On sets of points of approximate continuity and ϱ-upper continuity

In the paper some properties of sets of points of approximate continuity and ϱ-upper continuity are presented. We will show that for every Lebesgue measurable set E ⊂ ℝ there exists a function f : ℝ → ℝ which is approximately (ϱ-upper) continuous exactly at points from E. We also study properties of sets of points at which real function has Denjoy property. Some other related topics are discussed.

On Parseval Wavelet Frames via Multiresolution Analyses in

We give a characterization of all Parseval wavelet frames arising from a given frame multiresolution analysis. As a consequence, we obtain a description of all Parseval wavelet frames associated with a frame multiresolution analysis. These results are based on a version of Oblique Extension Principle with the assumption that the origin is a point of approximate continuity of the Fourier transform of the involved refinable functions. Our results are written for reducing subspaces.

Quasiconvexity and Density Topology

We prove that if f : ℝN → ℝ̄ is quasiconvex and U ⊂ ℝN is open in the density topology, then supU ƒ = ess supU f ; while infU ƒ = ess supU ƒ if and only if the equality holds when U = RN: The first (second) property is typical of lsc (usc) functions, and, even when U is an ordinary open subset, there seems to be no record that they both hold for all quasiconvex functions.This property ensures that the pointwise extrema of f on any nonempty density open subset can be arbitrarily closely approximated by values of ƒ achieved on “large” subsets, which may be of relevance in a variety of situations. To support this claim, we use it to characterize the common points of continuity, or approximate continuity, of two quasiconvex functions that coincide away from a set of measure zero.

Closure of dilates of shift-invariant subspaces

Let V be any shift-invariant subspace of square summable functions. We prove that if for some A expansive dilation V is A-refinable, then the completeness property is equivalent to several conditions on the local behaviour at the origin of the spectral function of V, among them the origin is a point of A*-approximate continuity of the spectral function if we assume this value to be one. We present our results also in a more general setting of A-reducing spaces. We also prove that the origin is a point of A*-approximate continuity of the Fourier transform of any semiorthogonal tight frame wavelet if we assume this value to be zero.

Approximate Fuzzy Continuity of Functions

The conventional continuity of a function was further advanced by the concept of approximate continuity introduced by Denjoy to solve some problems of differentiation and integration. According to this new type of continuity, the (classical) continuity conditions may be true not necessarily everywhere, but almost everywhere with respect to some measure, e.g., Borel measure or Lebesgue measure. However, functions that come from real life sources, such as measurement and computation, do not allow, in a general case, to test whether they are continuous or even approximately continuous in the strict mathematical sense. Hence, in this paper, the authors overcome these limitations by introducing and studying the more realistic concept of the approximate fuzzy continuity of functions.

DENJOY'S APPROXIMATE CONTINUITY FOR THE SOLUTIONS OF MULTIVALUED STOCHASTIC DIFFERENTIAL EQUATIONS

In this paper we prove Denjoy's approximate continuity for the solutions of multivalued stochastic differential equations.