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

Leviatan, D.

doi:10.1090/S0002-9947-1972-0296573-X

A representation theorem and approximation operators arising from inequalities involving differential operators
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by D. Leviatan

Trans. Amer. Math. Soc. 168 (1972), 85-99

DOI: https://doi.org/10.1090/S0002-9947-1972-0296573-X

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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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