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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.
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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 that there are no maximal almost disjoint families in Solovays model. We then use the ideas behind this proof to provide an extension of a dichotomy result by Rosenthal and by Bourgain, Fremlin and Talagrand to general pointwise bounded functions in Solovays model. We then show that the same conclusions can be drawn about the model obtained when we add a generic selective ultrafilter to the Solovay model.
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 questions raised by them in their paper (Ranks on the Baire class $xi$ functions, Trans. Amer. Math. Soc. 368(2016), 8111-8143). Specifically, we show that for any countable ordinal $xigeq1,$ the ranks $beta_{xi}^{ast}$ and $gamma_{xi}^{ast}$ are essentially equivalent, and that neither of them is essentially multiplicative. Since the rank $beta$ is not essentially multiplicative, we investigate further the behavior of this rank with respect to products. We characterize the functions $f$ so that $beta(fg)leq omega^{xi}$ whenever $beta(g)leqomega^{xi}$ for any countable ordinal $xi.$
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 one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index beta(f). We prove a refinement of Kuratowskis theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that beta_{Y}(f)<omega^{alpha}, alpha < omega_1, then f has an extension F onto X so that beta_X(F)is not more than omega^{alpha}. We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that beta_{X}(F)is not more than 3. An example is constructed to show that this result is optimal.
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 $p = d/(d-alpha)$. Under mild regularity assumptions on $K$ and $Phi$, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions $f$.
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, which turns the basis question into one about the localisation of the zeros and poles of a corresponding analytic multiplier. Our results improve upon various criteria formulated previously, which give sufficient conditions for invertibility of the multiplier in terms of sharp estimates on the Fourier coefficients. Our focus is on the concrete verification of the hypotheses by means of analytical or accurate numerical approximations. We then examine the basis question for profiles in a neighbourhood of a non-basis family generated by periodic jump functions. For one of these profiles, the $p$-sine functions, we determine a threshold for positive answer to the basis question which improves upon those found recently.
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