We discuss the reproducing kernel structure in shift-invariant spaces and the weighted shift-invariant spaces, and obtain the reconstruction formula in time-warped weighted shift-invariant spaces, then apply them to a spline subspace. In the spline subspace, we give a reconstruction formula in a time-warped spline subspace. 1. Introduction. The problem of reconstruction of a function f has been...

We prove that it is consistent with ZFC every unital endomorphism of the Calkin algebra $${\cal Q}(H)$$ unitarily equivalent to an which liftable a B}(H)$$ . use this result classify all endomorphisms up unitary equivalence by Fredholm index image unilateral shift. As further application, we show class C*-algebras embed into not closed under tensor product nor countable inductive limit.

In a regular category $$\mathbb {E}$$ , the direct image along epimorphism f of preorder is not in general. Set, its best approximation then cocartesian above f. category, existence such S actually equivalent to supremum $$R[f]\vee S$$ among preorders. We investigate here some conditions ensuring these images or equivalently suprema. They apply two very dissimilar contexts: any topos with supre...

We find necessary and sufficient conditions for a complete local (Noetherian) ring containing the rationals to be completion of countable excellent domain. Furthermore, we noncatenary domain, as well it unique factorization

We study the set of depths of relative algebras of countable Boolean algebras, in particular the extent to which this set may not be downward closed within the countable ordinals for a fixed countable Boolean algebra. Doing so, we exhibit a structural difference between the class of arbitrary rank countable Boolean algebras and the class of rank one countable Boolean algebras.

Given a countable set X (usually taken to be N or Z), an infinite permutation π of X is a linear ordering ≺π of X , introduced in [6]. This paper investigates the combinatorial complexity of infinite permutations on N associated with the image of uniformly recurrent aperiodic binary words under the letter doubling map. An upper bound for the complexity is found for general words, and a formula ...

In recent years, much work in descriptive set theory has been focused on the Borel complexity of naturally occurring classification problems, in particular, the study of countable Borel equivalence relations and their structure under the quasi-order of Borel reducibility. Following the approach of Louveau and Rosendal for the study of analytic equivalence relations, we study countable Borel qua...

A theory formulated in a countable predicate calculus can have at most 2א0 nonisomorphic countable models. In 1961 R. L. Vaught [9] conjected that if such a theory has uncountably many countable models, then it has exactly 2א0 countable models. This would of course follow immediately if one assumed the continuum hypothesis to be true. Almost ten years later, M. Morley [5] proved that if a count...

Reproducing systems of functions such as the wavelet and Gabor systems have been particularly successful in a variety of applications from both mathematics and engineering. In this chapter, we review a number of recent results in the study of such systems and their generalizations developed by the authors and their collaborators. We first describe the unified theory of reproducing systems. This...