Abstract
We prove a theorem on the relationship between the modulus of smoothness and the best approximation in L p , 0 < p < 1, and theorems on the extension of functions with preservation of the modulus of smoothness in L p , 0 < p < 1. We also give a complete description of multipliers of periodic functions in the spaces L p , 0 < p < 1.
Similar content being viewed by others
References
É. A. Storozhenko and P. Oswald, “Jackson theorem in L p (ℝk), 0 < p < 1,” Sib. Mat. Zh., 19, 888–901 (1978).
V. K. Runovskii, “On approximation by families of linear polynomial operators in the spaces L p , 0 < p < 1,” Mat. Sb., 185, 145–160 (1994).
R. K. S. Rathore, “The problem of A. F. Timan on the precise order of decrease of the best approximations,” J. Approxim. Theory, 77, 153–166 (1994).
V. K. Dzyadyk, “On the extension of functions that satisfy the Lipschitz condition in the metric of L p ,” Mat. Sb., 40, 239–242 (1956).
O. V. Besov, “Extension of functions with preservation of differential-difference properties in L p ,” Dokl. Akad. Nauk SSSR, 150, No. 5, 963–996 (1963).
R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer, New York (1993).
R. M. Trigub and E. S. Belinsky, Fourier Analysis and Approximation of Functions, Kluwer, Dordrecht (2004).
A. Zygmund, Trigonometric Series, [Russian translation], Vol. 1, Mir, Moscow (1965).
R. E. Edwards, Fourier Series. A Modern Introduction, Vol. 2, Springer, New York (1982).
V. I. Ivanov, “Direct and inverse theorems of approximation theory in the metric of L p for 0 < p < 1,” Mat. Zametki, 18, 641–658 (1975).
P. Oswald, “Approximation by splines in the metric of L p , 0 < p < 1,” Math. Nachr., 94, 69–96 (1980).
Yu. A. Brudnyi and V. K. Shalashov, Theory of Splines. A Tutorial [in Russian], Yaroslavl (1983).
É. A. Storozhenko, “Approximation of functions of the class L p , 0 < p < 1, by algebraic polynomials,” Izv. Akad. Nauk SSSR, Ser. Mat., 41, 652–662 (1977).
I. P. Irodova, “Properties of functions defined by the rate of decrease of piecewise-polynomial approximation,” in: Investigations on the Theory of Functions of Many Real Variables [in Russian], Yaroslavl (1980), pp. 92–117.
Yu. S. Kolomoitsev, “Description of a class of functions with condition \(\omega _r (f,h)_p \leqslant Mh^{r - 1 + \frac{1}{p}} \) for 0 < p < 1,” Vestn. Dnepropetr. Nats. Univ., Ser. Mat., Issue 8, 31–44 (2003).
A. F. Timan, Approximation Theory of Functions of Real Variables [in Russian], Fizmatgiz, Moscow (1960).
R. M. Trigub, “Multipliers in the Hardy space H p (D m) for p ∈ (0juvy 1] and approximation properties of summation methods for power series,” Mat. Sb., 188, No. 4, 145–160 (1997).
V. I. Ivanov and V. A. Yudin, “On a trigonometric system in L p , 0 < p < 1,” Mat. Zametki, 28, No. 6, 859–868 (1980).
E. M. Stein and G. Weiss, Introduction to Fourier Analysis of Euclidean Spaces [Russian translation], Mir, Moscow (1974).
Author information
Authors and Affiliations
Additional information
__________
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 9, pp. 1221–1238, September, 2007.
Rights and permissions
About this article
Cite this article
Kolomoitsev, Y.S. On moduli of smoothness and Fourier multipliers in L p , 0 < p< 1. Ukr Math J 59, 1364–1384 (2007). https://doi.org/10.1007/s11253-007-0092-2
Received:
Issue Date:
DOI: https://doi.org/10.1007/s11253-007-0092-2