For p 12 11 , the eigenfunctions of the non-linear eigenvalue problem for the p-Laplacian on the interval (0, 1) are shown to form a Riesz basis of L2(0, 1) and a Schauder basis of Lq(0, 1) whenever 1 < q < ∞.

We establish sufficient assumptions on sequences of Fucik eigenvalues the one-dimensional Laplacian which guarantee that corresponding eigenfunctions form a Riesz basis in $L^2(0,\pi)$.

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...

Discrete systems with long-range interactions are considered. Continuous medium models as continuous limit of discrete chain system are defined. Longrange interactions of chain elements that give the fractional equations for the medium model are discussed. The chain equations of motion with long-range interaction are mapped into the continuum equation with the Riesz fractional derivative. We fo...

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 :...

for all f ∈ H . The constant A (respectively, B) is a lower (resp. upper) frame bound for the frame. One of the most important frames for applications, especially signal processing, are the Weyl-Heisenberg frames. For g ∈ L(R) we define the translation parameter a > 0 and the modulation parameter b > 0 by: Embg(t) = e , Tnag(t) = g(t− na). For g ∈ L(R) and a, b > 0, we say for short that (g, 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...