Uncountable Frames in Non-Separable Hilbert Spaces and their Characterization

B. T. Bilalov, M. I. Ismailov, Y. I. Nasibov

Abstract


The concepts of Bessel families and frames in non-separable Hilbert spaces are introduced in this work. Besselianness criterion for a family is found. Similar to the usual case, analysis,
synthesis and frame operators are dened, their properties are studied. Many results related to
usual frames are extended to new case. Examples are given.


Keywords


non-separable space, Bessel family, uncountable frame

References


R. J. Duffin, A.C.Schaeffer, A class of nonharmonic Fourier series, Trans. Amer.

Math. Soc. 72(1952), 341-366.

Ch. Chui, Wavelets: a tutorial in theory and applications. Academic Press, Boston,

, 724 p.

Meyer Y. Wavelets and operators, Herman, Paris, 1990

I. Daubechies, Ten lectures on wavelets. SIAM, Philadelphia, 1992.

Mallat S. A wavelet tour of signal processing, Academic Press, San Diego, 1999

R. Young, An introduction to nonharmonic Fourier series, Academic Press, New

York, 1980.

Ch. Heil, A Basis Theory Primer, Springer, 2011, 534 p.

O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser Boston,

O. Christensen, Frames and bases. An introductory course, Birkhauser, Boston,

O. Christensen, D.T. Stoeva, p-frames in separable Banach spaces, Advances in

Computational Mathematics 18(2003), 17-126.

I.M. Dremin, O.V. Ivanov, V.A. Nechitailo, Wavelets and their uses. Usp. phisics

nauk, 171(5) (2001), 465-501.

A. Aldroubi, Q. Sun, W.Sh. Tang, p-frames and shift invariant subspaces of Lp , J.

Fourier Anal. Appl., 7(1) (2001), 1-22

N.K. Bari, Biorthogonal systems and bases in Hilbert space, Moscow Gos. Univ.

Uceneye Zapiski 148, Matematika 4 (1951), 69-107.

B.T. Bilalov, Z.G. Guseinov, p-Bessel and p-Hilbert systems and p-bases, Dokl.

NANA, LXIV(3) (2008), 3-8.

B.T. Bilalov, Z.G. Guseinov, K -Bessel and K -Hilbert systems and K - bases, Dok-

lady Mathematics, 80(3) (2009), 826-828.

B.T. Bilalov, Sh.M. Hashimov, On Decomposition In Banach Spaces, Proceedings

of the IMM of NAS of Azerbaijan, 40(2) (2014), 97-106.

S.R. Sadigova, On Frame Properties Of Degenerate System Of Exponents In

Hardy Classes, Caspian Journal of Applied Mathematics, Ecology and Economics,

(1)(2013), 97-103.

S.R. Sadigova, The general solution of the homogeneous Riemann problem in the

weighted Smirnov classes, Proc. of the IMM of NAS of Azerbaijan, 40(2) (2014),

-124.

S.R. Sadigova, I. Ismailov, On Frames of Double and Unary Systems in Lebesgue

Spaces, Pensee Journal, 76(4) (2014), 189-202.

S.R. Sadigova, Z.A. Kasumov, On atomic decomposition for Hardy classes with

respect to degenerate exponential systems, Proc. of the IMM of NAS of Azerbaijan,

(1) (2014), 55-67

A. Rahimi, B. Daraby, Z. Darvishi, Construction of Continuous Frames in Hilbert

spaces, Azerbaijan Journal of Mathematics, 7(1), 2017, 49-58.

Li S., Ogawa H. Pseudoframes for subspaces with applications, J. Fourier Anal.

Appl. 10 (2004) pp. 409-431

Feichtinger H.G., Grochening K.H. Banach spaces related to integrable group rep-

resentations and their atomic decompositions, I., J. of Func. Analysis. 86(2), 1989,

pp. 307-340

Feichtinger H.G., Grochening K.H. Banach spaces related to integrable group repre-

sentations and their atomic decompositions, II., Mh. Math. 108, 1989, pp.129{148.

Grochenig K.H. Describing Functions: Atomic Decompositions Versus Frames. Mh.

Math. 112, 1991, pp. 1-41.

Sun W. Stability of g-frames. Journal of Mathematical Analysis and Applications,

(2), 2007, pp. 858-868.

Sun W. G-frames and g-Riesz bases. Journal of Mathematical Analysis and Appli-

cations, 322 (1), 2006, pp. 437-452.

Bilalov B.T., Guliyeva F.A. Noetherian perturbation of frames. Pensee Journal

(ISSN: 0031-4773),Vol.75,Issue.12,2013,pp.425-431

Bilalov B.T., Guliyeva F.A. t-Frames and their Noetherian Perturbation. Complex

Analysis and Operator Theory, DOI 10.1007/s11785-014-0416-9

Bilalov B.T., Mamedova Z.V. On the frame properties of some degenerate trigono-

metric system, Dokl. Acad. Nauk, v.LXVIII, 5, 2012, pp. 14-18

Bilalov B.T., Guliyeva F.A. On The Frame Properties of Degenerate System of Sines,

Journal of Function Spaces and Applications, Vol. 2012 (2012), Article ID 184186,

pages, doi:10. 1155/2012/184186

Kadison R., Singer I. Extensions of pure states, Amer. J. Math. 81(1959), 547-564.

Casazza P.G., Vershynin R. Kadison-Singer meets Bourgain-Tzafriri, Preprint

Casazza P.G., Christensen O., Lindner A., Vershynin R. Frames and the Feichtinger

Conjecture, Proceedings of AMS, Vol. 133 No. 4 (2005) 1025-1033.

Bourgain J., Tzafriri L. On a problem of Kadison and Singer, J. Reine Angew. Math.

(1991), 1-43.

Marcus A., Spielman D. A., Srivastava N. Interlacing families II: Mixed characteris-

tic polynomials and the Kadison-Singer problem //arXiv preprint arXiv:1306.3969,

Weaver, Nik. "The Kadison-Singer problem in discrepancy theory." Discrete math-

ematics 278.1 (2004): 227-239

Weaver, Nik. "The Kadison-Singer problem in discrepancy theory II."

arXiv:1303.2405v1, 11 Mar 2013

Casazza P.G., Matthew F., Janet C.T., Eric W. "The Kadison-Singer problem in

mathematics and engineering: a detailed account." Contemporary Mathematics 414

(2006): 299.

Cruz-Uribe D.V., Fiorenza A. Variable Lebesgue Spaces: Foundations and Harmonic

Analysis. Springer, 2013, 312 p.

Kokilashvili V., Meshki A., Rafeiro H. Samko S. Integral operators in nonstandard

function spaces, Birkhauser-Springer, 2016, 1003 p.

Adams D.R. Morrey spaces. Birkhauser-Springer, 2015, 120 p.

Bardaro C., Musielak J., Vinti G. Nonlinear integral operators and applications,

Walter de Gruyter-Berlin-New York, 2013, 201 p.

Ismayilov M. I., Nasibov Y. I. One Generalization of Banach Frame. Azerb. J. of

Math., vol. 6, No. 2, 2016, pp.143-159

Ole Christensen, A Short Introduction To Frames, Gabor Systems, And Wavelet

Systems. Azerb. J. of Math., vol. 4, No. 1, 2014, pp.25-39

Ole Christensen, Maria I. Zakowicz , Paley-Wiener type perturbations of frames and

the deviation from perfect reconstruction . Azerb. J. of Math., vol. 7, No. 1, 2017,

pp.59-69

Kasumov Z.A., Hashimov Ch.M. On the equivalent bases of cosines in generalized

Lebesgue spaces. Proceedings of the Institute of Mathematics and Mechanics, Na-

tional Academy of Sciences of Azerbaijan Volume 41, Number 2, 2015, Pages 70{76


Refbacks

  • There are currently no refbacks.


free counters