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Cauchy multiplication of Euler summable series in ultrametric fields

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Deepa R. , Natarajan P. N. , Srinivasan V.

Commentationes Mathematicae

2013

tom Vol 53, No. 1

73--79

Euler summability method in a complete, non-trivially valued, ultrametric field of the characteristic zero was introduced by Natarajan in [7]. Some properties of the Euler summability method in such fields were studied in [2] and [7]. The purpose of the present note is to continue the study and to prove a pair of theorems on the Cauchy product of Euler summable sequences and series.

Some properties of the matrix class (ℓα, ℓα), 0 < α ⩽ 1

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Natarajan P. N.

Commentationes Mathematicae

2020

tom Vol. 60, No. 1/2

23--36

Throughout the present paper, entries of sequences, infinite series and infinite matrices are real or complex numbers. The class (ℓα, ℓα), 0 < α ⩽ 1, of all infinite matrices transforming sequences in ℓα to sequences in ℓα is characterized. The structure of (ℓα, ℓα), 0 < α ⩽ 1, is then discussed. Following Fridy [Properties of absolute summability matrices, Proc. Amer. Math. Soc. 24 (1970), 583-585], a Steinhaus type result involving the class (ℓα, ℓα) is also proved.