Systems of dilated functions: Completeness, minimality, basisness

The completeness, minimality, and basis property in L (2)[0, pi] and L (p)[0, pi], p not equal 2, are considered for systems of dilated functions u (n) (x) = S(nx), n a N, where S is the trigonometric polynomial S(x) = I pound (k=0) (m) a (k) sin(kx), a (0) a (m) not equal 0. A series of results are presented and several unanswered questions are mentioned.