Modular frames for Hilbert C*-modules and symmetric approximation of frames

We give a comprehensive introduction to a general modular frame construction in Hilbert C⁺-modules and to related linear operators on them. The Hilbert space situation appears as a special case. The reported investigations rely on the idea of geometric dilation to standard Hilbert C⁺-modules over unital C*-algebras that admit an orthonormal modular Riesz basis. Interrelations and applications to classical frame theory are indicated. Resorting to frames in Hilbert spaces we discuss some measures for pairs of frames to be close to one another. In particular, the existence and uniqueness of the closest tight frame to a given frame is investigated. For Riesz bases with certain restrictions the set of closest tight frames often contains a multiple of its symmetric orthogonalization.