We establish necessary and sufficient conditions for the validity of Bernstein-type inequalities for the fractional derivatives of trigonometric polynomials of several variables in spaces with integral metrics. The problem of sharpness of these inequalities is investigated.
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S. N. Bernstein, Collected Works [in Russian], Vol. 1, Akad. Nauk SSSR, Moscow (1952).
M. Riesz, “Formule d’interpolation pour la dérivée d’un polynome trigonométrique,” C. R. Acad. Sci., 158, 1152–1154 (1914).
A. Zygmund, Trigonometric Series [Russian translation], Vol. 2, Mir, Moscow (1965).
V. I. Ivanov, “Direct and inverse theorems of the approximation theory in the metric of L p for 0 < p < 1,” Mat. Zametki, 18, No. 5, 641–658 (1975).
É. A. Storozhenko, V. G. Krotov, and P. Oswald, “Direct and inverse Jackson theorems in the spaces L p , 0 < p < 1,” Mat. Sb., 98(140), No. 3(11), 395–415 (1975).
A. Mate and P. G. Nevai, “Bernstein’s inequality in L p for 0 < p < 1 and (C, 1) bounds for orthogonal polynomials,” Ann. Math., 111, No. 1, 145–154 (1980).
V. V. Arestov, “On the integral inequalities for trigonometric polynomials and their derivatives,” Izv. Akad. Nauk SSSR, Ser. Mat., 45, No. 1, 3–22 (1981).
P. I. Lizorkin, “Estimates for trigonometric integrals and the Bernstein inequality for fractional derivatives,” Izv. Akad. Nauk SSSR, Ser. Mat., 29, No. 1, 109–126 (1965).
E. Görlich, R. J. Nessel, and W. Trebels, “Bernstein-type inequalities for families of multiplier operators in Banach spaces with Cesáro decompositions. I. General theory,” Acta Sci. Math. (Szeged), 34, 121–130 (1973).
R. Taberski, “Approximation in the Fréchet spaces L p (0 < p < 1),” Funct. Approxim., 7, 105–121 (1979).
E. Belinsky and E. Liflyand, “Approximation properties in L p , 0 < p < 1,” Funct. Approxim., 22, 189–199 (1993).
K. Runovski and H.-J. Schmeisser, “On some extensions of Bernstein’s inequality for trigonometric polynomials,” Funct. Approxim., 29, 125–142 (2001).
S. A. Pichugov, “Inequalities for trigonometric polynomials in spaces with integral metric,” Ukr. Mat. Zh., 63, No. 12, 1657–1671 (2011); English translation: Ukr. Math. J., 63, No. 12, 1883–1899 (2012).
Yu. S. Kolomoitsev, “On the Bernstein-type inequalities for fractional derivatives in the classes φ(L),” in: Proc. of the Institute of Applied Mathematics and Mechanics, Ukrainian National Academy of Sciences [in Russian], 26, (2013), pp. 95–103.
E. M. Stein and G. Weiss, Introduction to Fourier Analysis of Euclidean Spaces, Princeton Univ. Press, Princeton (1971).
S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators [in Russian], Nauka, Moscow (1978).
R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions, Dordrecht, Kluwer (2004).
A. E. Ingham, “A note on Fourier transforms,” J. London Math. Soc., 1-9, No. 1, 29–32 (1934).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 1, pp. 42–56, January, 2014.
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Kolomoitsev, Y.S. Inequalities for the Fractional Derivatives of Trigonometric Polynomials in Spaces with Integral Metrics. Ukr Math J 67, 45–61 (2015). https://doi.org/10.1007/s11253-015-1064-6
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DOI: https://doi.org/10.1007/s11253-015-1064-6