On orthonormal bases and translates (Q906362)

\textit{T. E. Olson} and \textit{R. A. Zalik} [Lect. Notes Pure Appl. Math. 138, 401--408 (1992; Zbl 0764.42020)] showed that a system of translates of a single function cannot be an orthonormal basis, nor a Riesz one, in the space \(L^2(R)\). Moreover Christensen, Deng and Heil [\textit{O. Christensen} et al., Appl. Comput. Harmon. Anal. 7, No. 3, 292--304 (1999; Zbl 0960.42007)] showed that shifts of finitely many functions never generate even a frame. According to the author, H. Shapiro posed the question of whether there exists an orthonormal basis obtained by translations from a compact set of functions. In this note the author gives a positive answer to this question by proving a slightly stronger result. See also Freeman, Odell, Schlumprecht and Zsák [\textit{D. Freeman} et al., Isr. J. Math. 203, 189--209 (2014; Zbl 1305.42033)].