Some Tauberian theorems for the weighted mean methods of summability

Some Tauberian conditions obtained through weighted generator sequences

“…Recently, the concept of weighted generator sequence of a sequence has been introduced by Çanak and Totur [1]. They proved that certain conditions given in terms of weighted generator sequences are Tauberian conditions such that (1.3) implies (1.4).…”

“…The di erence between and its -th weighted mean (1) , ( ), which is called the weighted Kronecker identity, is given by the identity…”

“…The -th weighted mean of ( ) is de ned by (1) , (see [10]). If = 1 for all nonnegative , then the statistically ( , )-summability is reduced to statistically ( , 1)-summability.…”

“…Since ( ) is statistically ( , )-summable to , we have st − lim (1) , = , and, by Lemma 2.1, we have st − lim(2) , = . It follows from the weighted Kronecker identity that st − lim(1) , (Δ ) So…”

On Tauberian theorems for statistical weighted mean method of summability

“…Let u = (u n ) be a sequence of real numbers. The weighted means of (u n ) are defined by σ (1) n,p (u) := 1 P n n k=0 p k u k for all nonnegative integers n. The difference between u n and σ (1) n,p (u) which is called the weighted Kronecker identity is given by the identity…”