Let A ⊂ L2(R) be at most countable, and p, q ∈ N. We characterize various frame-properties for Gabor systems of the form G(1, p/q,A)= {e2 g(x − np/q) : m, n ∈ Z, g ∈ A} in terms of the corresponding frame properties for the row vectors in the Zibulski–Zeevi matrix. This extends work by [Ron and Shen, Weyl–Heisenberg systems and Riesz bases in L2(R d). Duke Math. J. 89 (1997) 237–282], who consi...

By the Riesz representation theorem, for every linear functional F : C[0; 1] → R there is a function g : [0; 1] → R of bounded variation such that F (h) = ∫ hdg (h ∈ C[0; 1]) . A computable version is proved in [Lu and Weihrauch(2007)]: a function g can be computed from F and its norm, and F can be computed from g and an upper bound of its total variation. In this article we present a much more...

We study closed subspaces of L(X ), where (X , μ) is a σ-finite measure space, that are invariant under the unitary representation associated to a measurable action of a discrete countable LCA group Γ on X . We provide a complete description for these spaces in terms of range functions and a suitable generalized Zak transform. As an application of our main result, we prove a characterization of...

In this paper, we give some sufficient conditions under which perturbations preserve Hilbert frames and near-Riesz bases. Similar results are also extended to frame sequences, Riesz sequences and Schauder frames. It is worth mentioning that some of our perturbation conditions are quite different from those used in the previous literature on this topic.

We construct a homeomorphism between the compact regular locale of integrals on a Riesz space and the locale of (valuations) on its spectrum. In fact, we construct two geometric theories and show that they are biinterpretable. The constructions are elementary and mostly consist of explicit manipulations on a distributive lattice associated to a given Riesz space.

In this paper, we consider the Riesz transform of higher order associated with the harmonic oscillator [Formula: see text], where Δ is the Laplacian on [Formula: see text]. Moreover, the boundedness of Riesz transforms of higher order associated with Hermite functions on the Hardy space is proved.

Abstract We provide the conditions for boundedness of Bochner–Riesz operator acting between two different Grand Lebesgue Spaces. Moreover we obtain a lower estimate constant appearing in Lebesgue–Riesz norm estimation and investigate convergence approximation spaces.

The purpose of this note is to announce a number of results concerning the various approaches to fractional integration on the real line E as due to H. Weyl [9], M. Riesz [7], W. Feller [4] and G. O. Okikiolu [ô]. Our principal contributions are on extensions of theorems of J. L. B. Cooper [3], on the interchange of the operations of fractional integration (differentiation) and the Hilbert tran...

The aim of this paper is to prove a generalization of a well-known convexity theorem of M. Riesz [8]. The Riesz theorem was originally deduced by "real-variable" techniques. Later, Thorin [10], Tamarkin and Zygmund [9], and Thorin [ll] introduced convexity properties of analytic functions in their study of Riesz's theorem. These ideas were put in especially suggestive form by A. P. Calderon and...