The Dual-Phase-Lagging (DPL) equation is formulated as an abstract differential equation. In the absence of a heat source term the DPL equation with homogeneous boundary conditions generates a contraction semigroup. The exact expression of the semigroup is achieved. It is proved that the associated eigenfunctions form a Riesz basis. The stability of semigroup is proved. Moreover, it is also sho...

We study the optimal approximation of the solution of an operator equation A(u) = f by linear mappings of rank n and compare this with the best n-term approximation with respect to an optimal Riesz basis. We consider worst case errors, where f is an element of the unit ball of a Hilbert space. We apply our results to boundary value problems for elliptic PDEs that are given by an isomorphism A :...

If the integer translates of a function φ with compact support generate a frame for a subspace W of L 2 (R), then it is automatically a Riesz basis for W, and there exists a unique dual Riesz basis belonging to W. We demonstrate that considerable freedom can be obtained by considering oblique duals, i.e., duals not necessarily belonging to W. For example, we present a condition for the existenc...

Let M ¢ 2 be a fixed positive integer. A family of closed subspaces Vj , j √ Z , of L 2 , the space of all square integrable functions on the real line, is said to be a multiresolution of L 2 if the following conditions hold: ( i) Vj , Vj/1 , and f √ Vj if and only if f ( Mr) √ Vj/1 for all j √ Z ; ( ii ) < j√Z Vj is dense in L 2 and > j√Z Vj Å M; ( iii ) there exists a function f in V0 such th...

Frames consisting of translates of a single function play an important role in sampling theory as well as in wavelet theory and Gabor analysis. We give a necessary and sufficient condition for a subfamily of regularly spaced translates of a function φ ∈ L(R), (τnbφ)n∈Λ, Λ ⊂ Z, to form a frame (resp. Riesz basis) for its closed linear span. One consequence is that if Λ ⊂ N , then this family is ...

A theorem of Luxemburg and Zaanen on normed Riesz spaces (Theorem 2.4 below) and one of Nakano (Theorem 2.3 below) have been extended by the author in [1] to metrizable locally solid linear topological Riesz spaces. This note gives an example which shows they cannot be further extended to nonmetrizable Hausdorff locally solid linear topological Riesz spaces. 1. Notation and basic concepts. For ...

Let G be a Lie group of polynomial growth. We prove that the second-order Riesz transforms on L 2 (G ; dg) are bounded if, and only if, the group is a local direct product of a compact group and a nilpotent group, in which case the transforms of all orders are bounded.