Frame for operators in finite dimensional hilbert space

Mohammad Janfada, Vahid Reza Morshedi, Rajabali Kamyabi Gol


In this paper, we study frames for operators ($K$-frames) in finite dimensional Hilbert spaces and express the dual of $K$-frames. Some properties of $K$-dual frames are investigated. Furthermore, the notion of their oblique $K$-dual and some properties are presented.


K-frame; K-dual; oblique K-dual

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F. Arabyani and A. A. Arefijamaal, Some constructions of $K$-frames and their duals}, To appear in Rocky Mountain J. Math.

J. J. Benedetto and M. Fickus, Finite normalized tight frames, Adv. Comput. Math., 18(2003), 357--385.

H. Bolcskei, F. Hlawatsch and H. G. Feichtinger, Frame-theoretic analysis of over-sampled filter banks, IEEE Trans. Signal Process. 46(1998), 3256--3268.

P. Casazza, G. Kutyniok and F. Philipp, Finite Frames: Theory and Applications. Birkhauser, Berlin, 2013.

O. Christensen, A. M. Powell and X. C. Xiao, A note on finite dual frame pairs, Proc. Amer. Math. Soc., 140(11) (2012), 3921--3930.

O. Christensen, Frames and Bases: An Introductory Course, Birkhauser, Boston 2008.

I. Daubechies, A. Grossmann and Y. Meyer, Painless non-orthogonal expansions, J. Math. Physics, 27(1986), 1271--1283.

R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72(1952), 341--366.

Y. C. Eldar and G. D. Forney Jr., Optimal tight frames and quantum measurement. IEEE Trans. Inform. Theory, 48(3) (2002), 599--610.

P. J. S. G. Ferreira, Mathematics for multimedia signal processing II: Discrete finite frames and signal reconstruction, In: Byrnes, J.S. (ed.) Signal processing for multimedia, IOS Press, Amsterdam (1999), 35--54.

L. Guavructa, Frames for operators, Appl. Comput. Harmon. Anal., 32(2012), 139--144.

L. Guavructa, Atomic decompositions for operators in reproducing kernel Hilbert spaces, Math. Rep., 17(3) (2015), 303--314.

K. Grochenig and C. Heil, Modulation spaces and pseudodifferential operators. Integral Equ. Oper. Theory 34(4) (1999), 439--457.

D. Han and D. Larson, Frames, Bases and group representations, Mem. Amer. Math. Soc., 147(697) 2000.

R. W. Heath Jr. and A. Paulraj, Linear dispersion codes for MIMO systems based on frame theory. IEEE Trans. Signal Process., 50(10) (2002), 2429--2441.

Zh. Q. Xiang and Y. M. Li, Frame sequences and dual frames for operators, ScienceAsia, 42(2016), 222--230.

X. Xiao, Y. Zhu and L. Guavructa, Some properties of $K$-frames in Hilbert spaces, Results. Math., 63(3-4) (2013), 1243--1255.

X. C. Xiao, Y. C. Zhu and X. M. Zeng, Oblique dual frames in finite-dimensional Hilbert spaces, Int. J. Wavelets Multiresolut. Inf. Process, 11(2) (2013), 1350011-1--1350011-14.


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