Woven (Weaving) Frames in Banach Spaces
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Authors: A. RAHIMI, S. BASATI, B. DARABY AND F. A. SHAH
DOI: 10.46793/KgJMat2402.181R
Abstract:
Banach frames are defined by the straightforward generalization of Hilbert space frames. Woven (weaving) frames are the recent generalization of standard frames which appeared in the applications of distributed signal processing. In this paper, we introduce the concepts of woven (weaving) Bessel and frame sequences in Banach spaces and characterize the woven frames in terms of bounded operators. We also give some equivalent conditions for woven Xd-frame in Banach spaces.
Keywords:
frame, woven frame, Banach frame, semi-inner product.
References:
[1] T. Bemrose, P. G. Casazza, K. Gröchenig, M. Lammers and R. Richard, Weaving frames, Oper. Matrices 10(4) (2016), 1093–1116.
[2] P. G. Casazza, The art of frame theory, Taiwanese J. Math. 4(2) (2000), 129–201.
[3] P. G. Casazza, Modern tools for Weyl-Heisenberg (Gabor) frame theory, Advances in Imaging and Electron Physics 115 (2001), 1–127.
[4] P. Casazza, O. Christensen and D. T. Stoeva, Frame expansions in separable Banach spaces, J. Math. Anal. Appl. 307(2) (2005), 710–723.
[5] P. G. Casazza, D. Freeman and R. G. Lynch, Weaving Schauder frames, J. Approx. Theory 211 (2016), 42–60. https://doi.org/10.1016/j.jat.2016.07.001
[6] P. G. Casazza and R. Lynch, Weaving properties of Hilbert space frames, 2015 International Conference on Sampling Theory and Applications (SampTA), 2015, 110–114. https://doi.org/10.1109/SAMPTA.2015.7148861
[7] P. G. Casazza, D. Han and D. Larson, Frames for Banach spaces, Contemp. Math. 247(4) (1999), 149–182.
[8] O. Christensen, An introduction to frames and Riesz bases, Springer, Boston, Basel, Berlin, 2016.
[9] O. Christensen and D. T. Stoeva, p-Frames in separable Banach spaces, Adv. Comput. Math. 18(2–4) (2003), 117–126.
[10] B. Dastourian and M. Janfada, Frames for operators in Banach spaces via semi-inner products, Int. J. Wavelets Multiresolut. Inf. Process. 14(3) (2016), Article ID 1650011. https://doi.org/10.1142/S0219691316500119
[11] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, Journal of Mathematical Physics 27(5) (1986), 1271–1283.
[12] S. S. Dragomir, Semi-Inner Products and Applications, Nova Science Publishers, Hauppauge, 2004.
[13] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Tran. Amer. Math. Soc. 72(2) (1952), 341–366.
[14] G. D. Faulkner, Representation of linear functionals in a Banach space, Rocky Mountain J. Math. 7(4) (1977), 789–792.
[15] H. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, J. Funct. Anal. 86(2) (1989), 307–340.
[16] J. R. Giles, Classes of semi-inner-product spaces, Tran. Amer. Math. Soc. 129(3) (1967), 436–446.
[17] K. Gröchenig, Describing functions: Atomic decompositions versus frames, Monatsh. Math. 112(1) (1991), 1–42.
[18] E. Guariglia, Fractional derivative of the Riemann zeta function, in: C. Cattani, H. M. Srivastava and X.-J. Yan (Eds.), Fractional Dynamics, De Gruyter, 2015, 357–368.
[19] E. Guariglia and S. Silvestrov, Fractional-wavelet analysis of positive definite distributions and wavelets on ????′(ℂ), in: Engineering Mathematics, II Edition, Springer, Switzerland, 2017.
[20] M. I. Ismailov and Y. I. Nasibov, On one generalization of Banach frame, Azerbaijan J. Math. 6(2) (2016), 143-159.
[21] D. Han and D. Larson, Frames, Bases and Group Representations, American Mathematical Society, Providence, Rhode Island, 2000.
[22] D. Li, J. Leng, T. Huang and X. Li, On weaving g-frames for Hilbert spaces, Complex Anal. Oper. Theory 14(2) (2020), 1–25.
[23] G. Lumer, Semi-inner-product spaces, Tran. Amer. Math. Soc. 100(1) (1961), 29–43.
[24] A. Rahimi, B. Daraby, Z. Darvishi, Construction of continuous frames in Hilbert spaces, Azerbaijan J. Math. 7(1) (2017), 49–58.
[25] A. Rahimi, Z. Samadzadeh and B. Daraby, Frame-related operators for woven frames, Int. J. Wavelets Multiresolut. Inf. Process. 17(3) (2019), Article ID 1950010. https://doi.org/10.1142/S0219691319500103
[26] R. Rezapour, A. Rahimi, E. Osgooei and H. Dehghan, Controlled weaving frames in Hilbert spaces, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 22(1) (2019), Artcle ID 1950003. https://doi.org/10.1142/S0219025719500036
[27] W. Rudin, Functional Analysis, McGraw-Hill Science, Singapure, 1991.
[28] I. Singer, Bases in Banach spaces II, Springer, Berlin, 1981.
[29] J. G. Stampfli, Adjoint abelian operators on Banach space, Canad. J. Math. 21 (1969), 505–512.
[30] D. T. Stoeva, Xd-Riesz bases in separable Banach spaces, Collection of papers dedicated to the 60th Anniv. of M. Konstantinov, BAS Publ. House, 2008.
[31] L. K. Vashisht and Deepshikha, Weaving properties of generalized continuous frames generated by an iterated function system, Journal of Geophysical Research 110 (2016), 282–295.
[32] L. K. Vashisht and Deepshikha, On continuous weaving frames, Adv. Pure Appl. Math. 8(1) (2017), 15–31. https://doi.org/10.1515/apam-2015-0077
[33] L. K. Vashisht, S. Garg, Deepshikha and P. K. Das, On generalized weaving frames in Hilbert spaces, Rocky Mountain J. Math. 48(2) (2018), 661–685. https://doi.org/10.1216/RMJ-2018-48-2-661
[34] L. K. Vashisht and G. Verma, Generalized weaving frames for operators in Hilbert spaces, Results Math. (2017), 1–23. https://doi.org/10.1007/s00025-017-0704-6
[35] H. Zhang and J. Zhang, Frames, Riesz bases, and sampling expansions in Banach spaces via semi-inner products, Appl. Comput. Harmon. Anal. 31(1) (2011), 1–25.
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