We discuss an operator equation Kf = g where X, Y are reflexive Banach spaces of functions in bounded domains Ω ⊂ RN and Ω0, and K : X −→ Y has no continuous inverse. Let V and V1 be another reflexive Banach spaces of functions in Ω and D ⊂ Ω respectively such that the embedding V ⊂ X is continuous. In order to stably reconstruct g by noise data gδ with ‖g − gδ‖Y ≤ δ: noise level, we consider t...

We study tne `q approximation of the sparsest solution of underdetermined linear systems. Mainly we present a condition on the matrix associated with an underdetermined linear system under which the solution of `q minimization is the sparsest solution of the system. Our condition generalizes a similar condition in [Candés, Romberg and Tao’06] ensuring that the solution of the `1 minimization is...

The series solution is widely applied to differential equations onR but is not found in q-differential equations. Applying the Taylor andmultiplication rule of two generalized polynomials, we develop a series solution of linear homogeneous q-difference equations. As an example, the series solution method is used to find a series solution of the second-order q-difference equation of Hermite’s type.

In this paper we study the reflexive subobject lattices and reflexive endomorphism algebras in a concrete category. For the category Set of sets and mappings, a complete characterization for both reflexive subobject lattices and reflexive endomorphism algebras is obtained. Some partial results are also proved for the category of abelian groups.