In [B. Han and Z. Shen, SIAM J. Math. Anal., 38 (2006), 530–556], a family of univariate short support Riesz wavelets was constructed from uniform B-splines. A bivariate spline Riesz wavelet basis from the Loop scheme was derived in [B. Han and Z. Shen, J. Fourier Anal. Appl., 11 (2005), 615–637]. Motivated by these two papers, we develop in this article a general theory and a construction meth...

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 of the conditions can be weakened. However, if the eigenvalues (co...

We develope a local theory for frames on finite dimensional Hilbert spaces. We show that for every frame (fi) m i=1 for an n-dimensional Hilbert space, and for every ǫ > 0, there is a subset I ⊂ {1, 2, . . . ,m} with |I| ≥ (1 − ǫ)n so that (fi)i∈I is a Riesz basis for its span with Riesz basis constant a function of ǫ, the frame bounds, and (‖fi‖) m i=1 , but independent of m and n. We also con...

In this paper, we introduce the concept of $g$-dual frames for Hilbert $C^{*}$-modules, and then the properties and stability results of $g$-dual frames  are given.  A characterization of $g$-dual frames, approximately dual frames and dual frames of a given frame is established. We also give some examples to show that the characterization of $g$-dual frames for Riesz bases in Hilbert spaces is ...

In this letter, the two wavelet families, biorthogonal and Riesz bases are introduced. Biorthogonality for two possible decompositions in these bases, The Riesz stability implies that there exist such that Biorthogonal wavelet bases are related to multiresolution approximations. The direct consequence of the above derivation is tradeoff made between the support size of a wavelet and its number ...

The concept of an MV-algebra was introduced by Chang [4] as an algebraic basis for many-valued logic. It turned out that MV-algebras are a subclass of a more general class of effect algebras [7, 6]. Namely, MV-algebras are in one-to-one correspondence with lattice ordered effect algebras satisfying the Riesz decomposition property [2], the latter are called MV-effect algebras. In the study of c...

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.