Geodesic
Frames in Banach Spaces
Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 3, pp. 50-62.

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The notion of a frame in a Banach space with respect to a model space of sequences is introduced. This notion is different from the notions of an atomic decomposition, Banach frame in the sense of Gröchenig, (unconditional) Schauder frame in the sense of Han and Larson, and other known definitions of frames for Banach spaces. The frames introduced in this paper are shown to play a universal role in the solution of the general problem of representation of functions by series. A projective description of these frames is given. A criterion for the existence of a linear frame expansion algorithm and an analogue of the extremality property for a frame expansion are obtained.
Keywords: frame, Banach frame, representation system, basis, projector, null series, complemented subspace.
Mots-clés : atomic decomposition, coefficient space 
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P. A. Terekhin. Frames in Banach Spaces. Funkcionalʹnyj analiz i ego priloženiâ, Tome 44 (2010) no. 3, pp. 50-62. http://geodesic.mathdoc.fr/item/FAA_2010_44_3_a3/

[1] R. J. Duffin, A. C. Schaeffer, “A class of nonharmonic Fourier series”, Trans. Amer. Math. Soc., 72 (1952), 341–366 | DOI | MR | Zbl

[2] M. A. Naimark, “Spektralnye funktsii simmetricheskogo operatora”, Izv. AN SSSR, ser. matem., 4:3 (1940), 277–318 | Zbl

[3] P. G. Casazza, D. Han, D. R. Larson, “Frames for Banach spaces”, Contemp. Math., 247 (1999), 149–182 | DOI | MR | Zbl

[4] B. S. Kashin, T. Yu. Kulikova, “Zamechanie ob opisanii freimov obschego vida”, Matem. zametki, 72:6 (2002), 941–945 | DOI | MR | Zbl

[5] R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980 | MR

[6] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., 61, SIAM, Philadelphia, PA, 1992 ; I. Dobeshi, Desyat lektsii po veivletam, RKhD, Izhevsk, 2001 | MR | Zbl

[7] O. Christensen, An Introduction to Frames and Riesz Bases, Appl. Numer. Harmon. Anal., Birkhäuser, Boston, MA, 2003 | MR

[8] I. Ya. Novikov, V. Yu. Protasov, M. A. Skopina, Teoriya vspleskov, Fizmatlit, M., 2005

[9] K. Gröchenig, “Describing functions: atomic decompositions versus frames”, Monatsh. Math., 112:1 (1991), 1–41 | DOI | MR

[10] D. Han, D. R. Larson, Frames, bases and group representations, Memoirs Amer. Math. Soc., 147, no. 697, 2000, x+94 pp. | DOI | MR

[11] P. G. Casazza, O. Christensen, D. T. Stoeva, “Frame expansions in separable Banach spaces”, J. Math. Anal. Appl., 307:2 (2005), 710–723 | DOI | MR | Zbl

[12] A. Aldroubi, Q. Sun, W.-S. Tang, “$p$-Frames and shift invariant subspaces of $L^p$”, J. Fourier Anal. Appl., 7:1 (2001), 1–21 | DOI | MR | Zbl

[13] O. Christensen, D. T. Stoeva, “$p$-Frames in separable Banach spaces”, Adv. Comput. Math., 18:2 (2003), 117–126 | DOI | MR | Zbl

[14] N. K. Bari, “O bazisakh v gilbertovom prostranstve”, DAN SSSR, 54 (1946), 383–386

[15] N. K. Bari, “Biortogonalnye sistemy i bazisy v gilbertovom prostranstve”, Matematika IV, Uchen. zap. MGU, 148, 1951, 69–107 | MR

[16] P. A. Terekhin, “Freimy v banakhovom prostranstve i ikh prilozheniya k postroeniyu vspleskov”, Issledovaniya po algebre, teorii chisel, funktsionalnomu analizu i smezhnym voprosam, Sb. nauch. trudov, vyp. 2, Izd-vo Sarat. un-ta, Saratov, 2003, 65–81

[17] P. A. Terekhin, “Sistemy predstavleniya i proektsii bazisov”, Matem. zametki, 75:6 (2004), 944–947 | DOI | MR | Zbl

[18] P. A. Terekhin, “Banakhovy freimy v zadache affinnogo sinteza”, Matem. sb., 200:9 (2009), 127–146 | DOI | MR | Zbl

[19] J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, Springer-Verlag, Berlin–Heidelberg–New York, 1977 | MR | Zbl

[20] B. S. Kashin, A. A. Saakyan, Ortogonalnye ryady, Izd-vo AFTs, M., 1999 | MR | Zbl

[21] P. A. Terekhin, “Neravenstva dlya komponentov summiruemykh funktsii i ikh predstavleniya po elementam sistemy szhatii i sdvigov”, Izv. vuzov, matem., 1999, no. 8, 74–81 | MR | Zbl

[22] A. Pelczynski, “Any separable Banach space with the bounded approximation property is a complemented subspace of a Banach space with a basis”, Studia Math., 40 (1971), 239–242 | DOI | MR

[23] W. B. Johnson, H. P. Rosenthal, M. Zippin, “On bases, finite dimensional decompositions and weaker structure in Banach spaces”, Israel J. Math., 9 (1971), 488–506 | DOI | MR | Zbl

[24] P. L. Ulyanov, “Predstavlenie funktsii ryadami i klassy $\varphi(L)$”, UMN, 27:2 (1972), 3–52 | MR | Zbl

[25] K. S. Kazarian, R. E. Zink, “Subsystems of the Schauder system that are quasibases for $L^p[0,1]$, $1\le p\infty$”, Proc. Amer. Math. Soc., 126:10 (1998), 2883–2893 | DOI | MR | Zbl

[26] T. P. Lukashenko, “O svoistvakh ortorekursivnykh razlozhenii po neortogonalnym sistemam”, Vestn. Mosk. un-ta, ser. 1 matem. mekh., 2001, no. 1, 6–10 | MR | Zbl

[27] V. P. Odinets, M. Ya. Yakubson, Proektory i bazisy v normirovannykh prostranstvakh, URSS, M., 2004 | MR | Zbl