Abstract
In this paper, we firstly study the properties of g-Bessel sequences in Hilbert spaces. Some more subtle properties are established. Then we study the properties of g-Riesz bases and g-orthonormal bases in Hilbert spaces. Especially we establish certain uniqueness on the dual g-frames of them. By using the disjoint g-frames as tools, we show that g-frame has a unique dual g-frame if and only if it is a g-Riesz basis. We also show that a normalized tight g-frame has a unique normalized tight dual g-frame, i.e., itself.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abdollahi A., Rahimi E.: Generalized frames on supper Hilbert spaces. Bull. Malays. Math. Sci. Soc. (2) 35(3), 807–818 (2012)
Arias M., Corach G., Pacheco M.: Characterization of Bessel sequences. Extr. Math. 22(1), 55–66 (2007)
Chui C.K.: An Introduction to Wavelets. Acadamic Press, New York (1992)
Chui C., Shi X.: Bessel Sequences and Affine Frames. Appl. Comput. Harmonic Anal. 1(1), 29C49 (1993)
Daubechies, I.: Ten Lectures on Wavelets. In: CBMS 61. SIAM, Philadelphia (1992)
Ding M., Zhu Y.: g-Besselian frames in hilbert spaces. Acta Math. 26, 2117–2130 (2010)
Duffin R.J., Shaffer A.C.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Guo, X.: g-Bases in Hilbert spaces, Abstr. Appl. Anal. 2012(article ID 923729), 1–14. doi:10.1155/2012/923729
Guo, X.: Operator characterizations and some properties of g-frames on Hilbert spaces. J. Funct. Spaces Appl. 2013(article ID 931367),1–9. doi:10.1155/2013/931367
Han, D., Larson, D.: Bases, frames and group representations, Memoirs Am. Math. Soc. 147(697) (2000)
Hernandez E., Weiss G.: A First Course on Wavelets. CRC Press, Boca Raton (1996)
Khosravi A., Musazadeh K.: Fusion frames and g-frames. J. Math. Anal. Appl. 342, 1068–1083 (2008)
Li J., Zhu Y.: Some equalities and inequalities for g-Bessel sequences in Hilbert spaces. Appl. Math. Lett. 25(11), 1601–1607 (2012)
Mallat S.: Multiresolution approximations and wavelet orthonormal basis of L2(R). Trans. Am. Math. Soc. 315, 69–87 (1989)
Najati A., Faroughi M., Rahimi A.: g-Frames and stability of g-frames in Hilbert spaces. Methods Funct. Anal. Topol. 14(3), 271–286 (2008)
Sun W.: g-Frames and g-Riesz bases. J. Math. Anal. Appl. 322, 437–452 (2006)
Sz-Nagy B.: Expansion theorems of Paley–Wiener type. Duke Math. J. 14, 975–978 (1947)
Young R.M.: An Introduction to Nonharmonic Fourier Series. Academic Press, New York (1980)
Zhu Y.: Characterizations of g-frames and g-Riesz bases in Hilbert spaces. Acta Math. Sin. 24, 1727–1736 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Guo, X. New Characterizations of g-Bessel Sequences and g-Riesz Bases in Hilbert Spaces. Results. Math. 68, 361–374 (2015). https://doi.org/10.1007/s00025-015-0444-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00025-015-0444-4