Communications in Mathematics and Applications Volume 3 (2012), Number 3, pp. 283–292 © RGN Publications http://www.rgnpublications.com On Weighted Banach Frames L.K. Vashisht and Shalu Sharma Abstract. We introduce and study weighted Banach frames in Banach spaces. Necessary and/or sufficient conditions for a weighted Banach frame to be exact are given. An application of weighted Banach frames is discussed. 1. Introduction Frames for Hilbert spaces were introduced by Duffin and schaeffer in [8] in 1952, while addressing some deep problems in non-harmonic Fourier series. Later, in 1986, Daubechies, Grossmann and Meyer [7] found new applications to wavelets and Gabor transforms in which frames played an important role. Today, frames play important roles in many applications in mathematics, science and engineering. In particular frames are widely used in sampling theory, wavelet theory, wireless communication, signal processing, image processing, differential equations, filter banks, geophysics, quantum computing, wireless sensor network, multiple-antenna code design and many more. Reason is that frames provide both great liberties in design of vector space decompositions, as well as quantitative measure on computability and robustness of the corresponding reconstructions. In the theoretical direction, powerful tools from operator theory and Banach spaces are being employed to study frames. For a nice and comprehensive survey on various types of frames, one may refer to [1, 5] and the references therein. Coifman and Weiss [6] introduced the notion of atomic decomposition for function spaces. Later, Feichtinger and Grochenig [10] extended this idea to Banach spaces. This concept was further is generalized by Grochenig [13] who introduced the notion of Banach frames for Banach Spaces. Casazza, Han and Larson [2] also carried out a study of atomic decompositions and Banach frames. In this paper we introduce and study weighted Banach frames in Banach spaces. Necessary and/or sufficient conditions for a weighted Banach frame to be exact are given. We know that if a Banach space has a Banach frame, then it can be 2010 Mathematics Subject Classification. 42C15, 42C30, 46B15. Key words and phrases. Frames; Banach frames.