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

 

An approximation of integrable functions by step functions with an application


Authors: M. G. Crandall and A. Pazy
Journal: Proc. Amer. Math. Soc. 76 (1979), 74-80
MSC: Primary 41A30
MathSciNet review: 534393
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Abstract: Let $ f \in {L^1}(0,\infty ),\delta > 0$ and $ ({G_\delta }f)(t) = {\delta ^{ - 1}}\smallint _t^\infty {e^{(t - s)/\delta }}f(s)ds$. Given a partition $ P = \{ 0 = {t_0} < {t_1} < \cdots < {t_i} < {t_{i + 1}} < \cdots \} $ of $ [0,\infty )$ where $ {t_i} \to \infty $, we approximate f by the step function $ {A_P}f$ defined by

$\displaystyle {A_P}f(t) = ({G_{{\delta _i}}}{G_{{\delta _{i - 1}}}} \cdots {G_{{\delta _i}}}f)(0)\quad {\text{for}}\;{t_{i - 1}} \leqslant t < {t_i},$

where $ {\delta _i} = {t_i} - {t_{i - 1}}$. The main results concern several properties of this process, with the most important one being that $ {A_P}f \to f$ in $ {L^1}(0,\infty )$ as $ \mu (P) = {\sup _i}{\delta _i} \to 0$. An application to difference approximations of evolution problems is sketched.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1979-0534393-0
PII: S 0002-9939(1979)0534393-0
Keywords: Step functions, approximation theory, accretive operators
Article copyright: © Copyright 1979 American Mathematical Society