Polynomials least deviating from zero on a square of the complex plane

E. B. Bayramov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 3, pp. 5-15

Résumé

The Chebyshev problem is studied on the square $\Pi=\left\{z=x+iy\in\mathbb{C}\colon\max\{|x|,|y|\}\le 1\right\}$ of the complex plane $\mathbb{C}$. Let $\mathfrak{P}_n$ be the set of algebraic polynomials of a given degree $n$ with the unit leading coefficient. The problem is to find the smallest value $\tau_n(\Pi)$ of the uniform norm $\|p_n\|_{C(\Pi)}$ of polynomials $p_n\in \mathfrak{P}_n$ on the square $\Pi$ and a polynomial with the smallest norm, which is called the Chebyshev polynomial (for the squire). The Chebyshev constant $\tau(Q)=\lim_{n\rightarrow\infty} \sqrt[n]{\tau_n(Q)}$ for the squire is found. Thus, the logarithmic asymptotics of the least deviation $\tau_n(\Pi)$ with respect to the degree of a polynomial is found. The problem is solved exactly for polynomials of degrees from 1 to 7. The class of polynomials in the problem is restricted; more exactly, it is proved that, for $n=4m+s$, $0\le s\le 3$, it is sufficient to solve the problem on the set of polynomials $z^sq_m(z)$, $q_m\in \mathfrak{P}_m$. Effective two-sided estimates for the value of the least deviation $\tau_n(\Pi)$ with respect to $n$ are obtained.

Détail

Mots-clés : algebraic polynomial
Keywords: uniform norm, square of the complex plane, Chebyshev polynomial.

Bibliographie

[1] Chebyshev P.L., “Teoriya mekhanizmov, izvestnykh pod nazvaniem parallelogramov”, Polnoe sobranie sochinenii P. L. Chebysheva, v 5 t., v. 2, Matematicheskii analiz, AN SSSR, M.; L., 1947, 23–51

[2] Smirnov V.I., Konstruktivnaya teoriya funktsii kompleksnogo peremennogo, Nauka, M.; L., 1964, 327 pp.

[3] Milovanovic G.V., Topics in polynomials: Extremal problems, inequalities, zeros, World Scientific Publ. Comp., Singapore, 1994, 821 pp.

[4] Thiran J.-P., “Chebyshev polynomials on circular arcs in the complex plane”, Progress in Approximation Theory, eds. P. Nevai, A. Pinkus, Acad. Press, Boston, 1991, 771–786

[5] Maergoiz L.S., “Mnogochleny Chebysheva s nulevym mnozhestvom na duge okruzhnosti”, Dokl. AN, 426:1 (2009), 26–28

[6] Lukashov A.L., “Ekstremalnye polinomy na dugakh okruzhnosti s nulyami na etikh dugakh”, Izv. NAN Armenii. Matematika, 2009, no. 3, 19–29

[7] Lukashov A.L., “Neravenstva dlya proizvodnykh ratsionalnykh funktsii”, Izv. RAN. Ser. matematicheskaya, 68:3 (2004), 115–138

[8] Arestov V.V., “O trigonometricheskikh polinomakh, naimenee uklonyayuschikhsya ot nulya”, Dokl. AN, 425:6 (2009), 733–736

[9] Arestov V.V., “Trigonometric polynomials that deviate the least from zero in measure and related problems”, J. Approx. Theory, 162:10 (2010), 1852–1878 | DOI

[10] Arestov V.V., “Algebraicheskie mnogochleny, naimenee uklonyayuschiesya ot nulya po mere na otrezke”, Ukr. mat. zhurn., 62:3 (2010), 292–301

[11] Babenko A.G., “Neravenstva slabogo tipa dlya trigonometricheskikh polinomov”, Tr. In-ta matematiki i mekhaniki UrO RAN, 2 (1992), 34–41

[12] Bernshtein S.N., Ekstremalnye svoistva polinomov, ONTI, M., 1937, 203 pp.

[13] Bairamov E.B., “O mnogochlenakh Chebysheva na kvadrate kompleksnoi ploskosti”, Sovremennye problemy matematiki i ee prilozhenii, tez. dokl. Mezhdunar. 49-i mol. shk.-konf., IMM UrO RAN, UrFU, Ekaterinburg, 2018, 72

[14] Goluzin G.M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, uchebnoe posobie, Nauka GITTL, M., L., 1952, 628 pp.

[15] Fekete M., “Uber die Verteilung der Wuzzeln bei gewissen algebraiscnen Gleichugen mit ganzzahligen Koeffizienten”, Math. Z., 17:1 (1923), 228–249

[16] Tobin A.D.,Lloyd N.T., Schwarz-Christoffel mapping: textbook, Cambridge Univ. Press, Cambridge, 2002, 132 pp.

[17] Ivanov V.I., Popov V.Yu., Konformnye otobrazheniya i ikh prilozheniya, Editorial URSS, M., 2002, 324 pp.

[18] Volkovyskii L.I., Lunts G.L., Aramanovich I.G., Sbornik zadach po teorii funktsii kompleksnogo peremennogo, Nauka, M., 1975, 320 pp.

[19] Fikhtengolts G.M., Kurs differentsialnogo i integralnogo ischisleniya, v 3 t., 2, Fizmatlit, M., 2001, 864 pp.