Abstract
We present a survey of new results related to the investigation of the rate of convergence of Fourier sums on the classes of functions defined by convolutions whose kernels have monotone Fourier coefficients.
Similar content being viewed by others
REFERENCES
A.N. Kolmogorov, “Zur Grössenordnung des Restliedes Fouriershen Reihen differenzierbaren Funktionen,” Ann. Math., 36, 521–526 (1935).
V. T. Pinkevich, “On the order of the remainder of the Fourier series of a function differentiable in the Weyl sense,” Izv. Akad. Nauk SSSR, Ser. Mat., 4, No. 5, 521–528 (1940).
S. M. Nikol'skii, “Approximation of periodic functions by trigonometric polynomials,” Tr. Inst. Mat. Akad. Nauk SSSR, 15, 1–76 (1945).
S. M. Nikol'skii, “Approximation of functions by trigonometric polynomials in the mean,” Izv. Akad. Nauk SSSR, Ser. Mat., 10, No. 3, 207–256(1946).
S. A. Telyakovskii, “On the norms of trigonometric polynomials and approximation of differentiable functions by linear means of their Fourier series. I,” Tr. Mat. Inst. Akad. Nauk SSSR, 62, 61–97 (1961).
A. V. Efimov, “Linear methods for approximation of continuous periodic functions,” Mat. Sb., 54, No. 1, 51–90 (1961).
A. V. Efimov, “Approximation of continuous periodic functions by Fourier sums,” Izv. Akad. Nauk SSSR, Ser. Mat., 24, No. 2, 243–296 (1960).
A. I. Stepanets, Classification and Approximation of Periodic Functions [in Russian], Naukova Dumka, Kiev (1987).
A. I. Stepanets, “Rate of convergence of Fourier series on the classes of \(\overline \psi \)-integrals,” Ukr. Mat. Zh., 49, No. 8, 1201–1251 (1997).
A. I. Stepanets, “Approximation of \(\overline \psi \)-integrals of periodic functions by Fourier sums (small smoothness). I,” Ukr. Mat. Zh., 50, No. 2, 314–333 (1998).
A. I. Stepanets, “Approximation of \(\overline \psi \)-integrals of periodic functions by Fourier sums (small smoothness). II,” Ukr. Mat. Zh., 50, No. 3, 442–454 (1998).
A. B. Stechkin, “Estimate of the remainder of Fourier series for differentiable functions,” Tr. Inst. Mat. Akad. Nauk SSSR, 145, 126–151 (1980).
A. I. Stepanets, Uniform Approximation by Trigonometric Polynomials [in Russian], Naukova Dumka, Kiev (1981).
A. I. Stepanets, “Solution of the Kolmogorov - Nikol'skii problem for Poisson integrals of continuous functions,” Mat. Sb., 191, No. 1, 113–138 (2001).
A. I. Stepanets and A. S. Serdyuk, “Lebesgue inequalities for Poisson integrals,” Ukr. Mat. Zh., 52, No. 6, 798–808 (2000).
A. I. Stepanets, “Lebesgue inequality on classes of (ψ, β)??-differentiable functions,” Ukr. Mat. Zh., 41, No. 4, 435–443 (1989).
A. I. Stepanets and A. S. Serdyuk, “Approximation by Fourier sums and best approximations on classes of analytic functions,” Ukr. Mat. Zh., 52, No. 3, 375–395 (2000).
A. V. Bushanskii, “On the best harmonic approximation in the mean for some functions,” in: Investigations on the Theory of Approximation of Functions and Their Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1978), pp. 29–37.
V. T. Shevaldin, “Widths of classes of convolutions with Poisson kernel,” Mat. Zametki, 51, No. 6, 611–617 (1992).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Stepanets', O.I. Approximation of Convolution Classes by Fourier Sums. New Results. Ukrainian Mathematical Journal 54, 713–740 (2002). https://doi.org/10.1023/A:1021623127852
Issue Date:
DOI: https://doi.org/10.1023/A:1021623127852