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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Pointwise convergence approximate identities of dilated radially decreasing kernels


Author: R. A. Kerman
Journal: Proc. Amer. Math. Soc. 101 (1987), 41-44
MSC: Primary 42B99; Secondary 44A35
MathSciNet review: 897067
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Abstract: Let $ \phi $ be integrable on $ {R^n}$ with $ \int_{{R^n}} {\phi (y)dy = 1} $. It is shown that

$\displaystyle {\lim _{\varepsilon \to 0 + }}(\phi _F^*f)(x) = {\lim _{\varepsil... ...varepsilon ^{ - n}}\int_{{R^n}} {(\frac{{x - y}} {\varepsilon })f(y)dy = f(x)} $

a.e. on $ {R^n}$, whenever the least radially decreasing majorant of $ \phi $, defined by $ \psi (x) = {\sup _{\vert y\vert \geqslant \vert x\vert}}\vert\phi (y)\vert$, is such that $ \vert x{\vert^n}\psi (x) = \vert x{\vert^n}\psi (\vert x\vert)$ is nonincreasing in $ \vert x\vert$ when $ \vert x\vert$ is large and $ (\psi _{{\varepsilon _0}}^*\vert f\vert)({x_0}) < \infty $ for some $ {x_0} \in {R^n}$ and $ {\varepsilon _0} > 0$.

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

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1987-0897067-7
PII: S 0002-9939(1987)0897067-7
Article copyright: © Copyright 1987 American Mathematical Society