Theorems of Paley–Wiener and Müntz–Szász type

In the paper a new integral representation of the well-known function classes $\mathscr H_2[\alpha]$ ($0<\alpha<1$) is established, which in the limiting case $\alpha=1$ goes into the Paley–Wiener theorem. A Hilbert space metric is introduced into the classes $\mathscr H_2[\alpha]$ ($0<\alpha<+\infty$), and a criterion for closedness of certain systems of functions in these spaces is established. In particular, a theorem of Müntz–Szász type in the complex domain is proved, and a complete intrinsic description is given of the corresponding nonclosed systems.
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