As a generalization of the concept of pseudo-biorthogonal bases (PBOB), we already presented in Ref. [3] the theory of the so-called extended pseudo-biorthogonal bases (EPBOB). We introduce in this paper two special types of EPBOB called EPBOB's of type O and of type L. This paper discusses characterizations, construction methods, inherent properties, and mutual relations of these types of EPBOB.

This paper introduces the concept of an extended pseudo-biorthogonal basis" (EPBOB), which is a generalization of the concepts of an orthonormal (OB), a biorthonormal (BOB), a pseudo-orthogonal (POB), and a pseudo-biorthogonal (PBOB) bases. Let HN be a subspace of a Hilbert space H. The concept of EPBOB says that we can always construct a set of 2M (MN) elements of H but not necessarily all in HN such that like BOB any element f in HN can be expressed by fMΣm=1(f,φ*m)φm. For a better understanding and a wide application of EPBOB, this paper provides their characterization and shows how they preserve the formalism of BOB. It also shows how to construct them.

The purpose of this paper is to deal with the problem of recovering a signal from its noisy version. One example is to restore old images degraded by noise. The recovery solution is given within the framework of series expansion and we shall show that for the general case the recovery functions have to be elements of an extended pseudo biorthogonal basis (EPBOB) in order to suppress efficiently the corruption noise. After we discuss the different situations of noise, we provide some methods to construct the optimal EPBOB in order to deal with these situations.