Fındık, GökşenÇanak, İbrahim2024-07-122024-07-122019Fındık, 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).978-0-7354-1816-5https://aip.scitation.org/doi/10.1063/1.5095103https://hdl.handle.net/20.500.12415/1903A 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).enCC0 1.0 Universalinfo:eu-repo/semantics/openAccessTauberian theoremsDouble sequencesCesaro meansWeighted meansConvergence in Pringsheim’s senseWeighted- ` Cesaro summabilityArticle030018-4030018-1