Xiao, Xuemei; Zhu, Yucan Duality principles of frames in Banach spaces. (Chinese. English summary) Zbl 1199.42137 Acta Math. Sci., Ser. A, Chin. Ed. 29, No. 1, 94-102 (2009). Summary: Duality principles in Gabor theory such as the Ron-Shen duality principle and the Wexler-Raz biorthogonality relations play a fundamental role for analyzing Gabor systems. For each sequence in a Banach space \(X\), we define a corresponding sequence dependent only on two \(p\)-Riesz bases in the Banach space \(X\). Then we characterize exactly properties of the first sequence in terms of the associated one. We generate some results that were obtained by P. G. Casazza, G. Kutyniok and M. C. Lammers [J. Fourier Anal. Appl. 10, No. 4, 383–408 (2004; Zbl 1058.42020)] about duality principles of frames in a separable Hilbert space \(H\). Cited in 1 ReviewCited in 11 Documents MSC: 42C15 General harmonic expansions, frames 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems Keywords:duality principles; \(p\)-Riesz basis; \(q\)-frame Citations:Zbl 1058.42020 PDFBibTeX XMLCite \textit{X. Xiao} and \textit{Y. Zhu}, Acta Math. Sci., Ser. A, Chin. Ed. 29, No. 1, 94--102 (2009; Zbl 1199.42137)