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Invariant Smooth Extensions

Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

* Corresponding Author
Annals of Communications in Mathematics 2024
, 7 (2),
Received: 12 April 2024 |
Accepted: 10 Jun 2024 |
Published: 30 Jun 2024


We 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.


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Invariant Smooth Extensions.

Annals of Communications in Mathematics,

7 (2):
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  • Copyright (c) 2023 by the Author(s). Licensee Techno Sky Publications. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (


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