Abstract
In this paper, we introduce the concept of (Ber)-convergence of bounded double sequences in the Fock space F(C²). We show that every (Ber)-convergent double sequence is Borel convergent. Namely, we prove the following theorem by using the Berezin symbol method: If the {x_{ij}}_{i,j=0}^{∞} is regularly convergent to x, then lim_{k,l→∞}e^{-k-l}∑_{i,j=0}^{∞}x_{ij}((k^{i}t^{j})/(i!j!))=x.
Highlights
Recall that a double sequence fxijg1 i;j=0 is said to be convergent in Pringsheim’s sense [7] if there exists a number x such that xij converges to x as both i and j tend to in...nity independently of one another lim xij = x; i;j!1 that is, if for every " > 0 there exists N = N (") 2 N such that jxij xj < " for every i; j N and x is said to the Pringsheim’s limit of xij
It is obvious that a double sequence is a Cauchy sequence if and only if it is convergent
A double sequence fxijg is bounded if there exists a positive number K such that jxijj K for every i and j, i.e., sup jxijj < 1
Summary
A double sequence fxijg is bounded if there exists a positive number K such that jxijj K for every i and j, i.e., sup jxijj < 1. A double sequence fxijg is said to be regularly convergent if it is convergent in Pringsheim’s sense and the following limits hold: lim xij = xj, (j = 1; 2; :::) and lim xij = xi, (i = 1; 2; :::) : i!1 j!1 Double sequence, Berezin symbol, Pringsheim’s sense, reproducing kernel.
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