Abstract
We present theorems showing when the discrete Abel mean and the Abel summability method are equivalent for bounded sequences and when two discrete Abel means are equivalent for bounded sequences.
Highlights
Introduction and notationThe well-known Abel summability method is a sequence-to-function transformation which is defined as follows: for a sequence s := {sn} of complex numbers, define ∞f (x) := (1 − x) skxk, k=0 (1.1)for all x for which the series converges
The discrete Abel mean is a sequence-to-sequence transformation given by the summability matrix Aλ whose nkth entry is
The main result of this section is that Aλ is equivalent to the Abel method for bounded sequences provided that λ(n + 1)/λ(n) → 1
Summary
Introduction and notationThe well-known Abel summability method is a sequence-to-function transformation which is defined as follows: for a sequence s := {sn} of complex numbers, define ∞f (x) := (1 − x) skxk, k=0 (1.1)for all x for which the series converges. The well-known Abel summability method is a sequence-to-function transformation which is defined as follows: for a sequence s := {sn} of complex numbers, define If f (x) exists for each x ∈ (0, 1) and limx→1− f (x) = L, the sequence s is Abel summable to L. The discrete Abel mean is a sequence-to-sequence transformation given by the summability matrix Aλ whose nkth entry is
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