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SETSCI - Volume 3 (2018)
ISAS2018-Winter - 2nd International Symposium on Innovative Approaches in Scientific Studies, Samsun, Turkey, Nov 30, 2018

Cauchy Multiplication by Nörlund Summability Generalized Nörlund Summability (ISAS2018-Winter_295)
Şaban Yılmaz1*
1Tokat Gaziosmanpaşa University, Tokat, Turkey
* Corresponding author: saban.yilmaz@gop.edu.tr
Published Date: 2019-01-14   |   Page (s): 1544-1545   |    14     5

ABSTRACT The definition set is called arrays of functions consisting of natural numbers. The sequence of the arrays is called a
convergent sequence to give a real number result, otherwise it is called a divergent array. In 1897, A. Tauber aimed to classify
the conditions under which the sequences are convergent. Tauber emphasized the conditions that provided convergence to his
work. Among the investigations, a remarkable feature is the convergence of divergent knees. In 1905, E. Cesàro, which was
the first to draw attention among these studies, revealed the theory of intergroup transformation. The basic principle of this
transformation was iy to convert the convergent sequence into a convergent sequence and to maintain its limit Bu. This clutch
is called regularity. Cesàro summability provides regularity and diverging some divergent sequences into convergent
sequences. In 1910, L. L. Silverman expressed and proved the theorem in 1913, which transformed the convergent sequence
into a convergent sequence and revealed its conditions. In 1911, M. Riesz defined a new transformation on any series with
positive terms. N. E. Nörlund described the Nörlund summability by making a similar transformation in 1920. In the following
years, generalization of Nörlund summability has emerged. In this study, Cauchy products of Nörlund summability generalized
with Nörlund summability were studied. It has been observed that the Cauchy product of the Nörlund summability generalized
with the Nörlund average transforms to the average of Nörlund.  
KEYWORDS Riesz Summability, Cesàro Summability, Generalied Nörlund Summability
REFERENCES [1] Mears, F.M., Absolute regularity and Nörlund means, Ann. Math.38 (3) (1937), 594-602.
[2] Nesin, A., Analiz II, Nesin Yayıncılık A.Ş., 2011, İstanbul.
[3] Nurcombe, J.R., Limitation and ineffectiveness theorems for generalized Nörlundsummability, Analysis 9 (1989), 347-356.
[4] Petersen, G.M., Regular Matrix Transformations, McGraw-Hill Publishing Company Limited, LondonNew York-Toronto-Sidney, 1966.

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