Abstract
In this work we will establish a sufficient condition under which the higher derivatives of 2Π-periodic absolutely continuous functions belong to the Orlicz classesϕ (L); ifϕ(2t)=O(ϕ(t)) (t → ∞), the condition is also necessary.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
F. Riesz, “Untersuchungen über Systeme integrierbaren Funktionen,” Math. Ann.,69, 449–497 (1910); see also R. Riesz, Gesammelte Arbeiten,1, 441–489 (1960).
Yu. T. Medvedev, “Generalization of a theorem of F. Riesz,” Usp. Matem. Nauk,8, No. 6, 115–118 (1953).
I. J. Schoenberg, “Spline interpolation and the higher derivatives,” Proc, Nat. Acad. Sci.,51, 24–28 (1964).
J. W. Jerome and L. L. Schumaker, “Characterizations of functions with higher-order derivatives in Lp,” Trans. Amer. Math. Soc.,143, 363–371 (1969).
H. W. Milnes, “Convexity of Orlicz spaces,” Pacific J. Math.,7, No. 3, 1451–1483 (1957).
A. O. Gel'fond, Calculus of Finite Differences [Russian translation], Moscow (1967).
A. Zygmund, Trigonometrical Series, Dover (1935).
Author information
Authors and Affiliations
Additional information
Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 479–486, October, 1973.
The author wishes to thank P. L. Ul'yanov for posing the problems in this paper and for helping to prepare the paper for publication.
Rights and permissions
About this article
Cite this article
Temirgaliev, N. Conditions under which higher derivatives belong to the classesϕ (L). Mathematical Notes of the Academy of Sciences of the USSR 14, 832–836 (1973). https://doi.org/10.1007/BF01108807
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01108807