American Institute of Mathematical Sciences

doi: 10.3934/mfc.2022005
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Behavior in $L^\infty$ of convolution transforms with dilated kernels

 Department of Mathematics, University of Connecticut, Storrs, CT 06269-1009, USA

Received  June 2021 Revised  January 2022 Early access February 2022

Assuming that $K(x)$ is in $L^1( {\mathbb R})$, $K_t(x) = t^{-1} K(x/t)$, and $f(x)$ is in $L^\infty( {\mathbb R})$, we study the behavior of the convolution $K_t*f(x)$ as the parameter $t$ tends to $\infty$. It turns out that the limit need not exist and, if it does exist, the limit is a constant independent of $x$. Situations where the limit exists and those where it fails to exist are identified. Several issues related to this are addressed, including the multivariate case. As one application, these results provide an accessible description of the behavior of bounded solutions to the initial value problem for the heat equation.

Citation: W. R. Madych. Behavior in $L^\infty$ of convolution transforms with dilated kernels. Mathematical Foundations of Computing, doi: 10.3934/mfc.2022005
References:
 [1] M. de Guzmán, Real Variable Methods in Fourier Analysis, North-Holland Mathematics Studies, 46. Notas de Matemtica [Mathematical Notes], 75. North-Holland Publishing Co., Amsterdam-New York, 1981. [2] F. John, Partial Differential Equations, Fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4684-9333-7. [3] B. F. Logan, Limits in $L^p$ of convolution transforms with kernels $aK(at)$, $a \to 0$, SIAM J. Math. Anal., 10 (1979), 733-740.  doi: 10.1137/0510068. [4] W. R. Madych, Limits of dilated convolution transforms, SIAM J. Math. Anal., 16 (1985), 551-558.  doi: 10.1137/0516041. [5] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970. [6] F. Zo, A note on approximation of the identity, Studia Math., 55 (1976), 111-122.  doi: 10.4064/sm-55-2-111-122.

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References:
 [1] M. de Guzmán, Real Variable Methods in Fourier Analysis, North-Holland Mathematics Studies, 46. Notas de Matemtica [Mathematical Notes], 75. North-Holland Publishing Co., Amsterdam-New York, 1981. [2] F. John, Partial Differential Equations, Fourth edition. Applied Mathematical Sciences, 1. Springer-Verlag, New York, 1982. doi: 10.1007/978-1-4684-9333-7. [3] B. F. Logan, Limits in $L^p$ of convolution transforms with kernels $aK(at)$, $a \to 0$, SIAM J. Math. Anal., 10 (1979), 733-740.  doi: 10.1137/0510068. [4] W. R. Madych, Limits of dilated convolution transforms, SIAM J. Math. Anal., 16 (1985), 551-558.  doi: 10.1137/0516041. [5] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N.J. 1970. [6] F. Zo, A note on approximation of the identity, Studia Math., 55 (1976), 111-122.  doi: 10.4064/sm-55-2-111-122.
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