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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Moduli of continuity for exponential Lipschitz classes


Author: Paul De Land
Journal: Trans. Amer. Math. Soc. 229 (1977), 175-189
MSC: Primary 26A15; Secondary 42A36
MathSciNet review: 0442157
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Abstract: Let $ \Psi $ be a convex function, and let f be a real-valued function on [0, 1]. Let a modulus of continuity associated to $ \Psi $ be given as

$\displaystyle {Q_\Psi }(\delta ,f) = \inf \left\{ {\lambda :\frac{1}{\delta }\i... ... f(x) - f(y)\vert}}{\lambda }} \right)}\;dx\;dy\; \leqslant \Psi (1)} \right\}.$

It is shown that $ \smallint _0^1{Q_\Psi }(\delta ,f)\;d\;( - {\Psi ^{ - 1}}(c/\delta )) < \infty $ guarantees the essential continuity of f, and, in fact, a uniform Lipschitz estimate is given. In the case that $ \Psi (u) = \exp \;{u^2}$ the above condition reduces to

$\displaystyle \int_0^1 {{Q_{\exp \;{u^2}}}\;(\delta ,f)\frac{{d\delta }}{{\delta \sqrt {\log (c/\delta )} }}\; < \infty .} $

This exponential square condition is satisfied almost surely by the random Fourier series $ {f_t}(x) = \Sigma _{n = 1}^\infty {a_n}{R_n}(t){e^{inx}}$, where $ \{ {R_n}\} $ is the Rademacher system, as long as

$\displaystyle \int_0^1 {\sqrt {a_n^2{{\sin }^2}(n\delta /2)} \frac{{d\delta }}{{\delta \sqrt {\log (1/\delta )} }}\; < \infty .} $

Hence, the random essential continuity of $ {f_t}(x)$ is guaranteed by each of the above conditions.

References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0442157-4
PII: S 0002-9947(1977)0442157-4
Article copyright: © Copyright 1977 American Mathematical Society