Geodesic
Concentration of the $L_1$-norm of trigonometric polynomials and entire functions
Sbornik. Mathematics, Tome 205 (2014) no. 11, pp. 1620-1649 Cet article a éte moissonné depuis la source Math-Net.Ru

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For any sufficiently large $n$, the minimal measure of a subset of $[-\pi,\pi]$ on which some nonzero trigonometric polynomial of order $\le n$ gains half of the $L_1$-norm is shown to be $\pi/(n+1)$. A similar result for entire functions of exponential type is established. Bibliography: 13 titles.
Keywords: trigonometric polynomials, entire functions, extremal problems
Mots-clés : $L_1$-norm. 
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Yu. V. Malykhin; K. S. Ryutin. Concentration of the $L_1$-norm of trigonometric polynomials and entire functions. Sbornik. Mathematics, Tome 205 (2014) no. 11, pp. 1620-1649. http://geodesic.mathdoc.fr/item/SM_2014_205_11_a3/

[1] Y. Benyamini, A. Kroó, A. Pinkus, “$L^1$-approximation and finding solutions with small support”, Constr. Approx., 36:3 (2012), 399–431 | DOI | MR | Zbl

[2] L. V. Taikov, “O nailuchshem priblizhenii yader Dirikhle”, Matem. zametki, 53:6 (1993), 116–121 | MR | Zbl

[3] L. V. Taikov, “Odin krug ekstremalnykh zadach dlya trigonometricheskikh polinomov”, UMN, 20:3(123) (1965), 205–211 | MR | Zbl

[4] D. V. Gorbachev, “Integralnaya zadacha Konyagina i $(C,L)$-konstanty Nikolskogo”, Teoriya funktsii, Sbornik nauchnykh trudov, Tr. IMM UrO RAN, 11, no. 2, 2005, 72–91 | MR | Zbl

[5] E. Nursultanov, S. Tikhonov, “A sharp Remez inequality for trigonometric polynomials”, Constr. Approx., 38:1 (2013), 101–132 | DOI | MR | Zbl

[6] Q. I. Rahman, G. Schmeisser, “$L^p$ inequalities for entire functions of exponential type”, Trans. Amer. Math. Soc., 320:1 (1990), 91–103 | DOI | MR | Zbl

[7] R. P. Boas, Jr., Entire functions, Academic Press Inc., New York, 1954, x+276 pp. | MR | Zbl

[8] P. Borwein, T. Erdélyi, Polynomials and polynomial inequalities, Grad. Texts in Math., 161, Springer-Verlag, New York, 1995, x+480 pp. | DOI | MR | Zbl

[9] N. P. Korneichuk, Tochnye konstanty v teorii priblizheniya, Nauka, M., 1987, 424 pp. ; N. Korneĭchuk, Exact constants in approximation theory, Encyclopedia Math. Appl., 38, Cambridge Univ. Press, Cambridge, 1991, xii+452 pp. | MR | Zbl | DOI | MR | Zbl

[10] A. G. Babenko, Yu. V. Kryakin, “Integralnoe priblizhenie kharakteristicheskoi funktsii intervala trigonometricheskimi polinomami”, Tr. IMM UrO RAN, 14:3 (2008), 19–37 ; A. G. Babenko, Yu. V. Kryakin, “Integral approximation of the characteristic function of an interval by trigonometric polynomials”, Proc. Steklov Inst. Math. (Suppl.), 264, suppl. 1 (2009), S19–S38 | MR | DOI

[11] A. G. Babenko, Yu. V. Kryakin, V. A. Yudin, “Ob odnom rezultate Geronimusa”, Tr. IMM UrO RAN, 16:4 (2010), 54–64 ; A. G. Babenko, Yu. V. Kryakin, V. A. Yudin, “On a result by Geronimus”, Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S37–S48 | Zbl | DOI

[12] F. Peherstorfer, “Trigonometric polynomials approximation in $L^1$-norm”, Math. Z., 169:3 (1979), 261–269 | DOI | MR | Zbl

[13] S. Ray, B. Lindsay, “The topography of multivariate normal mixtures”, Ann. Statist., 33:5 (2005), 2042–2065 | DOI | MR | Zbl