Abstract
We propose a new method for proving the classical inverse theorems of the theory of approximation by trigonometric polynomials and entire functions of exponential type. The method is based on the construction of identities expressing the derivatives of a function itself and of its trigonometrical conjugate in terms of convolution operators. As a consequence, we reduce the constants in the estimates of the norms of the derivatives in terms of best approximations.
Similar content being viewed by others
References
Akhiezer N. I., Lectures on Approximation Theory, Nauka, Moscow (1965) [Russian].
Timan A. F., Theory of Approximation of Functions of a Real Variable, Dover, New York (1994).
Wilmes G., “On Riesz-type inequalities and \( K \)-functionals related to Riesz potentials in \( ^{N} \),” Numer. Funct. Anal. Optim., vol. 1, no. 1, 57–77 (1979).
Stepanets A. I., Methods of Approximation Theory, VSP, Leiden (2005).
Bernstein S. N., “On the best approximation of continuous functions with polynomials of a given degree,” in: Collected Works. Vol. 1, Akad. Nauk SSSR, Moscow (1952), 11–104.
Sterlin M. D., “Inverse extremal problems in constructive function theory,” Soviet Math. Dokl., vol. 17, no. 3, 1065–1068 (1977).
Stechkin S. B., “On best approximation of conjugate functions by trigonometric polynomials,” Izv. Akad. Nauk SSSR Ser. Mat., vol. 20, 197–206 (1956).
Bari N. K. and Stechkin S. B., “Best approximations and differential properties of two conjugate functions,” Trudy Moskov. Mat. Obshch., vol. 5, 483–521 (1956).
Stepanets A. I., “Inverse theorems on approximation of periodic functions,” Ukrainian Math. J., vol. 47, no. 9, 1441–1448 (1995).
Simonov B. V. and Tikhonov S. Yu., “On embeddings of function classes defined by constructive characteristics,” Banach Center Publ., vol. 72, 285–307 (2006).
Simonov B. V. and Tikhonov S. Yu., “Embedding theorems in constructive approximation,” Sb. Math., vol. 199, no. 9, 1367–1407 (2008).
Potapov M. K., Simonov B. V., and Tikhonov S. Yu., Fractional Moduli of Smoothness, Maks Press, Moscow (2016) [Russian].
Taberski R., “Contribution to fractional analysis and exponential approximation,” Funct. Approx. Comment. Math., vol. 15, 81–106 (1986).
Timan A. F., “Best approximation and modulus of smoothness of functions prescribed on the entire real axis,” Izv. Vyssh. Uchebn. Zaved. Mat., vol. 6, 108–120 (1961).
Sterlin M. D., “Estimates of constants in inverse theorems of constructive function theory,” Soviet Math. Dokl., vol. 14, no. 6, 647–650 (1973).
Zhuk V. V., Approximation of Periodic Functions, Leningrad University, Leningrad (1982) [Russian].
Shapiro H. S., “Some Tauberian theorems with applications to approximation theory,” Bull. Amer. Math. Soc., vol. 74, no. 3, 500–504 (1968).
Vinogradov O. L., “On the constants in inverse theorems for the first-order derivative,” Vestnik St. Petersburg University, Math., vol. 54, no. 4, 334–344 (2021).
Stein E. M., Singular Integrals and Differentiability Properties of Functions, Princeton University, Princeton (1970).
Vinogradov O. L., “Sharp Jackson type inequalities for approximation of classes of convolutions by entire functions of finite degree,” St. Petersburg Math. J., vol. 17, no. 4, 593–633 (2006).
Natanson G. I., “An estimate for Lebesgue constants of the Vallée-Poussin sums,” in: Geometric Problems of the Theory of Functions and Sets, Kalinin University, Kalinin (1986), 102–107 [Russian].
Fikhtengolts G. M., A Course of Differential and Integral Calculus. Vol. 2, Fizmatgiz, Moscow (1959) [Russian].
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 3, pp. 531–544. https://doi.org/10.33048/smzh.2022.63.305
Rights and permissions
About this article
Cite this article
Vinogradov, O.L. On the Constants in the Inverse Theorems for the Norms of Derivatives. Sib Math J 63, 438–450 (2022). https://doi.org/10.1134/S0037446622030053
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0037446622030053