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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A representation theorem and approximation operators arising from inequalities involving differential operators


Author: D. Leviatan
Journal: Trans. Amer. Math. Soc. 168 (1972), 85-99
MSC: Primary 41A35
DOI: https://doi.org/10.1090/S0002-9947-1972-0296573-X
MathSciNet review: 0296573
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Abstract: A representation of functions as integrals of a kernel $ \psi (t;x)$, which was introduced by Studden, with respect to functions of bounded variation in $ [0,\infty )$ is obtained whenever the functions satisfy some conditions involving the differential operators $ (d/dt)\{ f(t)/{w_i}(t)\} ,i = 0,1,2, \ldots $. The results are related to the concepts of generalized completely monotonic functions and generalized absolutely monotonic functions in $ (0,\infty )$. Some approximation operators for the approximation of continuous functions in $ [0,\infty )$ arise naturally and are introduced; some sequence-to-function summability methods are also introduced.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1972-0296573-X
Keywords: Generalized completely monotonic functions, representation of functions as integrals of a kernel, approximation operators in $ [0,\infty )$, sequence-to-function summability methods
Article copyright: © Copyright 1972 American Mathematical Society