Four dimensional matrix characterization P-convergence fields of summability methods

In 1945 Agnew presented two dimensional matrix characterization of convergent fields. The goal of this paper is to present four dimensional matrix characterization of P-convergent field. This will be accomplished by the presentation of the following multiple dimensional analogues of Agnew's theorem. If A is a four-dimensional multiplicative with multiplier 0, then there are sequences 0=σ0<σ1<σ2<⋯ and 0=ρ0<ρ1<ρ2<⋯ of integers such that each bounded double sequence {xk,l} oscillating so slowly thatP-limm,nmaxσm⩽k,r⩽σm+1;ρn⩽l,s⩽ρn+1xk,l-xr,s=0is A summable to 0 in the Pringsheim sense. In addition, natural implications and variations are also presented.