Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
Mobile Device Pairing
Green Open Access
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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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?)

  • [1] A. P. Calderon and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85–139. MR 0052553 (14,637f)
  • [2] G. H. Hardy, Further researches in the theory of divergent series and integrals, Proc. Cambridge Philos. Soc. 21 (1908), 1-48.
  • [3] -, Fourier's double integral and the theory of divergent integrals, Proc. Cambridge Philos. Soc. 21 (1911), 427-451.
  • [4] -, Notes on some points in the integral calculus (XLII): On Weierstrass's singular integral and on a theorem of Lerch, Messenger Math. 46 (1917), 43-48.
  • [5] I. I. Hirschman and D. V. Widder, The convolution transform, Princeton University Press, Princeton, N. J., 1955. MR 0073746 (17,479c)
  • [6] E. C. Titchmarsh, Theory of Fourier integrals, 2nd ed., Oxford Univ. Press, 1962.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42B99, 44A35

Retrieve articles in all journals with MSC: 42B99, 44A35

Additional Information

PII: S 0002-9939(1987)0897067-7
Article copyright: © Copyright 1987 American Mathematical Society