JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2019, Vol. 54 ›› Issue (8): 76-80.doi: 10.6040/j.issn.1671-9352.0.2018.724

Previous Articles     Next Articles

On decompositions of continuous generalized frames in Hilbert spaces

ZHANG Wei1, FU Yan-ling2   

  1. 1. School of Mathematics and Information Sciences, Henan University of Economics and Law, Zhengzhou 450046, Henan, China;
    2. Department of Information Engineering, Henan Finance University, Zhengzhou 451464, Henan, China
  • Online:2019-08-20 Published:2019-07-03

PDF (PC)

547

Abstract: This paper establishes the characterization of continuous generalized frames,Parseval continuous g-frames,continuous generalized Riesz bases and continuous generalized orthonormal bases in term of the continuous generalized preframe operator. Using the established characterization results and decompositions of bounded operators, the representation of continuous generalized frames in term of linear combinations of simpler ones such as continuous generalized orthonormal bases, continuous generalized Riesz bases and Parseval continuous generalized frames is studied.

Key words: generalized frame, continuous generalized frame, continuous generalized orthonormal base, operator

CLC Number: 

  • O174.2
[1] DUFFIN R J, SCHAEFFER A C. A class of nonharmonic Fourier series[J]. Transactions of the American Mathematical Society, 1952, 72(2):341-366.
[2] DAUBECHIES I, GROSSMANN A, MEYER Y. Painless nonorthogonal expansions[J]. Journal of Mathematical Physics, 1986, 27(5):1271-1283.
[3] CHRISTENSEN O. An introduction to frames and Riesz bases [M]. Boston: Birkhäuser, 2016.
[4] ABDOLLAHPOUR M R, KHEDMATI Y. On some properties of continuous g-frames and Riesz-type continuous g-frames[J]. Indian Journal of Pure & Applied Mathematics, 2016, 48(1):1-16.
[5] 肖祥春,曾晓明. 连续g-框架的刻画[J]. 数学学报, 2012, 55(6):1131-1144. XIAO Xiangchun, ZENG Xiaoming. Characterization of continuous g-frames[J]. Acta Mathematica Sinica, 2012, 55(6):1131-1144.
[6] ALI S T, ANTOINE J P, GAZEAU J P. Continuous frames in Hilbert spaces[J]. Annals of Physics, 1993, 222:1-37.
[7] SUN Wenchang. G-frames and g-Riesz bases[J]. Journal of Mathematical Analysis and Applications, 2006, 322(1):437-452.
[8] ABDOLLAHPOUR M R, FAROUGHI M H. Continuous g-frames in Hilbert spaces[J]. Southeast Asian Bulletin Mathematics, 2008, 32(1):1-19.
[9] CASAZZA P G. Every frame is a sum of three(but not two)orthonormal bases—and other frame representations[J]. Journal of Fourier Analysis and Applications, 1998, 4(6):727-732.
[10] CONWAY J. A course in functional analysis[M]. New York: Springer, 1990.
[11] MADADIAN M, RAHMANI M. G-frame sequence operators, cg-Riesz bases and sum of cg-frames[J]. International Mathematical Forum, 2011,12(68):3357-3369.
[1] DAI Lei, HUANG Xiao-jing, GUO Qi. Property(ω1)and the single-valued extension property [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(8): 55-61.
[2] TAO Shuang-ping, YANG Yu-he. Weighted estimates of fractional maximal operator and its commutator on weighted λ-central Morrey spaces [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(8): 68-75.
[3] ZHANG Qian, LI Xuan, LI Xin, ZHENG Hui-hui, LI Lin-han, ZHANG Liang-yun. The construct of Rota-Baxter algebra on the Sweedler 4-dimensional Hopf algebra [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 47-52.
[4] WANG Tao. Study on the approximation properties of Gauss-Weierstrass operator [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(6): 96-98.
[5] FANG Xiao-zhen, SUN Ai-wen, WANG Min, SHU Li-sheng. Boundedness of generalized multilinear operators on variable exponent spaces [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(4): 6-16.
[6] ZHANG Shen-gui. Applications of variational method to impulsive differential systems with variable exponent [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(4): 22-28.
[7] HAN Qi, YIN Shi-de, CHEN Zhi-he. Related properties of rotation operator in quantum computing [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2019, 54(2): 121-126.
[8] TAO Shuang-ping, GAO Rong. Estimates of multilinear fractional integrals and maximal operators on weighted Morrey spaces [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 30-37.
[9] XIN Yin-ping, TAO Shuang-ping. Boundedness of Marcinkiewicz integrals operators with variable kernels on Herz-type Hardy spaces with variable exponent [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 38-43.
[10] . Representative forms of commutative BR0-algebras on a set by implication operator [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(6): 86-94.
[11] LI Chun-fang, MENG Bin. Some properties of measure frames [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(12): 99-104.
[12] PENG Jia-yin. Bidirectional controlled teleportation with a genuine five-qubit non-maximally entangled state as quantum channel [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(12): 105-113.
[13] ZHAO Huan, ZHOU Jiang. Commutators of multilinear Calderón-Zygmund operator on Herz-type Hardy spaces with variable exponent [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2018, 53(10): 42-50.
[14] LU Qiang-de, TAO Shuang-ping. Boundedness of commutators of Calderón-Zygmund operators and fractional integrals in homogeneous grand variable exponent Lebesgue spaces [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(9): 54-58.
[15] ZHANG Qian, LI Hai-yang. The iterative fraction thresholding algorithm in sparse information processing [J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2017, 52(9): 76-82.
Viewed
Full text


Abstract

Cited
  Shared   
  Discussed   
[1] TANG Feng-qin1, BAI Jian-ming2. The precise large deviations for a risk model with extended negatively upper orthant dependent claim  sizes[J]. J4, 2013, 48(1): 100 -106 .
[2] QIU Tao-rong, WANG Lu, XIONG Shu-jie, BAI Xiao-ming. A granular computing approach for knowledge hiding[J]. J4, 2010, 45(7): 60 -64 .
[3] XUE Qiu-fang1,2, GAO Xing-bao1*, LIU Xiao-guang1. Several equivalent conditions for H-matrix based on the extrapolated GaussSeidel iterative method[J]. J4, 2013, 48(4): 65 -71 .
[4] SHI Ai-ling1, MA Ming2*, ZHENG Ying2. Customer lifetime value and property with #br# homogeneous Poisson response[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 96 -100 .
[5] TIAN Xue-gang, WANG Shao-ying. Solutions to the operator equation AXB=C[J]. J4, 2010, 45(6): 74 -80 .
[6] PANG Guan-song, ZHANG Li-sha, JIANG Sheng-yi*, KUANG Li-min, WU Mei-ling. A multi-level clustering approach based on noun phrases for search results[J]. J4, 2010, 45(7): 39 -44 .
[7] ZHAO Jun1, ZHAO Jing2, FAN Ting-jun1*, YUAN Wen-peng1,3, ZHANG Zheng1, CONG Ri-shan1. Purification and anti-tumor activity examination of water-soluble asterosaponin from Asterias rollestoni Bell[J]. J4, 2013, 48(1): 30 -35 .
[8] YANG Yong-wei1, 2, HE Peng-fei2, LI Yi-jun2,3. On strict filters of BL-algebras#br#[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 63 -67 .
[9] LI Min1,2, LI Qi-qiang1. Observer-based sliding mode control of uncertain singular time-delay systems#br#[J]. JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE), 2014, 49(03): 37 -42 .
[10] WANG Qi,ZHAO Xiu-heng,LI Guo-jun . Embedding hypergraph in trees of rings[J]. J4, 2007, 42(10): 114 -117 .
Copyright 2018 © Editorial office of JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)
Supported by Beijing Magtech Co.Ltd