Abstract
We establish sharp estimates for the best approximations of convolution classes by entire functions of exponential type. To obtain these estimates, we propose a new method for testing Nikol’skiĭ-type conditions which is based on kernel periodization with an arbitrarily large period and ensuing passage to the limit. As particular cases, we obtain sharp estimates for approximation of convolution classes with variation diminishing kernels and generalized Bernoulli and Poisson kernels.
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References
Favard J., “Sur les méilleurs procédés d’approximation de certaines classes des fonctions par des polynomes trigonométriques,” Bull. Sci. Math., vol. 61, 209–224; 243–256 (1937).
Akhiezer N. I. and Kreĭn M. G., “Best approximation of differentiable periodic functions by trigonometric sums,” Dokl. Akad. Nauk SSSR, vol. 15, no. 3, 107–112 (1937).
Akhiezer N. I., Lectures on Approximation Theory [Russian], Nauka, Moscow 1965).
Pinkus A., n-Widths in Approximation Theory, Springer-Verlag, Berlin, Heidelberg, New York, and Tokyo (1985).
Stepanets A. I., Methods of Approximation Theory, Koninklijke Brill NV, Leiden 2005).
Nikol’skiĭ S. M., “Approximation of functions by trigonometric polynomials in the mean,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 10, no. 9, 207–256 (1946).
Nagy B., “Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. I. Periodischer Fall,” Ber. Verh. sächs. Akad. Wiss. Leipzig, vol. 90, 103–134 (1938).
Zastavnyi V. P., “Nikol’skii’s theorem for kernels satisfying the more general condition than A n*,” Tr. Inst. Prikl. Mat. Mekh. NAN Ukrainy, vol. 20, 75–85 (2010).
Vinogradov O. L., “Sharp Jackson-type inequalities for approximations of classes of convolutions by entire functions of exponential type,” St. Petersburg Math. J., vol. 17, no. 4, 593–633 (2006).
Lorenz G. G. and DeVore R. A., Constructive Approximation, Springer-Verlag, Berlin, Heidelberg, and New York 1993).
Akhiezer N. I., “On best approximation of a class of continuous periodic functions,” Dokl. Akad. Nauk SSSR, vol. 17, no. 9, 451–453 (1937).
Akhiezer N. I., “On best approximation of analytic functions,” Dokl. Akad. Nauk SSSR, vol. 18, no. 4–5, 241–244 (1938).
Kreĭn M. G., “On the theory of best approximation of periodic functions,” Dokl. Akad. Nauk SSSR, vol. 18, no. 4–5, 245–249 (1938).
Dzyadyk V. K., “On best approximation in classes of periodic functions defined by integrals of a linear combination of absolutely monotonic kernels,” Math. Notes, vol. 16, no. 5, 1008–1014 (1974).
Nguen Tkhi Tkh’eu Khoa, “Rolle’s theorem for differential operators, and some extremal problems in approximation theory,” Dokl. Akad. Nauk SSSR, vol. 295, no. 6, 1313–1318 (1987).
Nguen Tkhi Tkh’eu Khoa, “Oscillation properties of differential operators and convolution operators, and some applications,” Math. USSR-Izv., vol. 34, no. 3, 609–626 (1990).
Nagy B., “Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. II. Nichtperiodischer Fall,” Ber. Verh. sächs. Akad. Wiss. Leipzig, vol. 91, 3–24 (1939).
Serdyuk A. S., “Best approximations and widths of classes of convolutions of periodic functions of high smoothness,” Ukrainian Math. J., vol. 57, no. 7, 1120–1148 (2005).
Vinogradov O. L. and Gladkaya A. V., “A nonperiodic spline analog of the Akhiezer–Kreĭn–Favard operators,” J. Math. Sci. (New York), vol. 217, no. 1, 3–22 (2016).
Vinogradov O. L., “Sharp inequalities for approximations of classes of periodic convolutions by odd-dimensional subspaces of shifts,” Math. Notes, vol. 85, no. 4, 544–557 (2009).
Pinkus A., “On n-widths of periodic functions,” J. Anal. Math., vol. 35, 209–235 (1979).
Kreĭn M. G., “Best approximation of continuous differentiable functions on the real axis,” Dokl. Akad. Nauk SSSR, vol. 18, no. 9, 619–623 (1938).
Bushanskii A. V., “On the best harmonic approximation in the mean of some functions,” in: Studies of the Theory of Approximation of Functions and Their Applications [Russian], Inst. Math., Kiev, 1978, 29–37.
Shevaldin V. T., “Widths of classes of convolutions with Poisson kernel,” Math. Notes, vol. 51, no. 6, 611–617 (1992).
Serdyuk A. S., “Widths and the best approximations of classes of convolutions of periodic functions,” Ukr. Mat. Zh., vol. 51, no. 5, 674–687 (1999).
Baraboshkina N. A., “Approximation of harmonic functions by algebraic polynomials on a circle of radius smaller than one with constraints on the unit circle,” Trudy Inst. Mat. i Mekh. UrO RAN, vol. 19, no. 2, 71–78 (2013).
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St. Petersburg. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 58, No. 2, pp. 251–269, March–April, 2017; DOI: 10.17377/smzh.2017.58.202.
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Vinogradov, O.L. Sharp inequalities for approximations of convolution classes on the real line as the limit case of inequalities for periodic convolutions. Sib Math J 58, 190–204 (2017). https://doi.org/10.1134/S0037446617020021
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DOI: https://doi.org/10.1134/S0037446617020021