Moduli of continuity in 𝑅ⁿ and 𝐷⊂𝑅ⁿ

Ditzian, Z.

doi:10.1090/S0002-9947-1984-0732110-9

Moduli of continuity in $\textbf {R}^{n}$ and $D\subset \textbf {R}^{n}$
HTML articles powered by AMS MathViewer

by Z. Ditzian

Trans. Amer. Math. Soc. 282 (1984), 611-623

DOI: https://doi.org/10.1090/S0002-9947-1984-0732110-9

PDF | Request permission

Abstract:

The $r$ modulus of continuity for $f \in C({R^n})$ is expressed in terms of $r$ moduli of continuity in $n$ independent directions. Generalizations to other spaces of functions on ${R^n}$ or $D \subset {R^n}$ are also given.

References

Z. Ditzian, Some remarks on approximation theorems on various Banach spaces, J. Math. Anal. Appl. 77 (1980), no. 2, 567–576. MR 593235, DOI 10.1016/0022-247X(80)90248-6
V. P. Il′in, Conditions of validity of inequalities between $L_{p}$-norms of partial derivatives of functions of several variables, Proc. Steklov Inst. Math., No. 96 (1968): Automat. Programming, Numer. Methods and Functional Anal., Amer. Math. Soc., Providence, R.I., 1968, pp. 259–305. Trudy Mat. Inst. Steklov. 96 (1968), 205–242 (Russian); translated in. MR 0271724
H. Johnen and K. Scherer, On the equivalence of the $K$-functional and moduli of continuity and some applications, Constructive theory of functions of several variables (Proc. Conf., Math. Res. Inst., Oberwolfach, 1976) Lecture Notes in Math., Vol. 571, Springer, Berlin, 1977, pp. 119–140. MR 0487423
Robert C. Sharpley, Cone conditions and the modulus of continuity, Second Edmonton conference on approximation theory (Edmonton, Alta., 1982) CMS Conf. Proc., vol. 3, Amer. Math. Soc., Providence, RI, 1983, pp. 341–351. MR 729339
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
A. F. Timan, Theory of approximation of functions of a real variable, A Pergamon Press Book, The Macmillan Company, New York, 1963. Translated from the Russian by J. Berry; English translation edited and editorial preface by J. Cossar. MR 0192238