In this paper, the effective sequences in a Hilbert space are discussed and by introducing the adjoint sequence of a sequence, an equivalent characterization of an effective sequence is presented. Next, some properties of linear operators preserving the effectiveness of effective sequences are discussed. It is proved that a linear operator maps every effective sequence as an effective sequence if and only if it is a unitary. Finally, we give a necessary and sufficient condition for a sequence obtained by adding a unit vector into an orthonormal basis to be an effective sequence.
In this note we study the generalized property (ω), a variant of generalized Weyl's theorem, by means of the new spectrum. We establish for a bounded linear operator defined on a Banach space a sufficient and necessary condition for which generalized property (ω) holds. As a consequence of the main result, the stability of generalized property (ω) is considered.
Paraunitary matrices play a very important role in construction of wavelets, multiwavelets and frames. In this paper, we give an explicit algorithm for constructing paraunitary symmetric matrices (p.s.m. for short), whose components are symmetric or antisymmetric polynomials. Based on the constructed p.s.m. and the given orthogonal and symmetric multiwavelets (o.s.m. for short), parametrization of o.s.m. filter banks is given. Appropriately selecting some parameters, we can obtain o.s.m. with some additionally desirable properties such as Armlet. To embody our results, we construct a family of symmetric Chui--Lian Armlet filters.
In this paper, we use a new lemma to investigate the relation between solutions, their 1st, 2nd derivatives, differential polynomial of differential equations with function of small growth.