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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 $beta$ and the convergence index $gamma$. It is shown that these two indices are fully compatible in the following sense : a Baire-1 function $f$ satisfies $beta(f) leq omega^{xi_1} cdot omega^{xi_2}$ for some countable ordinals $xi_1$ and $xi_2$ if and only if there exists a sequence of Baire-1 functions $(f_n)$ converging to $f$ pointwise such that $sup_nbeta(f_n) leq omega^{xi_1}$ and $gamma((f_n)) leq omega^{xi_2}$. We also obtain an extension result for Baire-1 functions analogous to the Tietze Extension Theorem. Finally, it is shown that if $beta(f) leq omega^{xi_1}$ and $beta(g) leq omega^{xi_2},$ then $beta(fg) leq omega^{xi},$ where $xi=max{xi_1+xi_2, xi_2+xi_1}}.$ These results do not assume the boundedness of the functions involved.
We provide a proof that analytic almost disjoint families of infinite sets of integers cannot be maximal using a result of Bourgain about compact sets of Baire class one functions. Inspired by this and related ideas, we then provide a new proof of th
The theory of ordinal ranks on Baire class 1 functions developed by Kechris and Loveau was recently extended by Elekes, Kiss and Vidny{a}nszky to Baire class $xi$ functions for any countable ordinal $xigeq1$. In this paper, we answer two of the quest
A classical theorem of Kuratowski says that every Baire one function on a G_delta subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire
We study the inequalities of the type $|int_{mathbb{R}^d} Phi(K*f)| lesssim |f|_{L_1(mathbb{R}^d)}^p$, where the kernel $K$ is homogeneous of order $alpha - d$ and possibly vector-valued, the function $Phi$ is positively $p$-homogeneous, and $p = d/(
In this paper we formulate a concrete method for determining whether a system of dilated periodic functions forms a Riesz basis in $L^2(0,1)$. This method relies on a general framework developed by Hedenmalm, Lindqvist and Seip about 20 years ago, wh