×

New results on \(K\)-frames. (English) Zbl 1459.42047

In this research article, the author studied and discussed \(\mathit{K}\)-frames in Hilbert spaces. Some sufficient conditions for a given frame to produce an adjoint frame are given. The author also considered sum of frames and obtained few results in this direction.

MSC:

42C15 General harmonic expansions, frames
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] E. Alizadeh, A. Rahimi and M. Rahmani, The algebra of Bessel sequences and means of frames in Hilbert spaces, Miskolc Math. Notes (Accepted Manuscripts).
[2] Candés, E. J. and Donoho, D. L., New tight frames of curvelets and optimal representations of objects with piecewise \(C^2\) singularities, Commun. Pure Appl. Math.57 (2004) 219-266. · Zbl 1038.94502
[3] Christensen, O., Pairs of dual Gabor frame generators with compact support and desir frequency localization, Appl. Comput. Harmon. Anal.20 (2006) 403-410. · Zbl 1106.42030
[4] Christensen, O., An Introduction to Frames and Riesz Bases (Birkhauser, Boston, 2016). · Zbl 1348.42033
[5] Ding, M. L., Xiao, X. C. and Zeng, X. M., Tight \(\text{K} \)-frames in Hilbert spaces, Acta Math. Sin.56 (2013) 105-112(in Chinese). · Zbl 1289.42089
[6] Douglas, R. G., On majorization, factorization, and range inclusion of operators on Hilbert space, Proc. Am. Math. Soc.17(2) (1966) 413-415. · Zbl 0146.12503
[7] Duffin, R. J. and Schaeffer, A. C., A class of nonharmonic Fourier series, Trans. Am. Math. Soc.72 (1952) 341-366. · Zbl 0049.32401
[8] Fillmore, P. A. and Williams, J. P., On operator ranges, Adv. Math.7 (1971) 254-281. · Zbl 0224.47009
[9] Găvrutţa, L., Frames for operators, Appl. Comput. Harmon. Anal.32(1) (2012) 139-144.
[10] Găvrutţa, L., New results on frames for operators, Anal. Univ. Oradea Fasc. Mat. TomXIX(2) (2012) 55-61.
[11] Găvrutţa, L., Atomic decompositions for operators in reproducing kernel Hilbert spaces, Math. Rep.17(67) (2015) 303-314.
[12] He, M., Leng, J., Yu, J. and Xu, Y., On the sum of \(\text{K} \)-frames in Hilbert spaces, Mediterr. J. Math.17 (2020) 46.
[13] Jia, M. and Zhu, Y. C., Some results about the operator perturbation of a \(\text{K} \)-frame, Results Math.73 (2018) 138.
[14] Murphy, G. J., \( C^\ast \)-algebras and Operator Theory (Acadamic Press, 1990). · Zbl 0714.46041
[15] Najati, A., Abdollahpour, M. R., Osgooei, E. and Saem, M. M., More on sums of Hilbert space frames, Bull. Korean Math. Soc.50(6) (2013) 1841-1846. · Zbl 1283.41024
[16] Obeidat, S., Samarah, S., Casazza, P. G. and Tremain, J. C., Sums of Hilbert space frames, J. Math. Anal. Appl.351 (2009) 579-585. · Zbl 1170.42016
[17] Poon, C., A consistent and stable approach to generalized sampling, J. Fourier Anal. Appl.20 (2014) 985-1019. · Zbl 1320.94037
[18] Ramu, G. and Johnson, P. S., Frame operators of \(\text{K} \)-frames, SeMA J.73 (2016) 171-181. · Zbl 1349.42065
[19] Xiang, Z. Q. and Li, Y. M., Frame sequences and dual frames for operators, Sci. Asia42 (2016) 222-230.
[20] Xiao, X., Zhu, Y. and Gavruta, L., Some properties of \(K\)-frames in Hilbert spaces, Results Math.63(3-4) (2013) 1243-1255. · Zbl 1268.42067
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.