Approximately dual frames of vector-valued nonstationary Gabor frames and reconstruction errors

Nonstationary Gabor (NSG) frames for L2(Double-struck capital R)$$ {L}<^>2\left(\mathrm{\mathbb{R}}\right) $$ allow for flexible sampling and varying window functions and have found applications in adaptive signal analysis. Recently, the first author of this paper generalized the notion of NSG frames for L2(Double-struck capital R)$$ {L}<^>2\left(\mathrm{\mathbb{R}}\right) $$ to the space L2(Double-struck capital R,DOUBLE-STRUCK CAPITAL CK)$$ {L}<^>2\left(\mathrm{\mathbb{R}},{\mathrm{\mathbb{C}}}<^>K\right) $$ of vector-valued signals and investigated the existence of vector-valued NSG (VVNSG) frames. In this paper, we consider the reconstruction of any functions in L2(Double-struck capital R,DOUBLE-STRUCK CAPITAL CK)$$ {L}<^>2\left(\mathrm{\mathbb{R}},{\mathrm{\mathbb{C}}}<^>K\right) $$ from coefficients obtained from VVNSG frames. We provide different approximately dual frames of VVNSG frames and give the corresponding reconstruction error bounds. In particular, for a special class of VVNSG frames that are related to painless VVNSG frames, we show that the approximately dual frames carry the VVNSG structure, which is very important since there exist fast analysis and reconstruction algorithms for VVNSG frames. We also provide a method to construct the special class of VVNSG Bessel sequences. Finally, a constructive example is given to illustrate the theoretical results.