β-statistical core and ideal core of double sequences in 2-normed spaces via RH-regular families

Abstract

We investigated geometric limit sets of double sequences in finite dimensional 2-normed spaces when negligibility of index sets was measured by a density induced by a nonnegative Robison-Hamilton (RH)-regular family 3 of four-dimensional matrices. First, using the 3-density, we defined 3-statistical limit superior and limit inferior for the scalar reductions generated by the seminorms x 7 -> Hx, uH, and we studied the associated real cluster behavior. Next, we introduced 3-statistical cluster points and the 3-statistical core of a double sequence as the intersection of all closed convex sets that contained the sequence outside a 3-density zero set. We obtained a disk-type representation of the core via intersections of sets of the form {x is an element of X : Hx-z, uH <= r}, where the radii were governed by 3-statistical lim sup. We also proved a Knopp-type inclusion theorem: for a family of transforms satisfying a natural 3-regularity condition, the Knopp core of each transform was contained in the 3statistical core of the original sequence. Finally, replacing 3-density zero sets by a strongly admissible ideal I2 on N & times;N, we defined ideal cores, established disk representations, compared the density-based and ideal cores, and identified an explicit condition under which the I2-core and the I & lowast;2-core coincided.