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Öğe Cesàro integrability and tauberian theorems in quantum calculus(Sciendo, 2018) Çanak İ.; Totur Ü.; Sezer S.A.In this paper, we show some properties regarding Cesàro integrability method in quantum calculus. Moreover, we extend and generalize q-analogues of some classical Tauberian theorems given for Cesàro integrability by Fitouhi and Brahim [J. Nonlinear Math. Phys., 14 (3) (2007), 324–340]. © 2018, Sciendo. All rights reserved.Öğe Some tauberian conditions for the weighted mean method of summability(Sciendo, 2017) Totur Ü.; Çanak İ.Let p = (pn) be a sequence of nonnegative numbers and Pn:= ?n k=0 pk › ? as n › ?. Let the weighted general control modulo of the oscillatory behavior of integer order m ? 1 of a sequence (un) be denoted by (?n,p (m) (u)). We prove that if the weighted generator sequence of a sequence u = (un) of real numbers is summable to a finite number by the weighted mean method, (?(1) n,p(?(m-1) (u))) is increasing, the conditions (Formula presented) are satisfied, and certain conditions on (pn) are hold, then (un) is slowly oscillating. © 2018, Universitatii Al.I.Cuza din Iasi. All rights reserved.Öğe Some Tauberian conditions under which convergence follows from (C, 1, 1, 1) summability(Springer Science and Business Media B.V., 2020) Totur Ü.; Çanak İ.Given a real-valued integrable function f(x, y, z) which is integrable over [ 0 , ?) × [ 0 , ?) × [ 0 , ?) , let s(x, y, z) denote its integral over [ 0 , x] × [ 0 , y] × [ 0 , z] and let ?(x, y, z) denote its (C, 1, 1, 1) mean, the average of s(x, y, z) over [ 0 , x] × [ 0 , y] × [ 0 , z] , where x, y, z> 0. We give one-sided Tauberian conditions of Landau and Hardy type under which convergence of s(x, y, z) follows from (C, 1, 1, 1) summability of s(x, y, z). We obtain convergence of s(x, y, z) from its (C, 1, 1, 1) summability provided that s(x, y, z) is slowly oscillating in certain senses. Furthermore, we extend a Tauberian theorem given by Móricz (Stud Math 138(1):41–52, 2000) for improper double integrals to improper triple integrals. © 2019, Forum D'Analystes, Chennai.Öğe Some Tauberian theorems for iterations of Hölder integrability method(Birkhauser Verlag AG, 2019) Önder Z.; Çanak İ.Let f be a real or complex-valued function on [1 , ?) which is continuous over every finite interval [1, x) for 1 < x< ?. We set s(x):=?1xf(t)dt and define ? k (s(x)) by ?k(s(x))={1x?1x?k-1(s(t))dt,k?1s(x),k=0for each nonnegative integer k. An improper integral ?1?f(x)dxis said to be integrable to a finite number µ by the k-th iteration of Hölder or Cesàro mean method of order one, or for short, the (H, k) integrable to µ if limx›??k(s(x))=µ.In this case, we write s(x)›µ(H,k). It is clear that the (H, k) integrability method reduces to the ordinary convergence for k= 0 and the (H, 1) integrability method is (C, 1) integrability method. It is known that lim x › ? s(x) = µ implies lim x › ? ? k (s(x) ) = µ. But the converse of this implication is not true in general. In this paper, we obtain some Tauberian conditions for the iterations of Hölder integrability method under which the converse implication holds. © 2019, Springer Nature Switzerland AG.Öğe Statistical Extensions of Some Classical Tauberian Theorems for Cesàro Summability of Triple Sequences(Birkhauser Verlag AG, 2016) Çanak İ.; Önder Z.; Totur Ü.In (Çanak and Totur, Georgian Math J 23(1):33–42, 2016), Çanak and Totur have extended some classical Tauberian theorems for single sequences to triple sequences. In (Fridy and Khan, Proc Am Math Soc 128:2347–2355, 2000), Fridy and Khan obtained statistical extensions of some classical Tauberian theorems. The concept of statistical convergence for triple sequences has been introduced by Şahiner et al. (Selcuk J Appl Math 8(2):49–55, 2007). In this paper, we investigate Tauberian conditions for the statistical convergence and statistical (C,1,1,1) summability of triple sequences. © 2016, Springer International Publishing.Öğe Tauberian conditions for the (C, ?) integrability of functions(Birkhauser Verlag AG, 2017) Totur Ü.; Çanak İ.For a real-valued continuous function f(x) on [ 0 , ?) , we define (Fomula presented.) for x> 0. We say that ?0?f(u)du is (C, ?) integrable to L for some ?> - 1 if the limit lim x › ???(x) = L exists. It is known that lim x › ?s(x) = L implies lim x › ???(x) = L for all ?> - 1. The aim of this paper is twofold. First, we introduce some new Tauberian conditions for the (C, ?) integrability method under which the converse implication is satisfied, and improve classical Tauberian theorems for the (C, ?) integrability method. Next we give short proofs of some classical Tauberian theorems as special cases of some of our results. © 2016, Springer International Publishing.Öğe Tauberian conditions under which convergence follows from cesàro summability of double integrals over r2+(University of Nis, 2019) Fındık G.; Çanak İ.For a real-or complex-valued continuous function f over R2+:= [0, ?) × [0, ?), we denote its integral over [0, u] × [0, v] by s(u, v) and its (C, 1, 1) mean, the average of s(u, v) over [0, u] × [0, v], by ?(u, v). The other means (C, 1, 0) and (C, 0, 1) are defined analogously. We introduce the concepts of backward differences and the Kronecker identities in different senses for double integrals over R2+. We give one-sided and two-sided Tauberian conditions based on the difference between double integral of s(u, v) and its means in different senses for Cesàro summability methods of double integrals over [0, u] × [0, v] under which convergence of s(u, v) follows from integrability of s(u, v) in different senses. © 2019, University of Nis. All rights reserved.Öğe Tauberian conditions underwhich convergence follows from statistical summability byweighted means(Springer Singapore, 2018) Önder Z.; Çanak İ.Let (pn) be a sequence of nonnegative numbers such that p0 > 0 and. Let (sn) be a sequence of real and complex numbers. The weighted mean of (sn) is defined by. We obtain some sufficient conditions, under which the existence of the limit lim sn = µ follows from that of st-lim tn = µ, where µ is a finite number. If (sn) is a sequence of real numbers, then these Tauberian conditions are one-sided. If (sn) is a sequence of complex numbers, these Tauberian conditions are two-sided. These Tauberian conditions are satisfied if (sn) satisfies the one-sided condition of Landau type relative to (Pn) in the case of real sequences or if (sn) satisfies the two-sided condition of Hardy type relative to (Pn) in the case of complex numbers. © Springer Nature Singapore Pte Ltd. 2018.Öğe Tauberian remainder theorems for iterations of methods of weighted means(Academic Publishing House, 2019) Sezer S.A.; Çanak İ.In this paper we prove some Tauberian remainder theorems on ?-bounded sequences for iterations of methods of weighted means. © 2019, Academic Publishing House. All rights reserved.Öğe A Tauberian theorem for the generalized Nörlund summability method(Walter de Gruyter GmbH, 2018) Çanak İ.; Braha N.L.; Totur Ü.Let (pn) and (qn) be any two non-negative real sequences, with R n := ? k = 0 n n p k q n - k ? 0 (n ? N). Let ?? k=0 a k be a series of real or complex numbers with partial sums (sn), and set tn p , q := 1 Rn ?n k = 0 p k qn k sk for n ? N. In this paper, we present the necessary and sufficient conditions under which the existence of the limit limn›? sn = L follows from that of lim n›? tn p , q = L. These conditions are one-sided or two-sided if (sn) is a sequence of real or complex numbers, respectively. © 2018 Walter de Gruyter GmbH, Berlin/Boston.Öğe Tauberian theorems for (N¯ , p, q) summable double sequences of fuzzy numbers(Springer, 2020) Totur Ü.; Çanak İ.In this paper, we define the weighted mean method (N¯ , p, q) of double sequences of fuzzy numbers and give necessary and sufficient Tauberian conditions under which convergence in Pringsheim’s sense of a double sequence of fuzzy numbers follows from its (N¯ , p, q) summability. These conditions are weaker than the weighted analogues of Landau’s conditions and Schmidt’s slow oscillation condition in some senses for two-dimensional case. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.Öğe Tauberian theorems for (N¯ , p, q) summable double sequences of fuzzy numbers(Springer Verlag, 2019) Totur Ü.; Çanak İ.In this paper, we define the weighted mean method (N¯ , p, q) of double sequences of fuzzy numbers and give necessary and sufficient Tauberian conditions under which convergence in Pringsheim’s sense of a double sequence of fuzzy numbers follows from its (N¯ , p, q) summability. These conditions are weaker than the weighted analogues of Landau’s conditions and Schmidt’s slow oscillation condition in some senses for two-dimensional case. © 2019, Springer-Verlag GmbH Germany, part of Springer Nature.Öğe Tauberian theorems for statistically (C, 1, 1) summable double sequences(Birkhauser Verlag AG, 2019) Önder Z.; Çanak İ.In this paper, we obtain some Tauberian conditions in terms of slow oscillation and slow decreasing in certain senses, under which convergence of a double sequence in Pringsheim’s sense follows from its statistical (C, 1, 1) summability. © 2019, Springer Nature Switzerland AG.Öğe Tauberian theorems for the product of Borel and Hölder summability methods(Sciendo, 2017) Çanak İ.In this paper we prove some Tauberian theorems for the product of Borel and Hölder summability methods which improve the classical Tauberian theorems for the Borel summability method. © 2017, Sciendo. All rights reserved.Öğe Two theorems on the product of Abel and Cesàro summability methods(Academic Publishing House, 2015) Erdem Y.; Çanak İ.; Allahverdiev B.P.In this paper, we introduce one-sided Tauberian conditions to recover (C, ?) summability and convergence of a sequence from its (A)(C, ?) summability. © 2015 Academic Publishing House. All rights reserved.
