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Ricardo Estrada
Open AccessArticle

Invariant Smooth Extensions

Annals of Communications in Mathematics 2024

, 7 (2)

, 91-94

DOI: https://doi.org/10.62072/acm.2024.070201

AbstractWe study continuous extension operators for smooth functions from [0, ∞) to R. If A is a topological vector space of smooth functions in R, let us denote by A[0, ∞) the space of restrictions of functions of A to [0, ∞). We show that when A is any of the standard test function spaces D, S, or K then there is a continuous linear operator E from A[0, ∞) to A that satisfies that E (φ) (t) = φ (t) for t ≥ 0 and that satisfies the invariant condition E {ϕ (λx) ;t} = E {ϕ (x) ; λt} , for λ ≥ 0 . However, we show that when A is E, the space of all smooth functions, then such an operator E does not exist.
Open AccessArticle

On Iyengar’s and Ostrowski’s Integral Means

Annals of Communications in Mathematics 2024

, 7 (4)

, 393-400

DOI: https://doi.org/10.62072/acm.2024.070407

AbstractFrullani’s Integral Formula is an old formula that was known to hold under strict conditions. Iyengar, and later Ostrowski, provided necessary and sufficient conditions for the existence of the Frullani Integral Formula. Their conditions were different but equivalent. In this article, we identify other conditions that are equivalent. We show that these conditions are, in fact, solutions to a family of linear differential equations of the first order. We study the limiting behavior of these solutions at zero and infinity, and in doing so, arrive at a new proof of the equivalence of Iyengar’s and Ostrowski’s conditions. Lastly, we provide applications of our results.