On Frames in Banach Spaces

L. K. Vashisht


Banach frames of type $\omega P^*$, shrinking Banach frames and retro shrinking Banach frames in Banach spaces have been introduced and studied. Necessary and sufficient conditions for a Banach frame (retro shrinking Banach frame) to be shrinking are given. Relation between various types of Banach frames are discussed.


Frames; Banach frames; Retro Banach frames; Schauder frames

Full Text:



P.G. Casazza, The art of frame theory,Taiwanese J. Math.4(2) (2000), 129–201.

P.G. Casazza, D. Han and D.R. Larson, Frames for Banach spaces, Contemp. Math. 247 (1999), 149–182.

P.G. Casazza and O. Christensen, Perturbation of operator and applications to frame theory, J. Fourier Anal. Appl. 3 (1997), 543–557.

O. Christensen and C. Heil, Perturbation of Banach frames and atomic decompositions, Math. Nachr 185 (1997), 33–47.

O. Christensen, Frames and bases (An introductory course), Birkhäuser, Boston (2008).

R.R. Coifman and G.Weiss, Expensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569–645.

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

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

N. Dunford and J.T. Schwartz, Linear Operator, Vol. I, Interscience Publishers, New York, London, 1958.

H.G.Feichtinger and K. Gröchenig,Aunified approach to atomic decompositons via inegrable group representations, Lecture Notes in Mathematics 1302

(Springer, Berlin, 1988), 52–73.

S.J. Favier and R.A. Zalik, On the stablity of frames and Riesz bases, Appl. Comp. Harm. Anal. 2 (1995), 160–173.

D. Gabor, Theory of communicatons, Jour. Inst. Elec. Engg. 93 (1946), 429–457.

K. Gröchenig, Describing functions: Atomic decompositions versus frames, Monatsh. Math. 112 (1991), 1–41.

D. Han and D.R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (697) (2000), 1–91.

C. Heil, ABasis Theory Primer, Birkhäuser (expanded edition)(1998).

H.Heuser, Functional Analysis, JohnWiley and Sons, NewYork (1982).

P.K. Jain, S.K. Kaushik and L.K.Vashisht, Banach frames for conjugate Banach spaces, Zeit. Anal. Anwendungen, 23 (4) (2004), 713–720.

P.K. Jain, S.K. Kaushik and L.K. Vashisht, On perturbation of Banach frames, Int. J. Wavelets, Multi. Infor. Processing (IJWMIP) 4 (3) (2006), 559–565.

P.K. Jain, S.K. Kaushik and L.K. Vashisht,On Banach frames,Indian J. Pure Appl. Math. 37 (5) (2006), 265–272.

P.K. Jain, S.K. Kaushik and Nisha Gupta, On frame systems in Banach spaces, Int. J. Wavelets Multi. Info. Processing 7 (1) (2009), 1-7.

S.K. Kaushik, Ageneralization of frames in Banach spaces J. Contemp. Math. Anal. 44 (4) (2009), 212–218.

S.K. Kaushik, Some results concerning frames in Banach spaces, Tamking J. Math. 38 (3) (2007), 267–276.

S.K. Kaushik, A note on exact Banach frames, Int. J. Pure Appl. Math. 31 (2) (2006), 279–286.

S.K. Kaushik, Ageneralization of frames in Banach frames, J. Contemp. Math. Anal. 44 (4) (2009), 212–218.

S.K. Kaushik and L.K. Vashisht,On strong retro Banach frames, communicated.

R. Liu and B. Zheng, Acharacterization of Schauder frames which are near-Schauder bases, J.Fourier Anal. Appl. 16 (2010), 791–803.

R. Liu, On shrinking and boundedly complete Schauder frames of Banach spaces, J. Math. Anal. Appl. 365 (2010), 385–398.

I. Singer, Basic sequences and reflexivity of Banach spaces, Studia Math. 21 (1962), 351–369.

I. Singer, Bases in Banach Spaces. II (Springer, NewYork, 1981).

L.K. Vashisht, On retro Banach frames of type P, Azerbaijan J. Math., 2 (1) (2012), 82-89.

L.K. Vashisht, On F-Schauder Frames, TWMS J. Appl. Eng. Math., 2 (1) (2012), 116-120.

K. Yoshida, Functional Analysis, (Berlin-Heidelber-New York, Springer-Verlag, 1966).

R. Young, A Introduction to Non-harmonicFourier Series, Academic Press, NewYork, 1980 (revised first edition 2001).

DOI: http://dx.doi.org/10.26713%2Fcma.v3i3.215


  • There are currently no refbacks.

eISSN 0975-8607; pISSN 0976-5905