The space D(R) of test functions (infinitely differentiable functions with compact support) is an important example of a non-metrizable (LF)-space, and the domain of distributions (generalized functions). E. Bishop suggested in [1, Appendix A] and [2, Chapter 7, Notes] that the completeness of D(R) and the weak completeness of its dual space would not hold in Bishop’s constructive mathematics. ...

In this paper we analyze the approximation of stable linear time-invariant systems, like the Hilbert transform, by sampling series for bandlimited functions in the Paley–Wiener space PW π . It is known that there exist systems and functions such that the approximation process is weakly divergent, i.e., divergent for certain subsequences. Here we strengthen this result by proving strong divergen...

We investigate the computable content of the Uniform Boundedness Theorem and of the closely related Banach-Steinhaus Theorem. The Uniform Boundedness Theorem states that a pointwise bounded sequence of bounded linear operators on Banach spaces is also uniformly bounded. But, given the sequence, can we also effectively find the uniform bound? It turns out that the answer depends on how the seque...

H. Steinhaus has asked whether inside each acute triangle there is a point from which perpendiculars to the sides divide the triangle into three parts of equal areas. We present two solutions of Steinhaus’ problem. The n-dimensional case of Theorem 1 below was proved in [6], see also [2] and [4, Theorem 2.1, p. 152]. For an earlier mass-partition version of Theorem 1, for bounded convex masses ...

A Steinhaus matrix is a binary square matrix of size n which is symmetric, with diagonal of zeros, and whose upper-triangular coefficients satisfy ai,j = ai−1,j−1+ai−1,j for all 2 6 i < j 6 n. Steinhaus matrices are determined by their first row. A Steinhaus graph is a simple graph whose adjacency matrix is a Steinhaus matrix. We give a short new proof of a theorem, due to Dymacek, which states...

We show that a real sequence x is convergent if and only there exist regular matrix A an Fσδ-ideal I on N such the set of subsequences y for which Ay I-convergent second Baire category. This includes cases where ideal asymptotic density zero sets, Banach finite sets. The latter recovers old result given by Keogh Petersen in (1958) [26]. Our proofs are different nature rely recent results contex...

The conclusion of the classical ham sandwich theorem of Banach and Steinhaus may be strengthened: there always exists a common bisecting hyperplane that touches each of the sets, that is, intersects the closure of each set. Hence, if the knife is smeared with mayonnaise, a cut can always be made so that it will not only simultaneously bisect each of the ingredients, but it will also spread mayo...