To answer a question in [24], we propose \documentclass{aastex}
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\begin{document}
$$\mathcal{U}\mathcal{L}\mathcal{P}(\mathbb{R}^ + ,H)$$
\end{document}, the space of uniform limit power functions and \documentclass{aastex}
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$$\mathcal{L}\mathcal{P}_2$$
\end{document}, the space of limit power functions. We show that \documentclass{aastex}
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\begin{document}
$$\mathcal{U}\mathcal{L}\mathcal{P}(\mathbb{R}^ + ,H)$$
\end{document} and \documentclass{aastex}
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$$\mathcal{L}\mathcal{P}_2$$
\end{document} have properties respectively similar to that of \documentclass{aastex}
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\begin{document}
$$\mathcal{A}\mathcal{P}(\mathbb{R}^ + ,H)$$
\end{document}, the space of almost periodic functions and to that of
B2
, Besicovitch’s space. Finally, we point out that \documentclass{aastex}
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$$\mathcal{L}\mathcal{P}_2$$
\end{document} is the largest among those Hilbert spaces in limit power function set whose members have associated Fourier series (in the sense of a new basis) and satisfy Parseval’s equality.
Uniform limit power-type function spaces