astr. Soc., 33:3 (1973), 347–366 | DOI | Zbl

[12] G. Pólya and G. Szegő, Problems and Theorems in Analysis, II: Theory of Functions, Zeros, Polynomials, Determinants, Number Theory, Geometry, Springer, Berlin, 1998 | MR

[13] Prony R., “Essai expérimental et analytique 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. Éc. polytech., 2 (1795), 24–76

[14] S. B. Stechkin, “Some extremal properties of positive trigonometric polynomials”, Math. Notes, 7:4 (1970), 248–255 | DOI | MR | Zbl

[15] Sylvester J.J., “On a remarkable discovery in the theory of canonical forms and of hyperdeterminants”, Philos. Mag., 2:12 (1851), 391–410 | DOI

[16] D. G. Vasilchenkova and V. I. Danchenko, “Extraction of harmonics from trigonometric polynomials by amplitude and phase operators”, Algebra Anal., 32:2 (2020), 21–44