Weak nonhomogeneous wavelet dual frames for Walsh reducing subspace of L2(ℝ+)

In wavelet analysis, refinable functions are the bases of extension principles for constructing (weak) dual wavelet frames for [Formula: see text] and its reducing subspaces. This paper addresses refinable function-based dual wavelet frames construction in Walsh reducing subspaces of [Formula: see text]. We obtain a Walsh–Fourier transform domain characterization for weak [Formula: see text]-adic nonhomogeneous dual wavelet frames; and present a mixed oblique extension principle for constructing weak [Formula: see text]-adic nonhomogeneous dual wavelet frames in Walsh reducing subspaces of [Formula: see text].

Partial Gabor frames and dual frames

The theory of Gabor frames has been extensively investigated. This paper addresses partial Gabor systems. We introduce the concepts of partial Gabor system, frame and dual frame. We present some conditions for a partial Gabor system to be a partial Gabor frame, and using these conditions, we characterize partial dual frames. We also give some examples. It is noteworthy that the density theorem does not hold for general partial Gabor systems.

Dual finite frames for vector spaces over an arbitrary field with applications

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.