Functions of Baire class one
| dc.creator | Leung, Denny H. | |
| dc.creator | Tang, Wee-Kee | |
| dc.date | 2000-05-02 | |
| dc.date.accessioned | 2026-07-07T04:34:57Z | |
| dc.date.available | 2026-07-07T04:34:57Z | |
| dc.description | Let $K$ be a compact metric space. A real-valued function on $K$ is said to be of Baire class one (Baire-1) if it is the pointwise limit of a sequence of continuous functions. In this paper, we study two well known ordinal indices of Baire-1 functions, the oscillation index $β$ and the convergence index $γ$. It is shown that these two indices are fully compatible in the following sense : a Baire-1 function $f$ satisfies $β(f) \leq ω^{ξ_1} \cdot ω^{ξ_2}$ for some countable ordinals $ξ_1$ and $ξ_2$ if and only if there exists a sequence of Baire-1 functions $(f_n)$ converging to $f$ pointwise such that $\sup_nβ(f_n) \leq ω^{ξ_1}$ and $γ((f_n)) \leq ω^{ξ_2}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $β(f) \leq ω^{ξ_1}$ and $β(g) \leq ω^{ξ_2},$ then $β(fg) \leq ω^ξ,$ where $ξ=\max\{ξ_1+ξ_2, ξ_2+ξ_1}\}.$ These results do not assume the boundedness of the functions involved. | |
| dc.identifier | https://arxiv.org/abs/math/0005013 | |
| dc.identifier | http://arxiv.org/abs/math/0005013 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/59103 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Functional Analysis | |
| dc.subject | 26A21, 03E15, 54C30 | |
| dc.title | Functions of Baire class one | |
| dc.type | text |