On Borel convergence of double sequences

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.

Keywords

• Borel convergence,Berezin symbol,Pringsheim's sense,double sequence,reproducing kernel

References

1. Aronzajn, N., Theory of reproducing kernels, Trans. Amer. Math. Soc., 68(1950), 337-404.
2. Berezin, F.A., Covariant and contravariant symbols for operators, Math. USSR-Izv., 6(1972), 1117-1151.
3. Garayev, M.T., Gürdal, M. and Yamanci, U., Berezin symbols and Borel Summability, Quaest. Math., 40(3)(2017), 403-411.
4. Hardy, G.H., On the convergence of certain multiplie series, Proc. Cambridge Philos. Soc., 19 (1916-1919), 86-95.
5. Karaev, M.T. and Zelster, M., On Abel convergence of double sequences, Numer. Funct. Anal. Optim., 31(10)(2010), 1185-1189. Nordgren, E. and Rosenthal, P., Boundary Values of Berezin symbols, Oper. Theory Adv. Appl., 73 (1994), 362-368.
6. Pringsheim, A., Elementare theorie der unendliche doppelreihen . Sitsungs Berichte der Math. Akad. der Wissenschafften zu Münch. Ber., 7(1898), 101-153.
7. Saitoh, S., Theory of reproducing kernels and its applications, Pitman Research Notes in Mathematics Series, v.189, 1988.
8. Sawyer, B. and Watson, B., Borel's Methods of Summability: Theory and Applications, Oxford University Press Inc., New York, 1994.

9. Yamancı, U. and Gürdal, M., Statistical convergence and operators on Fock space, New York J. Math., 22 (2016), 199-207.
10. Zelster, M., Investigation methods for summability of double sequences, Ph.D thesis, Tallin, 2001.