Abstract/Overview
A bounded operator with the spectrum lying in a compact set V ⊂ R, has C∞ (V) functional calculus. On the other hand, an operator H acting on a Hilbert space H, admits a C(R) functional calculus if H is self-adjoint. So in a Banach space setting, we really desire a large enough intermediate topological algebra A, with C∞ 0 (R) ⊂ A ⊆ C(R) such that spectral operators or some sort of their restrictions, admit a A functional calculus. In this paper we construct such an algebra of smooth functions on R that decay like (√ 1 + x2) β as |x| → ∞, for some β < 0. Among other things, we prove that C∞ c (R) is dense in this algebra. We demonstrate that important functions like x 7→ e x are either in the algebra or can be extended to functions in the algebra. We
O.</div>, < (2024). The algebra of smooth functions of rapid descent. Afribary. Retrieved from https://afribary.com/works/the-algebra-of-smooth-functions-of-rapid-descent
O.</div>, <div>Oleche "The algebra of smooth functions of rapid descent" Afribary. Afribary, 04 Jun. 2024, https://afribary.com/works/the-algebra-of-smooth-functions-of-rapid-descent. Accessed 16 Sep. 2024.
O.</div>, <div>Oleche . "The algebra of smooth functions of rapid descent". Afribary, Afribary, 04 Jun. 2024. Web. 16 Sep. 2024. < https://afribary.com/works/the-algebra-of-smooth-functions-of-rapid-descent >.
O.</div>, <div>Oleche . "The algebra of smooth functions of rapid descent" Afribary (2024). Accessed September 16, 2024. https://afribary.com/works/the-algebra-of-smooth-functions-of-rapid-descent