The authors consider constructions of dual pairs of frames in the setting of the Hilbert space of periodic functions \(L^2(0,2\pi)\). The frames constructed are generalized shift-invariant systems, whose generators are given explicitly as trigonometric polynomials which provides an efficient computation procedure of the coefficients in the series expansions.
The main result (Theorem 3.1) is the following general setup for constructions of dual pairs of frames for \(L^2(0,2\pi)\): Let \(\{\psi_k\}_{k \in I}\) be a collection of trigonometric polynomials with real-valued Fourier coefficients, and consider any sequence \(\{L_k\}_{k\in I}\) of positive integers. Assume that \(\{T_{2\pi/L_k}^\ell \psi_k\}_{k \in I,\ell=1,\dotsc,L_k}\) forms a Bessel sequence. Under a partition of unity condition \(\sum_k \sqrt{L_k}\hat{\psi_k}(n)=1\), \(n \in \mathbb{Z}\), and two assumptions on the support sets \(S_k:=\operatorname{supp}\,\hat{\psi_k} \subset \mathbb{Z}\) (bounding the number of overlapping \(S_k\)’s and bounding a growth rate of certain unions of the \(S_k\)’s), the systems \(\{T_{2\pi/L_k}^\ell \psi_k\}_{k \in I,\ell=1,\dotsc,L_k}\) and \(\{T_{2\pi/L_k}^\ell \widetilde{\psi}_k\}_{k \in I,\ell=1,\dotsc,L_k}\) form a pair of dual frames, whenever \(\{T_{2\pi/L_k}^\ell \widetilde{\psi}_k\}_{k \in I,\ell=1,\dotsc,L_k}\) is a Bessel sequence. Here, \(\widetilde{\psi}_k\) is defined as a finite linear combination of the \(\psi_k\); in particular, \(\widetilde{\psi}_k\) is also a trigonometric polynomial for each \(k \in I\). The result is similar in spirit of those in [O. Christensen, Appl. Comput. Harmon. Anal. 20, No. 3, 403–410 (2006; Zbl 1106.42030)], where dual pairs of Gabor frames are constructed for \(L^2(\mathbb{R})\). Note, however, that the technical Bessel condition on the dual frame in the above general setup is not necessary in the setting of, e.g., Gabor and wavelet systems (cf. Corollary 3.1 and 3.2).
Finally, the construction procedure is applied to constructions of periodic frames of various types, including nonstationary wavelet systems, Gabor systems and hybrids of them.