We construct locally supported basis functions which are biorthogonal to conforming nodal finite element basis functions of degree p in one dimension. In contrast to earlier approaches, these basis functions have the same support as the nodal finite element basis functions and reproduce the conforming finite element space of degree p − 1. Working with Gauß–Lobatto nodes, we find an interesting ...

This work presents an adaptive block Lanczos method for large-scale non-Hermitian Eigenvalue problems (henceforth the ABLE method). The ABLE method is a block version of the non-Hermitian Lanczos algorithm. There are three innovations. First, an adaptive blocksize scheme cures (near) breakdown and adapts the blocksize to the order of multiple or clustered eigenvalues. Second, stopping criteria ...

Derivation of the self-consistent multimode laser equations. The Green function which forms the kernel of the fundamental self-consistent equation Eq. 1 has a non-hermitian spectral representation of the form (S1) G(x, x ′ ; k) = ∞ m=1 ϕ m (x, k) ¯ ϕ * m (x ′ , k) 2 k a (k − k m (k)). (S1) where the CF states ϕ m (x, k) satisfy the equation − 1 ǫ(x) ∇ 2 ϕ m (x, k) = k 2 m (k) ϕ m (x, k) (S2) wi...

Orthonormal bases of compactly supported wavelet bases correspond to subband coding schemes with exact reconstruction in which the analysis and synthesis filters coincide. We show here that under fairly general conditions, exact reconstruction schemes with synthesis filters different from the analysis filters give rise: to two dual Riesz bases of compactly supported wavelets. We give necessary ...

We introduce the sequence of generalized Gon£arov polynomials, which is a basis for the solutions to the Gon£arov interpolation problem with respect to a delta operator. Explicitly, a generalized Gon£arov basis is a sequence (tn(x))n≥0 of polynomials de ned by the biorthogonality relation εzi(d (tn(x))) = n!δi,n for all i, n ∈ N, where d is a delta operator, Z = (zi)i≥0 a sequence of scalars, a...

Originating from automated theorem proving, deduction modulo removes computational arguments from proofs by interleaving rewriting with the deduction process. From a proof-theoretic point of view, deduction modulo defines a generic notion of cut that applies to any first-order theory presented as a rewrite system. In such a setting, one can prove cut-elimination theorems that apply to many theo...

Multiresolution analysis and synthesis for discretetime signals is described in this paper. Concepts of scale and resolution are first reviewed in discrete time. The resulting framework allows one to treat the discrete wavelet transform, octave-band perfect reconstruction filter banks, and pyramid transforms from a unified standpoint. This approach is very close to previous work on multiresolut...