Frames of Eigenfunctions Associated with a Boundary Value Problem

We introduce and study a redundant system of retro Banach frames consisting of eigenfunctions associated with a given boundary value problem. 1. Introduction Duffin and Schaeffer in [1], while addressing some deep problems in nonharmonic Fourier series, abstracted Gabor’s method [2], of time-frequency atomic decomposition for signal processing to define frames for Hilbert spaces. A sequence in a real (or complex) separable Hilbert space with inner product is a frame (or Hilbert frame) for if there exist finite positive constants and such that The positive constants and are called lower and upper bounds of the frame, respectively. The inequality (1) is called the frame inequality of the frame. The operator given by is called the synthesis operator or the preframe operator of the frame. The adjoint operator of is called the analysis operator. More precisely, is given by Composing and , we obtain the frame operator which is given by The frame operator is a positive continuous invertible linear operator from to . Every vector can be written as The series converges unconditionally and is called the reconstruction formula for the frame. The representation of in reconstruction formula need not be unique. Thus, frames are redundant systems in a Hilbert space which yield one natural representation for every vector in the given Hilbert space, but which may have infinitely many different representations for a given vector. Frames provide an appropriate mathematical framework for redundant signal expansions [3, 4]. Moreover, frames find many applications in mathematics, science, and engineering. In particular, frames are widely used in nonuniform sampling [5], wavelet theory [6, 7], wireless communication, signal processing [3, 8], filter banks [9], and many more. The reason is that frames provide both great liberties in design of vector space decompositions and 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 introduction to various types of frames with applications, one may refer to [10–13]. In 1986, Daubechies et al. [6] found new applications to wavelets and Gabor transforms in which frames played an important role. Coifman and Weiss [14] introduced the notion of atomic decomposition for function spaces. Feichtinger and Gr？chenig [15] studied the atomic decomposition via integrable group representation. Casazza et al. [16] also carried out a study of atomic decompositions and Banach frames.