The paper is concerned with integrability of the Fourier sine transform function when $f\in {\mathrm{BV}}_0\left(ℝ\right)$, where ${\mathrm{BV}}_0\left(ℝ\right)$ is the space of bounded variation functions vanishing at infinity. It is shown that for the Fourier sine transform function of $f$ to be integrable in the Henstock-Kurzweil sense, it is necessary that $f/x\in L^1\left(ℝ\right)$. We prove that this condition is optimal through the theoretical scope of the Henstock-Kurzweil integration theory.

For a fusion Banach frame $(\{G_n,v_n\},S)$ for a Banach space $E$, if $(\{v_n^{*}\left(E^{*}\right),v_n^{*}\},T)$ is a fusion Banach frame for $E^{*}$, then $(\{G_n,v_n\},S;\{v_n^{*}\left(E^{*}\right),v_n^{*}\},T)$ is called a fusion bi-Banach frame for $E$. It is proved that if $E$ has an atomic decomposition, then $E$ also has a fusion bi-Banach frame. Also, a sufficient condition for the existence of a fusion bi-Banach frame is given. Finally, a characterization of fusion bi-Banach frames is given.

We consider the subspace of L²(ℝ) spanned by the integer shifts of one function ψ, and formulate a condition on the family ${\psi (·-n)}_{n=-\infty }^{\infty }$, which is equivalent to the weight function ${\sum }_{n=-\infty }^{\infty }\left|\psi ̂(·+n)\right|²$ being > 0 a.e.

Let $f\left(x\right)\sim \Sigma a_n{e}^{2\pi inx},f*\left(x\right)\sim {\sum }_{n=0}^{\infty }a{*}_n\cos 2\pi nx$, where the $a{*}_n$ are the numbers $\left|a_n\right|$ rearranged so that $a_n^{*}\searrow 0$. Then for any convex increasing $\psi$, $\parallel \psi \left(\right|f{|}^2{\parallel }_1\le \parallel \psi \left(20\right|f*{|}^2{\parallel }_1$. The special case $\psi \left(t\right)=t^{q/2}$, $q\ge 2$, gives $\parallel f{\parallel }_q\le 5\parallel f*{\parallel }_q$ an equivalent of Littlewood.