Duality principles in frame theory (Q1882380)

Some of the most important results in Gabor analysis concern duality. The aim of the present paper is to introduce duality in the study of general frames in Hilbert spaces. Given a sequence \(\{f_j\}\) of elements in a Hilbert space \(H\), a dual sequence \(\{w_j^f\}\) is introduced, based on a choice of two orthonormal bases for \(H\). It is proved that several properties of \(\{f_j\}\) can be formulated in terms of \(\{w_j^f\}\); for example, \(\{f_j\}\) is a frame sequence if and only if \(\{w_j^f\}\) is a Riesz sequence, and \(\{f_j\}\) is a tight frame with frame bound one if and only if \(\{w_j^f\}\) is an orthonormal system. Furthermore, two frames \(\{f_j\}\) and \(\{g_j\}\) are dual if and only if \(\{w_j^f\}\) and \(\{w_j^g\}\) are biorthogonal. The results are related to the known duality results for Gabor systems.