Vol 215, No 3 (2024)

The closed linear span of the Rademacher functions in L2[0,1] contains functions with arbitrarily large distribution, provided that the ratio of this distribution to the distribution of a standard normal variable tends to zero. A similar result is also obtained for some classes of Λ(2)-spaces.

We develop a quantified version of the propositional modal logic BK from an article by Odintsov and Wansing, which is based on the (non-modal) Belnap–Dunn system; we denote this version by QBK. First, by using the canonical model method we prove that QBK, as well as some important extensions of it, is strongly complete with respect to a suitable possible world semantics. Then we define translations (in the spirit of Gödel–McKinsey–Tarski) that faithfully embed the quantified versions of Nelson's constructive logics into suitable extensions of QBK. In conclusion, we discuss interpolation properties for QBK-extensions.

Gabor frames generated by the Gaussian function are considered. The localization of the window functions of dual frames is estimated in terms of the uncertainty constants, it its dependence on the relation between the parameters of the time-frequency window and the degree of overcompleteness. It is shown that localization worsens rapidly with the increasing disproportion in the parameters of the window. On the other hand, the higher the system of functions forming the frame is overdetermined, the better the window function of the dual frame is localized. For a tight frame the localization of the window function with the same set of parameters is much better than that for the dual frame. This problem is closely related to the problem of interpolation by we have uniform shifts of the Gaussian function. Both the nodal interpolation function and the window function of the dual frame are constructed from the same coefficients. These coefficients play an important role also in the derivation of formulae for the uncertainty constants. This is why their properties related to sign alternation and the monotonicity of decrease of the absolute value are considered in the paper.

A point on the surface of a convex body and a supporting plane to the body at this point are under consideration. A plane parallel to this supporting plane and cutting off part of the surface is drawn. The limiting behaviour of the cut-off part of the surface as the cutting plane approaches the point in question is investigated. More precisely, the limiting behavior of the appropriately normalized surface area measure in S2 generated by this part of the surface is studied. The cases when the point is regular and singular (a conical or a ridge point) are considered. The supporting plane can be positioned in different ways with respect to the tangent cone at the point: its intersection with the cone can be a vertex, a line (if a ridge point is considered), a plane angle (which can degenerate into a ray or a half-plane), or a plane (if the point is regular and, correspondingly, the cone degenerates into a half-space). In the case when the intersection is a ray, the plane can be tangent (in a one- or two-sided manner) or not tangent to the cone.
It turns out that the limiting behaviour of the measure can be different. In the case when the intersection of the supporting plane and the cone is a vertex or in the case of a (one- or two-sided) tangency, the weak limit always exists and is uniquely determined by the plane and the cone. In the case when the intersection is a line or a ray with no tangency, there may be no limit at all. In this case all possible weak partial limits are characterized