×

Frames in Banach spaces. (English. Russian original) Zbl 1271.42043

Funct. Anal. Appl. 44, No. 3, 199-208 (2010); translation from Funkts. Anal. Prilozh. 44, No. 3, 50-62 (2010).
Summary: 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.

MSC:

42C15 General harmonic expansions, frames
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] R. J. Duffin and A. C. Schaeffer, ”A class of nonharmonic Fourier series,” Trans. Amer. Math. Soc., 72 (1952), 341–366. · Zbl 0049.32401 · doi:10.1090/S0002-9947-1952-0047179-6
[2] M. A. Naimark, ”Spectral functions of a symmetric operator,” Izv. Akad. Nauk SSSR, Ser. Mat., 4:3 (1940), 277–318. · JFM 66.0549.02
[3] P. G. Casazza, D. Han, and D. R. Larson, ”Frames for Banach spaces,” Contemp. Math., 247 (1999), 149–182. · Zbl 0947.46010 · doi:10.1090/conm/247/03801
[4] B. S. Kashin and T. Yu. Kulikova, ”A note on the description of frames of general form,” Mat. Zametki, 72:6 (2002), 941–945; English transl.: Math. Notes, 72:6 (2002), 863–867. · Zbl 1044.42026 · doi:10.4213/mzm675
[5] R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980. · Zbl 0493.42001
[6] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math., vol. 61, SIAM, Philadelphia, PA, 1992.
[7] O. Christensen, An Introduction to Frames and Riesz Bases, Appl. Numer. Harmon. Anal., Birkhauser, Boston, MA, 2003. · Zbl 1017.42022
[8] I. Ya. Novikov, V. Yu. Protasov, and M. A. Skopina, Wavelet Theory [in Russian], Fizmatlit, Moscow, 2005.
[9] K. Gröchenig, ”Describing functions: atomic decompositions versus frames,” Monatsh. Math., 112:1 (1991), 1–41. · Zbl 0736.42022
[10] D. Han and D. R. Larson, ”Frames, bases and group representations,” Memoirs Amer. Math. Soc, 147:697 (2000), 1–91.
[11] P. G. Casazza, O. Christensen, and D. T. Stoeva, ”Frame expansions in separable Banach spaces,” J. Math. Anal. Appl., 307:2 (2005), 710–723. · Zbl 1091.46007 · doi:10.1016/j.jmaa.2005.02.015
[12] A. Aldroubi, Q. Sun, and W.-S. Tang, ”p-Frames and shift invariant subspaces of L p” J. Fourier Anal. Appl., 7:1 (2001), 1–22. · Zbl 0983.46027 · doi:10.1007/s00041-001-0001-2
[13] O. Christensen and D. T. Stoeva, ”p-Frames in separable Banach spaces,” Adv. Comput. Math., 18:2 (2003), 117–126. · Zbl 1012.42024 · doi:10.1023/A:1021364413257
[14] N. K. Bari, ”On bases in a Hilbert space,” Dokl. Akad. Nauk SSSR, 54 (1946), 383–386.
[15] N. K. Bari, ”Biortogonal systems and bases in a Hilbert space,” Uchenye Zapiski Moskov. Gos. Univ., Mat. IV, 148 (1951), 69–107.
[16] P. A. Terekhin, ”Frames in Banach spaces and Their Applications to Construction of Wavelets,” in: Research in Algebra, Number Theory, Functional Analysis, and Related Topics. Collection of Scientific Works [in Russian], vol. 2, Izd. Saratov Univ., Saratov, 2003, 65–81.
[17] P. A. Terekhin, ”Representation systems and projections of bases,” Mat. Zametki, 75:6 (2004), 944–947; English transl.: Math. Notes, 75:6 (2004), 881–884. · Zbl 1085.46007 · doi:10.4213/mzm562
[18] P. A. Terekhin, ”Banach frames in the affine synthesis problem,” Mat. Sb., 200:9 (2009), 127–146; English transl.: Russian Acad. Sci. Sb. Math., 200:9 (2009), 1383–1402. · doi:10.4213/sm5655
[19] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I: Sequence spaces, Springer-Verlag, Berlin-Heidelberg-New York, 1977. · Zbl 0362.46013
[20] B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Izd. AFTs, Moscow, 1999. · Zbl 1188.42010
[21] P. A. Terekhin, ”Inequalities for components of summable functions and their representations by elements of a system of compressions and shifts,” Izv. Vyssh. Uchebn. Zaved. Mat., 1999:8, 74–81; English transl.: Russian Math., 43:8 (1999), 70–77. · Zbl 1004.42006
[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.
[23] W. B. Johnson, H. P. Rosenthal, and M. Zippin, ”On bases, finite dimensional decompositions and weaker structures in Banach spaces,” Israel J. Math., 9 (1971), 488–506. · Zbl 0217.16103 · doi:10.1007/BF02771464
[24] P. L. Ul’yanov, ”Representation of functions by series and classes (L)” Uspekhi Mat. Nauk, 27:2 (1972), 3–52; English transl.: Russian Math. Surveys, 27:2 (1972), 1–54.
[25] K. S. Kazarian and R. E. Zink, ”Subsystems of the Schauder system that are quasibases for L p[0,1], 1 p < ” Proc. Amer. Math. Soc, 126:10 (1998), 2883–2893. · Zbl 0904.46008 · doi:10.1090/S0002-9939-98-04388-3
[26] T. P. Lukashenko, ”Properties of orthorecursive expansions in nonorthogonal systems,” Vestnik Moscov. Univ. Ser. I Mat. Mekh., 2001:1, 6–10; English transl.: Mosc. Univ. Math. Bull., 56:1 (2001), 5–9. · Zbl 1023.42014
[27] V. P. Odinets and M. Ya. Yakubson, Projectors and Bases in Normed Spaces [in Russian], URSS, Moscow, 2004. · Zbl 1057.46013
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.