Let φ be a compactly supported refinable function in L2(R) such that the shifts of φ are stable and φ̂(2ξ) = â(ξ)φ̂(ξ) for a 2π-periodic trigonometric polynomial â. A wavelet function ψ can be derived from φ by ψ̂(2ξ) := e−iξ â(ξ + π)φ̂(ξ). If φ is an orthogonal refinable function, then it is well known that ψ generates an orthonormal wavelet basis in L2(R). Recently, it has been shown in the liter...

We study linear combinations of exponentials en, λn ∈ Λ in the case where the distance between some points λn tends to zero. We suppose that the sequence Λ is a finite union of uniformly discrete sequences. In (Avdonin and Ivanov, 2001), necessary and sufficient conditions were given for the family of divided differences of exponentials to form a Riesz basis in space L(0, T ). Here we prove tha...

We consider the natural generating system for a cyclic subspace of a Hilbert space generated by a dual integrable unitary representation of a countable abelian group. We prove, under mild hypothesis, that whenever the generating system is a quasi-greedy basis it must also be an unconditional Riesz basis. A number of applications to Gabor systems and to general Vilenkin systems are considered. I...

Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a Riesz basis (or unconditional basis) of the Hilbert space. Furthermore, we show that none the conditions can be weakened.

We introduce Riesz Logic, whose models are abelian lattice ordered groups, which generalise Riesz spaces (vector lattices), and show soundness and completeness. Our motivation is to provide a logic for distributional semantics of natural language, where words are typically represented as elements of a vector space whose dimensions correspond to contexts in which words may occur. This basis prov...

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...

In this paper we investigate a new notion of bases in Hilbert spaces and similar to fusion frame theory we introduce fusion bases theory in Hilbert spaces. We also introduce a new denition of fusion dual sequence associated with a fusion basis and show that the operators of a fusion dual sequence are continuous projections. Next we dene the fusion biorthogonal sequence, Bessel fusion basis, Hil...

Various equivalent conditions under which a frame is a Riesz basis of a separable Hilbert space are known. See [8] among others, for instance. We add two new conditions to the list. They are inspired by the projection method proposed in [2], which approximates frame coefficients by using finite subsets of a frame. Our main approach is to transcribe anything involving a frame in a Hilbert space ...

A sequence of vectors {f1, f2, f3, . . . } in a separable Hilbert space H is said to be a Schauder basis for H if every element f ∈ H has a unique norm-convergent expansion f = ∑ cnfn. If, in addition, there exist positive constants A and B such that A ∑ |cn| ≤ ∥∥∥∑ cnfn∥∥∥2 ≤ B∑ |cn|, then we call {f1, f2, f3, . . . } a Riesz basis. In the first half of this paper, we show that every Schauder ...