Four dimensional matrix that induces the double Gibbs phenomenon

Author： Patterson R.   Rhoades B.  

Publisher： Springer Publishing Company

ISSN： 0236-5294

Source： Acta Mathematica Hungarica, Vol.129, Iss.1-2, 2010-10, pp. : 142-152

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Abstract

In 1930 Knopp presented the following matrix characterization for the core of ordinary sequences. If A is a nonnegative regular matrix then the core of [Ax] is contained in the core of [x], provided that [Ax] exists. Patterson in 1999 extended Knopp’s results to double sequences via four dimensional matrices. In a manner similar to the Knopp’s and Patterson’s results we present multidimensional extensions of Bustoz’s singular dimensional Gibbs phenomenon results. These results include a notion of what it means for a four dimensional matrix transformation to induce the double Gibbs phenomenon in s. In addition, necessary and sufficient conditions for a four dimensional matrix to induce the double Gibbs phenomenon is also presented.

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