Tauberian theorems for the product of weighted and cesaro summability methods of double sequences
Küçük Resim Yok
Tarih
2019
Yazarlar
Dergi Başlığı
Dergi ISSN
Cilt Başlığı
Yayıncı
Maltepe Üniversitesi
Erişim Hakkı
CC0 1.0 Universal
info:eu-repo/semantics/openAccess
info:eu-repo/semantics/openAccess
Özet
A double sequence u = (umn) is called convergent in Pringsheim’s sense (in short P-convergent) to s [? ], if for a given ? > 0 there exists a positive integer N0 such that |umn ? s| < ? for all nonnegative integers m, n ? N0. (C, 1, 1) means of (umn) are defined by ? (11) mn (u) := 1 (m + 1)(n + 1) ?m i=0 ?n j=0 ui j for all nonnegative integers m and n. Similarly, (C, 1, 0) and (C, 0, 1) means of (umn) are defined respectively by ? (10) mn (u) := 1 m + 1 ?m i=0 uin, ?(01) mn (u) := 1 n + 1 ?n j=0 um j for all nonnegative integers m and n. A sequence (umn) is said to be (C, ?, ?) summable to s if lim m,n?? ? (??) mn (u) = s, where (?, ?) = (1, 1), (1, 0) and (0, 1). In this case, we write umn ? s (C, ?, ?). Let p := {pk} ? k=0 and q := {ql} ? l=0 be sequences of nonnegative real numbers with p0, q0 > 0 such that Pm := ?m k=0 pk , 0 for all m ? 0 and Qn := ?n l=0 ql , 0 for all n ? 0. The weighted means t (??) mn of a double sequence (umn), in short, the ( N, p, q; ?, ?) means, are defined respectively by tmn(u) = t (11) mn (u) := 1 PmQn ?m k=0 ?n l=0 pkqlukl, t (10) mn (u) = 1 Pm ?m k=0 pkukn, t (01) mn (u) = 1 Qn ?n l=0 qluml where m, n ? 0. A sequence (umn) is said to be summable by the weighted mean method determined by the sequences p and q, in short, summable (N, p, q; ?, ?) where (?, ?) = (1, 1), (1, 0), (0, 1) if lim m,n?? t (??) mn = s. In this case, we write umn ? s ( N, p, q; ?, ?) . The product of ( N, p, q; 1, 1 ) and (C, ?, ?) summability is defined by ( N, p, q; 1, 1 ) (C, ?, ?) summability, where (?, ?) = (1, 1), (1, 0), (0, 1).
Açıklama
Anahtar Kelimeler
Tauberian theorems, Double sequences, Cesaro means, Weighted means, Convergence in Pringsheim’s sense, Weighted- ` Cesaro summability
Kaynak
International Conference of Mathematical Sciences
WoS Q Değeri
Scopus Q Değeri
Cilt
Sayı
Künye
Fındık, G., Çanak, İ. (2019). Tauberian theorems for the product of weighted and cesaro summability methods of double sequences. International Conference of Mathematical Sciences. 030018(1)-030018(4).
