Bernstein–Szegö inequality for fractional derivatives of trigonometric polynomials

V. V. Arestov ; P. Yu. Glazyrina

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 20 (2014) no. 1, pp. 17-31

Résumé

On the set $\mathscr F_n$ of trigonometric polynomial of degree $n\ge1$ with complex coefficients, we consider the Szegö operator $D^\alpha_\theta$ defined by the relation $D^\alpha_\theta f_n(t)=\cos\theta D^\alpha f_n(t)-\sin\theta D^\alpha\widetilde f_n(t)$ for $\alpha,\theta\in\mathbb R$, $\alpha\ge0$; where $D^\alpha f_n$ and $D^\alpha\widetilde f_n$ are the Weyl fractional derivatives of (real) order $\alpha$ of the polynomial $f_n$ and its conjugate polynomial $\widetilde f_n$. In particular, we prove that, if $\alpha\ge n\ln2n$, then, for any $\theta\in\mathbb R$, the sharp inequality $\|\cos\theta D^\alpha f_n-\sin\theta D^\alpha\widetilde f_n\|_{L_p}\le n^\alpha\|f_n\|_{L_p}$ holds in the spaces $L_p$ for all $p\ge0$ on the set $\mathscr F_n$. For classical derivatives (of integer order $\alpha\ge1$), this inequality was obtained by Szegö (1928) in the uniform norm $(p=\infty)$ and by Zygmund (1931–1935) for $1\le p\infty$. A. I. Kozko (1998) proved this inequality for fractional derivatives of (real) order $\alpha\ge1$ and $1\le p\le\infty$.

Détail

Keywords: trigonometric polynomial, Weyl fractional derivative, Bernstein inequality, Szegö inequality.

Bibliographie

[1] Bernshtein S. N., “O nailuchshem priblizhenii nepreryvnykh funktsii posredstvom mnogochlenov dannoi stepeni”, Sobr. soch., v 4 t., v. 1, Konstruktivnaya teoriya funktsii, Izd.-vo AN SSSR, M., 1952, 11–104

[2] Bernstein S., “Sur l'ordre de la meilleure approximation des fonctions continues par des polynomes de degré donné”, Mémoires de l'Académie Royale de Belgique, 2:4 (1912), 1–103

[3] Bernshtein S. N., “Avtorskie kommentarii”, Sobr. soch., v 4 t., v. 1, Konstruktivnaya teoriya funktsii, Izd.-vo AN SSSR, M., 1952, 526–562

[4] Bernstein S., Leçons sur les propriétés extrémales et la meilleure approximation des fonctions analytiques d'une variable réelle, Collection Borel, Gauthier-Villar, Paris, 1926, 213 pp. | Zbl

[5] Riesz M., “Formule d'interpolation pour la dérivée d'un polynome trigonométrique”, C. R. Acad. Sci., 158 (1914), 1152–1154 | Zbl

[6] Riesz M., “Eine trigonometrische Interpolationsformel und einige Ungleichungen für Polynome”, Jahresbericht der Deutschen Mathematiker-Vereinigung, 23 (1914), 354–368 | Zbl

[7] Szegö G., “Über einen Satz des Herrn Serge Bernstein”, Schriften der Königsberger Gelehrten Gesellschaft, 5:4 (1928), 59–70 | Zbl

[8] Zigmund A., Trigonometricheskie ryady, v 2 t., v. 1, Mir, M., 1965, 616 pp. ; т. 2, 538 с. | MR

[9] Lizorkin P. I., “Otsenki trigonometricheskikh integralov i neravenstvo Bernshteina dlya drobnykh proizvodnykh”, Izv. AN SSSR. Ser. matematicheskaya, 29:1 (1965), 109–126 | MR | Zbl

[10] Bang T., “Une inégalité de Kolmogoroff et les fonctions presquepériodiques”, Kongelige Danske Videnskabernes Selskab. Matematisk-fysiske Meddenlelser, 19:4 (1941), 1–28 | MR | Zbl

[11] Geisberg S. P., “Analogi neravenstv S. N. Bernshteina dlya drobnoi proizvodnoi”, Voprosy prikladnoi matematiki i matematicheskogo modelirovaniya, Kratkie soderzhaniya dokl. 25-i nauch. konf. (24 yanv. – 4 fevr. 1967 g.), Leningr. inzh.-stroit. in-t, L., 1967, 5–10

[12] Samko S. G., Kilbas A. A., Marichev O. I., Integraly i proizvodnye drobnogo poryadka i nekotorye ikh prilozheniya, Nauka i tekhnika, Minsk, 1987, 688 pp. | MR | Zbl

[13] Wilmes G., “On Riesz-type inequalities and $K$-functionals related to Riesz potentials in $\mathbb R^N$”, N. Numer. Funct. Anal. Optim., 1:1 (1979), 57–77 | DOI | MR | Zbl

[14] Kozko A. I., “The exact constants in the Bernstein–Zygmund–Szegö inequalities with fractional derivatives and the Jackson–Nikolskii inequality for trigonometric polynomials”, East J. Approx., 4:3 (1998), 391–416 | MR | Zbl

[15] Arestov V. V., Glazyrina P. Yu., “Sharp integral inequalities for fractional derivatives of trigonometric polynomials”, J. Approx. Theory, 164:11 (2012), 1501–1512 | DOI | MR | Zbl

[16] Arestov V. V., Glazyrina P. Yu., “Integralnye neravenstva dlya algebraicheskikh i trigonometricheskikh polinomov”, Dokl. AN, 442:6 (2012), 727–731 | MR | Zbl

[17] Arestov V. V., “S. N. Bernstein inequalities for algebraic and trigonometric polynomials”, Dokl. AN SSSR, 246:6 (1979), 1289–1292 | MR | Zbl

[18] Arestov V. V., “Ob integralnykh neravenstvakh dlya trigonometricheskikh polinomov i ikh proizvodnykh”, Izv. AN SSSR. Ser. matematicheskaya, 45:1 (1981), 3–22 | MR | Zbl

[19] Glazyrina P. Yu., “Necessary conditions for metrics in integral Bernstein-type inequalities”, J. Approx. Theory, 162:6 (2010), 1204–1210 | DOI | MR | Zbl

[20] Arestov V. V., “Neravenstvo Sege dlya proizvodnykh sopryazhennogo trigonometricheskogo polinoma v $L_0$”, Mat. zametki, 56:6 (1994), 10–26 | MR | Zbl

[21] Belinsky E. S., Liflyand E. R., “Approximation properties in $L_p$, $0

1$”, Funct. Approx. Comment. Math., 22 (1993), 189–199 | MR

[22] Runovski K., Schmeisser H.-J., “On some extensions of Bernstein's inequality for trigonometric polynomials”, Funct. Approx. Comment. Math., 29 (2001), 125–142 | MR

[23] Leonteva A. O., “Neravenstvo Bernshteina dlya proizvodnoi nulevogo poryadka trigonometricheskikh polinomov v $L_0$”, Tr. In-ta matematiki i mekhaniki UrO RAN, 19, no. 2, 2013, 216–223

[24] Taikov L. V., “O sopryazhennykh trigonometricheskikh polinomakh”, Mat. zametki, 48:4 (1990), 110–114 | MR | Zbl

[25] Arestov V. V., “Integralnye neravenstva dlya algebraicheskikh mnogochlenov na edinichnoi okruzhnosti”, Mat. zametki, 48:4 (1990), 7–18 | MR | Zbl

[26] Polia G., Sege G., Zadachi i teoremy iz analiza, v 2 t., v. 1, Nauka, M., 1978, 392 pp.; т. 2, 432 с.

[27] Marden M., Geometry of polynomials, Math. surveys, 3, AMS, Providence, 1966, 243 pp. | MR | Zbl

[28] De Bruijn N. G., Springer T. A., “On the zeros of composition-polynomials”, Nederl. Akad. Wetensch. Proc., 50 (1947), 895–903, (= Indag. Math., 9 (1947) 406–414) | MR | Zbl

[29] De Bruijn N. G., “Inequalities concerning polynomials in the complex domain”, Nederl. Akad. Wetensch. Proc., 50 (1947), 1265–1272, (= Indag. Math., 9 (1947) 591–598) | MR | Zbl

[30] Pólya G., “Über die Nullstellen gewisser ganzer Funktionen”, Math. Zeitschrift., 2:3–4 (1918), 352–383 | DOI | MR | Zbl

[31] Markushevich A. I., Teoriya analiticheskikh funktsii, v. 2, Nauka, M., 1968, 624 pp. | Zbl