Abstract
Moduli of p-continuity provide a measure of fractional smoothness of functions via p-variation. We prove a sharp estimate of the modulus of p-continuity in terms of the modulus of q-continuity (1<p<q<∞).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
V. A. Andrienko, Necessary conditions for imbedding the function classes \(H_{p}^{\omega}\), Mat. Sb. (N.S.), 78(120), (1969), 280–300; English transl.: Math. USSR Sb., 7 (1969), 273–292.
G. Bourdaud, M. Lanza de Cristoforis and W. Sickel, Superposition operators and functions of bounded p-variation II, Nonlinear Anal., 62 (2005), 483–517.
R. A. DeVore and G. G. Lorentz, Constructive Approximation, Springer-Verlag (Berlin, Heidelberg, 1993).
B. I. Golubov, Criteria for the compactness of sets in spaces of functions of bounded generalized variation, Izv. Akad. Nauk Armjan. SSR Ser. Mat., 3 (1968), 409–416 (in Russian).
G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Z., 28 (1928), 612–634.
V. I. Kolyada, On relations between moduli of continuity in different metrics, Trudy Mat. Inst. Steklov, 181 (1988), 117–136; English transl. in Proc. Steklov Inst. Math. (1989), 127–147.
V. I. Kolyada and M. Lind, On functions of bounded p-variation, J. Math. Anal. Appl., 356 (2009), 582–604.
E. R. Love, A generalization of absolute continuity, J. Lond. Math. Soc., 26 (1951), 1–13.
J. Musielak and W. Orlicz, On generalized variation (I), Studia Math., 18 (1959), 11–41.
K. I. Oskolkov, Approximation properties of integrable functions on sets of full measure, Mat. Sb., 103 (1977), 563–589; English transl. in Math. USSR Sb., 32 (1977), 489–514.
J. Peetre, Espaces d’interpolation et théorème de Soboleff, Ann. Inst. Fourier (Grenoble), 16 (1966), 279–317.
A. P. Terehin, Approximation of functions of bounded p-variation, Izv. Vyssh. Uchebn. Zaved. Mat. (1965), 171–187 (in Russian).
A. P. Terehin, Integral smoothness properties of periodic functions of bounded p-variation, Mat. Zametki, 2 (1967), 289–300 (in Russian).
A. P. Terehin, Functions of bounded p-variation with a given modulus of continuity, Mat. Zametki, 53 (1972), 523–530; English transl. in Math. Notes, 12 (1972), 751–755.
A. F. Timan, Theory of Approximation of Functions of a Real Variable, Dover Publications, Inc. (New York, 1994).
P. L. Ul’yanov, Absolute and uniform convergence of Fourier series, Mat. Sb., 72 (1967), 193–225.
P. L. Ul’yanov, Imbeddings of certain function classes \(H_{p}^{\omega}\), Izv. Akad. Nauk SSSR, Ser. Mat., 32 (1968), 649–686; English transl. in Math. USSR Izv., 2 (1968).
P. L. Ul’yanov, Imbedding theorems and relations between best approximation in different metrics, Mat. Sb., 81(123) (1970), 104–131; English transl. in Math. USSR Sb., 70 (1970).
N. Wiener, The quadratic variation of a function and its Fourier coefficients, J. Math. Phys., 3 (1924), 72–94.
Author information
Authors and Affiliations
Corresponding author
Additional information
Corresponding author.
Rights and permissions
About this article
Cite this article
Kolyada, V.I., Lind, M. On moduli of p-continuity. Acta Math Hung 137, 191–213 (2012). https://doi.org/10.1007/s10474-012-0246-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-012-0246-z