Invariance of shift-invariant spaces (Q1934433)

A characterization of shift-invariant subspaces which are also invariant under additional noninteger translations is given. A shift-invariant subspace of \(L^p(\mathbb R)\) is, by definition, a closed subspace which is invariant under translations by an integer. The authors give necessary and sufficient conditions for the noninteger translation (shift) invariance. Shift invariant subspaces of \(L^p(\mathbb R)\) were characterized, in particular, by Jia and Michelli in 1991 in a theorem, where they showed that the absolute value of the Fourier transform of the translated (by integer) instances of the scaling function ranges within some finite nonvanishing lower-upper bounds. For noninteger translations, in this paper and for the first time, their result on integer translations is generalized by the authors to noninteger translations. The authors show that there exist some shift-invariant subspaces for some \(1/n\)-translations for each integer \(n\geq 2\). This invariance under some noninteger translations could be an interesting useful tool for approximation theory, wavelet theory and many other applications. In particular, this noninteger shift invariance could open some new perspectives in sampling theory, and compression, by sampling at rationals.