We introduce Euler summability method for sequences of fuzzy numbers and state a Tauberian theorem concerning Euler summability method, of which proof provides an alternative to that of K. Knoop[Über das Eulersche Summierungsverfahren II, Math. Z. 18 (1923)] when the sequence is of real numbers. As corollaries, we extend the obtained results to series of fuzzy numbers.
AltayB. and BaşarF. and
MursaleenM., On the Euler sequence spaces which include the spaces ℓp and ℓ∞, Information Sciences176 (2006), 1450–1462.
2.
AltinY., MursaleenM. and AltinokH., Statistical summability (C; 1)-for sequences of fuzzy real numbers and a Tauberian theorem, Journal of Intelligent and Fuzzy Systems21 (2010), 379–384.
3.
AltinokH. and ÇolakR. and
AltinY., On the class of λ-statistically convergent difference sequences of fuzzy numbers, Soft Computing16(6) (2012), 1029–1034.
4.
BedeB. and GalS.G., Almost periodic fuzzy-number-valued functions, Fuzzy Sets and Systems147 (2004), 385–403.
5.
BoosJ., Classical and Modern Methods in Summability, 2000–Oxford University Press.
6.
BoydJ.P., Sum-accelerated pseudospectral methods: The Euler-accelerated sine algorithm, Applied Numerical Mathematics7 (1991), 287–296.
7.
BoydJ.P., A proof, based on the Euler sum acceleration, of the recovery of an exponential (geometric) rate of convergence for the Fourier series of a function with Gibbs phenomenon, in Springer, Spectral and High Order Methods for Partial Differential Equations2011–.
8.
ChandraP., Multipliers for the absolute Euler summability of Fourier series, Proceedings of the Indian Academy of Sciences-Mathematical Sciences111(2) (2001), 203–219.
9.
Çanakİ., On the Riesz mean of sequences of fuzzy real numbers, Journal of Intelligent and Fuzzy Systems26(6) (2014), 2685–2688.
10.
Çanakİ., On Tauberian theorems for Cesàro summability of sequences of fuzzy numbers, Journal of Intelligent and Fuzzy Systems30(5) (2016), 2657–2662.
11.
DikshitG.D., Absolute euler summability of fourier series, Journal of Mathematical Analysis and Applications220 (1998), 268–282.
12.
DrummondJ.E., Convergence speeding, convergence and summability, Journal of Computational and Applied Mathematics11(2) (1984), 145–159.
13.
EstradaR. and VindasJ., Exterior Euler summability, Journal of Mathematical Analysis and Applications388 (2012), 48–60.
14.
KabardovM.M., On analytic continuation of a hypergeometric series using the euler-knopp transformation, Vestnik St Petersburg University: Mathematics42(3) (2009), 169–174.
15.
KnoppK., Über das eulersche summierungsverfahren, Mathematische Zeitschrift15 (1922), 226–253.
16.
KnoppK., Über das eulersche summierungsverfahren II, Mathematische Zeitschrift18 (1923), 125–156.
17.
MeronenO. and TammeraidI., Generalized euler-knopp method and convergence acceleration, Mathematical Modelling and Analysis11(1) (2006), 87–94.
18.
MursaleenM. and BaşarF. and
AltayB., On the Euler sequence spaces which include the spaces ℓp and ℓ∞, Nonlinear Analysis65 (2006), 707–717.
OouraT., A generalization of the continuous Euler transformation and its application to numerical quadrature, Journal of Computational and Applied Mathematics157 (2003), 251–259.
21.
ÖnderZ., SezerS.A. and Çanakİ., A Tauberian theorem for the weighted mean method of summability of sequences of fuzzy numbers, Journal of Intelligent and Fuzzy Systems28 (2015), 1403–1409.
22.
SezerS.A. and Çanakİ., Power series methods of summability for series of fuzzy numbers and related Tauberian Theorems, Soft Copmuting (2015). Doi: 10.1007/s00500-015-1840-0
23.
SondowJ., Analytic continuation of Riemann’s zeta function and values at negative integers via Euler’s transformation of series, Proceedings of the American Mathematical Society120(2) (1994), 421–424.
24.
StojakovićM. and StojakovićZ., Addition and series of fuzzy sets, Fuzzy Sets and Systems83 (1996), 341–346.
25.
StojakovićM. and StojakovićZ., Series of fuzzy sets, Fuzzy Sets and Systems160 (2009), 3115–3127.
26.
SubrahmanyamP.V., Cesàro summability of fuzzy real numbers, Journal of Analysis7 (1999), 159–168.
27.
TaloÖ. and ÇakanC., On the Cesàro convergence of sequences of fuzzy numbers, Applied Mathematics Letters25 (2012), 676–681.
28.
TaloÖ. and BaşarF., On the slowly decreasing sequences of fuzzy numbers, Abstract and Applied Analysis (2013), 1–7.
29.
TaloÖ., KadakU. and BaşarF., On series of fuzzy numbers, Contemporary Analysis and Applied Mathematics4(1) (2016), 132–155. 573 207.
30.
TripathyB.C. and BaruahA., Nörlund and Riesz mean of sequences of fuzzy real numbers, Applied Mathematics Letters23 (2010), 651–655.
31.
TripathyB.C. and DasP.C., On convergence of series of fuzzy real numbers, Kuwait Journal of Science and Engineering39(1A) (2012), 57–70.
32.
TripathyB.C. and SenM., On fuzzy I-convergent difference sequence space, Journal of Intelligent and Fuzzy Systems25(3) (2013), 643–647.
33.
TripathyB.C., BrahaN.L. and DuttaA.J., A new class of fuzzy sequences related to the ℓp space defined by Orlicz function, Journal of Intelligent and Fuzzy Systems26(3) (2014), 1273–1278.
34.
WalkH., Almost sure Cesàro and Euler summability of sequences of dependent random variables, Archiv der Mathematik89 (2007), 466–480.
35.
YavuzE. and ÇoşkunH., Tauberian theorems for Abel summability of sequences of fuzzy numbers, AIP Conference Proceedings1676 (2015). doi: 10.1063/1.4930505
36.
YavuzE. and ÇoşkunH., On the logarithmic summability method for sequences of fuzzy numbers, Soft Computing (2016). doi: 10.1007/s00500-016-2156-4
37.
ZadehL.A., Fuzzy sets, Information and Control8 (1965), 29–44.
38.
ZubkovA.M. and SerovA.A., A complete proof of universal inequalities for the distribution function of the binomial law, Theory of Probability & Its Applications57(3) (2013), 539–544.
