Paley and Wiener have formulated a criterion for a set of functions {gn} to be "near" a given orthonormal set {/w}. The interest of this criterion is that it guarantees the set {gn} to have expansion properties similar to an orthonormal set. In particular, they show that the set \gn) approximately satisfies Parseval's formula. In the first part of this paper we show that, conversely, if a set {...

We describe some constructions of orthonormal lattices in totally real subfields of cyclotomic fields, obtained by endowing their ring of integers with a trace form. We also describe constructions of quaternion division algebras over such fields. Orthonormal lattices and quaternion division algebras over totally real fields find use in wireless networks in ultra wideband communication, and we d...

We construct a uniformly bounded orthonormal almost greedy basis for L p ([0, 1]), 1 < p < ∞. The example shows that it is not possible to extend Orlicz's theorem, stating that there are no uniformly bounded orthonormal unconditional bases for L p ([0, 1]), p = 2, to the class of almost greedy bases.

Any non-complete orthonormal system in a Hilbert space can be transformed into basis by small perturbations.

We investigate properties of subspaces of L2 spanned by subsets of a finite orthonormal system bounded in the L∞ norm. We first prove that there exists an arbitrarily large subset of this orthonormal system on which the L1 and the L2 norms are close, up to a logarithmic factor. Considering for example the Walsh system, we deduce the existence of two orthogonal subspaces of L2 , complementary to...

We use the matrix-valued Fejér–Riesz lemma for Laurent polynomials to characterize when a univariate shift-invariant space has a local orthonormal shift-invariant basis, and we apply the above characterization to study local dual frame generators, local orthonormal bases of wavelet spaces, and MRA-based affine frames. Also we provide a proof of the matrixvalued Fejér–Riesz lemma for Laurent pol...