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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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On bounded functions satisfying averaging conditions. II
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by Rotraut Goubau Cahill PDF
Trans. Amer. Math. Soc. 223 (1976), 295-304 Request permission

Abstract:

Let $S(f)$ denote the subspace of ${L^\infty }({R^n})$ consisting of those real valued functions f for which \[ \lim \limits _{r \to 0} \frac {1}{{|B(x,r)|}} {\int } _{B(x,r)}f(y)dy = f(x)\] for all x in ${R^n}$ and let $L(f)$ be the subspace of $S(f)$ consisting of the approximately continuous functions. A number of results concerning the existence of functions in $S(f)$ and $L(f)$ with special properties are obtained. The extreme points of the unit balls of both spaces are characterized and it is shown that $L(f)$ is not a dual space. As a preliminary step, it is shown that if E is a ${G_\delta }$ set of measure 0 in ${R^n}$, then the complement of E can be decomposed into a collection of closed sets in a particularly useful way.
References
  • Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • Zygmunt Zahorski, Über die Menge der Punkte in welchen die Ableitung unendlich ist, Tôhoku Math. J. 48 (1941), 321–330 (German). MR 27825
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 223 (1976), 295-304
  • MSC: Primary 26A69; Secondary 31B05
  • DOI: https://doi.org/10.1090/S0002-9947-1976-0422539-6
  • MathSciNet review: 0422539