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Journal articles on the topic 'Sequences, Series, Summability'

Author: Grafiati

Published: 4 June 2021

Last updated: 30 January 2023

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1

Yildiz, Sebnem. "An absolute matrix summability of infinite series and Fourier series." Boletim da Sociedade Paranaense de Matemática 38, no. 7 (October 14, 2019): 49–58. http://dx.doi.org/10.5269/bspm.v38i7.46594.

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The aim of this paper is to generalize a main theorem concerning weighted mean summability to absolute matrix summability which plays a vital role in summability theory and applications to the other sciences by using quasi-$f$-power sequences.

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2

Özarslan, H. S. "On a new application of quasi power increasing sequences." Carpathian Mathematical Publications 11, no. 1 (June 30, 2019): 152–57. http://dx.doi.org/10.15330/cmp.11.1.152-157.

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In the present paper, absolute matrix summability of infinite series has been studied. A new theorem concerned with absolute matrix summability factors, which generalizes a known theorem dealing with absolute Riesz summability factors of infinite series, has been proved under weaker conditions by using quasi $\beta$-power increasing sequences. Also, a known result dealing with absolute Riesz summability has been given.

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3

Bor, Hüseyin. "On quasi-monotone sequences and their applications." Bulletin of the Australian Mathematical Society 43, no. 2 (April 1991): 187–92. http://dx.doi.org/10.1017/s0004972700028951.

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In this paper using δ-quasi-monotone sequences a theorem on summability factors of infinite series, which generalises a theorem of Mazhar [7] on |C, 1|k summability factors of infinite series, is proved. Also we apply the theorem to Fourier series.

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4

Karakaş, Ahmet. "A new application of almost increasing sequences." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 18, no. 1 (December 1, 2019): 59–65. http://dx.doi.org/10.2478/aupcsm-2019-0005.

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Abstract In this paper, a known result dealing with | ̄N, pn|k summability of infinite series has been generalized to the φ − | ̄N, pn; δ|k summability of infinite series by using an almost increasing sequence.

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5

Yıldız, Şebnem. "An application of quasi-monotone sequences to absolute matrix summability." Filomat 32, no. 10 (2018): 3709–15. http://dx.doi.org/10.2298/fil1810709y.

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Recently, Bor [5] has obtained two main theorems dealing with |?N,pn|k summability factors of infinite series and Fourier series. In the present paper, we have generalized these theorems for |A,?n|k summability method by using quasi-monotone sequences.

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6

Bor, Hüseyin. "Almost increasing sequences and their new applications II." Filomat 28, no. 3 (2014): 435–39. http://dx.doi.org/10.2298/fil1403435b.

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7

Yanetz (Chachanashvili), Sh. "Absolute Summability Factors and Absolute Tauberian Theorems for Double Series and Sequences." gmj 6, no. 6 (December 1999): 591–600. http://dx.doi.org/10.1515/gmj.1999.591.

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Abstract Let 𝐴 and 𝐵 be the linear methods of the summability of double series with fields of bounded summability and , respectively. Let 𝑇 be certain set of double series. The condition 𝑥 ∈ 𝑇 is called 𝐵𝑏-Tauberian for 𝐴 if . Some theorems about summability factors enable one to find new 𝐵𝑏-Tauberian conditions for 𝐴 from the already known 𝐵𝑏-Tauberian conditions for 𝐴.

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8

Yıldız, Şebnem. "On the generalizations of some factors theorems for infinite series and fourier series." Filomat 33, no. 14 (2019): 4343–51. http://dx.doi.org/10.2298/fil1914343y.

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Quite recently, Bor [Quaest. Math. (doi.org/10.2989/16073606.2019.1578836, in press)] has proved a new result on weighted arithmetic mean summability factors of non decreasing sequences and application on Fourier series. In this paper, we establish a general theorem dealing with absolute matrix summability by using an almost increasing sequence and normal matrices in place of a positive non-decreasing sequence and weighted mean matrices, respectively. So, we extend his result to more general cases.

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9

Özarslan, H. S., and A. Keten. "A New Application of Almost Increasing Sequences." Annals of the Alexandru Ioan Cuza University - Mathematics 61, no. 1 (January 1, 2015): 153–60. http://dx.doi.org/10.2478/aicu-2013-0049.

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Abstract Bor has proved a main theorem dealing with | N̄ , pn|k summability factors of infinite series. In this paper, we have generalized this theorem to the φ − |A, pn|k summability factors, under weaker conditions by using an almost increasing sequence instead of a positive monotonic non-decreasing sequence.

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10

Rhoades, B. E., and Ekrem Savaş. "A summability factor theorem for absolute summability involving almost increasing sequences." International Journal of Mathematics and Mathematical Sciences 2004, no. 69 (2004): 3793–97. http://dx.doi.org/10.1155/s0161171204310173.

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11

KARTAL, Bağdagül. "Ces\`aro Summability Involving $\delta$-Quasi-Monotone and Almost Increasing Sequences." Journal of New Theory, no. 41 (December 31, 2022): 94–99. http://dx.doi.org/10.53570/jnt.1185603.

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This paper generalises a well-known theorem on ${\mid{C},\rho\mid}_\kappa$ summability to the $\varphi-{\mid{C},\rho;\beta\mid}_\kappa$ summability of an infinite series using an almost increasing and a $\delta$-quasi monotone sequence.

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12

Bor, Hüseyin. "A new note on generalized absolute Cesàro summability factors." Filomat 32, no. 9 (2018): 3093–96. http://dx.doi.org/10.2298/fil1809093b.

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Quite recently, in [10], we have proved a theorem dealing with the generalized absolute Ces?ro summability factors of infinite series by using quasi monotone sequences and quasi power increasing sequences. In this paper, we generalize this theorem for the more general summability method. This new theorem also includes some new and known results.

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13

Tuncer, A. Nihal. "Factors for absolute Cesaro summability." Tamkang Journal of Mathematics 32, no. 1 (March 31, 2001): 21–25. http://dx.doi.org/10.5556/j.tkjm.32.2001.363.

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In this paper using $ \delta $-quasi-monotone sequences a theorem on $ | C, \alpha, \beta; \delta |_k $ summability factors of infinite series, which generalizes a theorem of Mazhar[6] on $ | C, 1 |_k $ summability factors, has been proved.

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14

Bor, Hüseyin. "On two summability methods." Mathematical Proceedings of the Cambridge Philosophical Society 97, no. 1 (January 1985): 147–49. http://dx.doi.org/10.1017/s030500410006268x.

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Let Σan be a given infinite series with partial sums sn, and rn = nan. By and we denote the nth Cesáro means of order α (α –1) of the sequences (sn) and (rn), respectively. The series Σan is said to be absolutely summable (C, a) with index k, or simply summable |C, α|k, k ≥ 1, if

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15

Bor, Hüseyin. "A new application of quasi monotone sequences and quasi power increasing sequences to factored infinite series." Filomat 31, no. 16 (2017): 5105–9. http://dx.doi.org/10.2298/fil1716105b.

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In this paper, we generalize a known theorem under more weaker conditions dealing with the generalized absolute Ces?ro summability factors of infinite series by using quasi monotone sequences and quasi power increasing sequences. This theorem also includes some new results.

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16

Özarslan, Hikmet Seyhan. "A new result on the quasi power increasing sequences." Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica 19, no. 1 (December 1, 2020): 95–103. http://dx.doi.org/10.2478/aupcsm-2020-0008.

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AbstractThis paper presents a theorem dealing with absolute matrix summability of infinite series. This theorem has been proved taking quasi β-power increasing sequence instead of almost increasing sequence.

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17

Totura, Ümit, and İbrahim Çanak. "On regularly weighted generated sequences." Filomat 31, no. 7 (2017): 2167–73. http://dx.doi.org/10.2298/fil1707167t.

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We introduce a new concept which is called a regularly weighted generated sequence for sequences of real numbers. Moreover, we obtain some Tauberian conditions in terms of regularly weighted generated sequences for the power series method of summability, and generalize some classical Tauberian theorems given by Tietz [Acta Sci. Math. 54 (3-4) 355-365 (1990)].

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18

Yıldız, Şebnem. "A general matrix application of convex sequences to Fourier series." Filomat 32, no. 7 (2018): 2443–49. http://dx.doi.org/10.2298/fil1807443y.

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By using a convex sequence Bor [H. Bor, Local properties of factored Fourier series, Appl. Math. Comp. 212 (2009) 82-85] has obtained a result dealing with local property of factored Fourier series for weighted mean summability. The purpose of this paper is to extend this result to more general cases by taking normal matrices in place of weighted mean matrices.

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19

Bor, Hüseyin. "Quasimonotone and Almost Increasing Sequences and Their New Applications." Abstract and Applied Analysis 2012 (2012): 1–6. http://dx.doi.org/10.1155/2012/793548.

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Recently, we have proved a main theorem dealing with the absolute Nörlund summability factors of infinite series by using -quasimonotone sequences. In this paper, we prove that result under weaker conditions. A new result has also been obtained.

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20

Sahani, Suresh Kumar, Vishnu Narayan Mishra, and Narayan Prasad Pahari. "Some Problems on Approximations of Functions (Signals) in Matrix Summability of Legendre Series." Nepal Journal of Mathematical Sciences 2, no. 1 (April 30, 2021): 43–50. http://dx.doi.org/10.3126/njmathsci.v2i1.36566.

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In this paper, we prove a main theorem dealing the matrix summability of Legendre series using non-negative monotonic non-increasing sequences of function. This paper is more general than [9], [12] and [22].

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21

Buntinas, Martin, and Naza Tanović-Miller. "Strong Boundedness and Strong Convergence in Sequence Spaces." Canadian Journal of Mathematics 43, no. 5 (October 1, 1991): 960–74. http://dx.doi.org/10.4153/cjm-1991-053-8.

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AbstractStrong convergence has been investigated in summability theory and Fourier analysis. This paper extends strong convergence to a topological property of sequence spaces E. The more general property of strong boundedness is also defined and examined. One of the main results shows that for an FK-space E which contains all finite sequences, strong convergence is equivalent to the invariance property E = ℓ ν0. E with respect to coordinatewise multiplication by sequences in the space ℓν0 defined in the paper. Similarly, strong boundedness is equivalent to another invariance E = ℓν.E. The results of the paper are applied to summability fields and spaces of Fourier series.

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22

Bachmayr, Markus, Albert Cohen, and Giovanni Migliorati. "Sparse polynomial approximation of parametric elliptic PDEs. Part I: affine coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 51, no. 1 (December 23, 2016): 321–39. http://dx.doi.org/10.1051/m2an/2016045.

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We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the affine form a = a(y) = ā + ∑j≥1yjψj for some parameter vector y = (yj)j ≥ 1 ∈ U = [−1,1]N. We study the summability properties of polynomial expansions of the solution map y → u(y) ∈ V := H01(D) . We consider both Taylor series and Legendre series. Previous results [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47] show that, under a uniform ellipticity assuption, for any 0 <p< 1, the ℓp summability of the (∥ψj∥L∞)j ≥ 1 implies the ℓp summability of the V-norms of the Taylor or Legendre coefficients. Such results ensure convergence rates n− s of polynomial approximations obtained by best n-term truncation of such series, with s = (1/p)−1 in L∞(U,V) or s = (1/p)−(1/2) in L2(U,V). In this paper we considerably improve these results by providing sufficient conditions of ℓp summability of the coefficient V-norm sequences expressed in terms of the pointwise summability properties of the (|ψj|)j ≥ 1. The approach in the present paper strongly differs from that of [A. Cohen, R. DeVore and C. Schwab, Anal. Appl. 9 (2011) 11–47], which is based on individual estimates of the coefficient norms obtained by the Cauchy formula applied to a holomorphic extension of the solution map. Here, we use weighted summability estimates, obtained by real-variable arguments. While the obtained results imply those of [7] as a particular case, they lead to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, these results imply that for all 0 <p< 2, the ℓp summability of the coefficient V-norm sequences follows from the weaker assumption that (∥ψj∥L∞)j ≥ 1 is ℓq summable for q = q(p) := 2p/(2−p) . We provide a simple analytic example showing that this result is in general optimal and illustrate our findings by numerical experiments. The analysis in the present paper applies to other types of linear PDEs with similar affine parametrization of the coefficients, and to more general Jacobi polynomial expansions.

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23

Li, Jinlu. "Matrix transformations from absolutely convergent series to convergent sequences as general weighted mean summability methods." International Journal of Mathematics and Mathematical Sciences 24, no. 8 (2000): 533–38. http://dx.doi.org/10.1155/s0161171200003732.

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We prove the necessary and sufficient conditions for an infinity matrix to be a mapping, from absolutely convergent series to convergent sequences, which is treated as general weighted mean summability methods. The results include a classical result by Hardy and another by Moricz and Rhoades as particular cases.

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24

Bor, Hüseyin. "A new theorem on the absolute Riesz summability factors." Filomat 28, no. 8 (2014): 1537–41. http://dx.doi.org/10.2298/fil1408537b.

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In [5], we proved a main theorem dealing with absolute Riesz summability factors of infinite series using a quasi-?-power increasing sequence. In this paper, we generalize that theorem by using a general class of power increasing sequences instead of a quasi-?-power increasing sequence. This theorem also includes some new and known results.

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25

Kranz, Radosława, Aleksandra Rzepka, and Ewa Sylwestrzak-Maślanka. "Degrees of the approximations by some special matrix means of conjugate Fourier series." Demonstratio Mathematica 52, no. 1 (March 5, 2019): 116–29. http://dx.doi.org/10.1515/dema-2019-0014.

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Abstract In this paper we will present the pointwise and normwise estimations of the deviations considered by W. Łenski, B. Szal, [Acta Comment. Univ. Tartu. Math., 2009, 13, 11-24] and S. Saini, U. Singh, [Boll. Unione Mat. Ital., 2016, 9, 495-504] under general assumptions on the class considered sequences defining the method of the summability. We show that the obtained estimations are the best possible for some subclasses of Lp by constructing the suitable type of functions.

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26

Şevli̇, H., and L. Leindler. "On the absolute summability factors of infinite series involving quasi-power-increasing sequences." Computers & Mathematics with Applications 57, no. 5 (March 2009): 702–9. http://dx.doi.org/10.1016/j.camwa.2008.11.007.

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27

Yıldız, Şebnem. "Absolute Matrix Summability Factors of Fourier Series with Quasi-f-Power Increasing Sequences." Electronic Notes in Discrete Mathematics 67 (June 2018): 37–41. http://dx.doi.org/10.1016/j.endm.2018.05.007.

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28

Gökçe, Fadime, and Mehmet Ali Sarigöl. "Extension of Maddox’s space l(μ) with Nörlund means." Asian-European Journal of Mathematics 12, no. 06 (October 14, 2019): 2040005. http://dx.doi.org/10.1142/s1793557120400057.

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In this paper, we introduce a new series space [Formula: see text] as the set of all series summable by the absolute Nörlund summability method, which includes the spaces [Formula: see text] and [Formula: see text] of Maddox [Spaces of strongly summable sequences, Quart. J. Math. 18(1) (1967) 345–355], Sarıgöl [Spaces of series summable by absolute Cesàro and matrix operators, Comm. Math Appl. 7(1) (2016) 11–22], Hazar and Sarıgöl [On absolute Nörlund spaces and matrix operators, Acta Math. Sinica, (English Ser.) 34(5) (2018) 812–826], respectively. Also, we study its some algebraic and topological structures such as isomorphism, the [Formula: see text], [Formula: see text], [Formula: see text] duals, Schauder basis, and characterize certain matrix transformations on that space.

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29

Çanak, İbrahim, Naim L. Braha, and Ümit Totur. "A Tauberian theorem for the generalized Nörlund summability method." Georgian Mathematical Journal 27, no. 1 (March 1, 2020): 31–36. http://dx.doi.org/10.1515/gmj-2017-0062.

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AbstractLet {(p_{n})} and {(q_{n})} be any two non-negative real sequences, with {R_{n}:=\sum_{k=0}^{n}{p_{k}q_{n-k}}\neq 0} ({n\in\mathbb{N}}). Let {\sum_{k=0}^{\infty}a_{k}} be a series of real or complex numbers with partial sums {(s_{n})}, and set {t_{n}^{p,q}:=\frac{1}{R_{n}}\sum_{k=0}^{n}{p_{k}q_{n-k}s_{k}}} for {n\in\mathbb{N}}. In this paper, we present the necessary and sufficient conditions under which the existence of the limit {\lim_{n\to\infty}{s_{n}}=L} follows from that of {\lim_{n\to\infty}t_{n}^{p,q}=L}. These conditions are one-sided or two-sided if {(s_{n})} is a sequence of real or complex numbers, respectively.

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30

Móricz, Ferenc. "Statistical convergence of sequences and series of complex numbers with applications in Fourier Analysis and Summability." Analysis Mathematica 39, no. 4 (October 24, 2013): 271–85. http://dx.doi.org/10.1007/s10476-013-0403-9.

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31

Bachmayr, Markus, Albert Cohen, Ronald DeVore, and Giovanni Migliorati. "Sparse polynomial approximation of parametric elliptic PDEs. Part II: lognormal coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 51, no. 1 (December 23, 2016): 341–63. http://dx.doi.org/10.1051/m2an/2016051.

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We consider the linear elliptic equation − div(a∇u) = f on some bounded domain D, where a has the form a = exp(b) with b a random function defined as b(y) = ∑ j ≥ 1yjψj where y = (yj) ∈ ℝNare i.i.d. standard scalar Gaussian variables and (ψj)j ≥ 1 is a given sequence of functions in L∞(D). We study the summability properties of Hermite-type expansions of the solution map y → u(y) ∈ V := H01(D) , that is, expansions of the form u(y) = ∑ ν ∈ ℱuνHν(y), where Hν(y) = ∏j≥1Hνj(yj) are the tensorized Hermite polynomials indexed by the set ℱ of finitely supported sequences of nonnegative integers. Previous results [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826] have demonstrated that, for any 0 <p ≤ 1, the ℓp summability of the sequence (j ∥ψj ∥L∞)j ≥ 1 implies ℓp summability of the sequence (∥ uν∥V)ν ∈ ℱ. Such results ensure convergence rates n− s with s = (1/p)−(1/2) of polynomial approximations obtained by best n-term truncation of Hermite series, where the error is measured in the mean-square sense, that is, in L2(ℝN,V,γ) , where γ is the infinite-dimensional Gaussian measure. In this paper we considerably improve these results by providing sufficient conditions for the ℓp summability of (∥uν∥V)ν ∈ ℱ expressed in terms of the pointwise summability properties of the sequence (|ψj|)j ≥ 1. This leads to a refined analysis which takes into account the amount of overlap between the supports of the ψj. For instance, in the case of disjoint supports, our results imply that, for all 0 <p< 2 the ℓp summability of (∥uν∥V)ν ∈ ℱfollows from the weaker assumption that (∥ψj∥L∞)j ≥ 1is ℓq summable for q := 2p/(2−p) . In the case of arbitrary supports, our results imply that the ℓp summability of (∥uν∥V)ν ∈ ℱ follows from the ℓp summability of (jβ∥ψj∥L∞)j ≥ 1 for some β>1/2 , which still represents an improvement over the condition in [V.H. Hoang and C. Schwab, M3AS 24 (2014) 797−826]. We also explore intermediate cases of functions with local yet overlapping supports, such as wavelet bases. One interesting observation following from our analysis is that for certain relevant examples, the use of the Karhunen−Loève basis for the representation of b might be suboptimal compared to other representations, in terms of the resulting summability properties of (∥uν∥V)ν ∈ ℱ. While we focus on the diffusion equation, our analysis applies to other type of linear PDEs with similar lognormal dependence in the coefficients.

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32

Kiesel, R., and U. Stadtmüller. "Tauberian theorems for general power series methods." Mathematical Proceedings of the Cambridge Philosophical Society 110, no. 3 (November 1991): 483–90. http://dx.doi.org/10.1017/s0305004100070560.

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Let us assume throughout that (pn) denotes a sequence of reals which satisfiesFor real sequences (sn) with increments an = sn – sn−1 for n ≥ 0,(where s−1 = 0), we consider the power seriesmethod of summability (P), where we sayThe power series methods (P) containthe so-called (Jp)-methods (R = 1)and the Borel-type methods (Bp)(R = ∞). We consider only regular (P)-methods, i.e. sn → s implies sn → s(P). By theorem 5 in [5], p.49, we have regularity if and only ifHere we are interested in the converse conclusion, namely sn → s(P) implies sn → s, which can only be validiffurther conditions, so-called Tauberian conditions are satisfied by (sn). These so-called Tauberian theorems for power series methods have a long history; see e.g. the books [5, 14, 23], and they found new attentionrecently in the papers [6, 18, 19, 20] and [8, 9, 10, 11, 12]. The latter papers contain certain o- Tauberian theorems for all power series methods in question and O-Tauberian theorems, if the weight sequence (pn) can be interpolated by alogarithmico-exponential function g(·)(see e.g. [4]), i.e.

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33

Misra, Mahendra, B. P. Padhy, Santosh Kumar Nayak, and U. K. Misra. "Index Summability of an Infinite Series Using δ-Quasi Monotone Sequence." Bulletin of Mathematical Sciences and Applications 11 (February 2015): 21–29. http://dx.doi.org/10.18052/www.scipress.com/bmsa.11.21.

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34

Sonker, Smita, Rozy Jindal, and Lakshmi Narayan Mishra. "Almost Inceasing Sequence for Absolute Riesz Summable Factor." Nepali Mathematical Sciences Report 38, no. 2 (December 31, 2021): 20–26. http://dx.doi.org/10.3126/nmsr.v38i2.42704.

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35

KAMA, RAMAZAN, and BILAL ALTAY. "Some sequence spaces and completeness of normed spaces." Creative Mathematics and Informatics 26, no. 3 (2017): 281–87. http://dx.doi.org/10.37193/cmi.2017.03.05.

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In this paper we introduce new sequence spaces obtained by series in normed spaces and Cesaro summability method. We prove that completeness ´ and barrelledness of a normed space can be characterized by means of these sequence spaces. Also we establish some inclusion relationships associated with the aforementioned sequence spaces.

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36

Kadak, Uğur, and Hakan Efe. "Matrix Transformations between Certain Sequence Spaces over the Non-Newtonian Complex Field." Scientific World Journal 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/705818.

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In some cases, the most general linear operator between two sequence spaces is given by an infinite matrix. So the theory of matrix transformations has always been of great interest in the study of sequence spaces. In the present paper, we introduce the matrix transformations in sequence spaces over the fieldC*and characterize some classes of infinite matrices with respect to the non-Newtonian calculus. Also we give the necessary and sufficient conditions on an infinite matrix transforming one of the classical sets overC*to another one. Furthermore, the concept for sequence-to-sequence and series-to-series methods of summability is given with some illustrated examples.

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37

Sonker, Smita, Rozy Jindal, and Lakshmi Mishra. "Application of quasi-f -power increasing sequence in absolute ph - |C, a, b; d; l| of infinite series." Mathematica Moravica 25, no. 2 (2021): 1–11. http://dx.doi.org/10.5937/matmor2102001s.

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An increasing quasi-f-power sequence of a wider class has been used to establish a universal theorem on a least set of conditions, which is sufficient for an infinite series to be generalized ph-|C, a, b; d; l| k summable. Further, a set of new and well-known arbitrary results have been obtained by using the main theorem. Considering suitable conditions a previous result has been obtained, which validates the current findings. In this way, Bounded Input Bounded Output(BIBO) stability of impulse has been improved by finding a minimal set of sufficient condition for absolute summability because absolute summable is the necessary and sufficient conditions for BIBO stability.

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38

Yıldız, Şebnem. "A matrix application of power increasing sequences to infinite series and Fourier series." Ukrains’kyi Matematychnyi Zhurnal 72, no. 5 (April 29, 2020). http://dx.doi.org/10.37863/umzh.v72i5.6016.

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UDC 517.54 The aim of the paper is a generalization, under weaker conditions, of the main theorem on quasi- σ -power increasing sequences applied to | A , θ n | k summability factors of infinite series and Fourier series. We obtain some new and known results related to basic summability methods.

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39

Azevedo, Douglas, and Thiago P. Andrade. "Summability characterizations of positive sequences." Communications in Mathematics Volume 30 (2022), Issue 1 (May 12, 2022). http://dx.doi.org/10.46298/cm.9290.

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In this paper, we propose extensions for the classical Kummer test, which is a very far-reaching criterion that provides sufficient and necessary conditions for convergence and divergence of series of positive terms. Furthermore, we present and discuss some interesting consequences and examples such as extensions of the Olivier's theorem and Raabe, Bertrand and Gauss's test.

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40

Yıldız, Şebnem. "A new factor theorem on absolute matrix summability method." Journal of Applied Analysis, September 25, 2021. http://dx.doi.org/10.1515/jaa-2021-2060.

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Abstract In this paper, we have a new matrix generalization with absolute matrix summability factor of an infinite series by using quasi-β-power increasing sequences. That theorem also includes some new and known results dealing with some basic summability methods

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41

Yıldız, Şebnem. "A new result for weighted arithmetic mean summability factors of infinite series involving almost increasing sequences." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, March 29, 2019, 1611–20. http://dx.doi.org/10.31801/cfsuasmas.546583.

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Chalmoukis, N., and G. Stylogiannis. "Quasi-nilpotency of Generalized Volterra Operators on Sequence Spaces." Results in Mathematics 76, no. 4 (August 2, 2021). http://dx.doi.org/10.1007/s00025-021-01482-7.

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AbstractWe study the quasi-nilpotency of generalized Volterra operators on spaces of power series with Taylor coefficients in weighted $$\ell ^p$$ ℓ p spaces $$1<p<+\infty $$ 1 < p < + ∞ . Our main result is that when an analytic symbol g is a multiplier for a weighted $$\ell ^p$$ ℓ p space, then the corresponding generalized Volterra operator $$T_g$$ T g is bounded on the same space and quasi-nilpotent, i.e. its spectrum is $$\{0\}.$$ { 0 } . This improves a previous result of A. Limani and B. Malman in the case of sequence spaces. Also combined with known results about multipliers of $$\ell ^p$$ ℓ p spaces we give non trivial examples of bounded quasi-nilpotent generalized Volterra operators on $$\ell ^p$$ ℓ p . We approach the problem by introducing what we call Schur multipliers for lower triangular matrices and we construct a family of Schur multipliers for lower triangular matrices on $$\ell ^p, 1<p<\infty $$ ℓ p , 1 < p < ∞ related to summability kernels. To demonstrate the power of our results we also find a new class of Schur multipliers for Hankel operators on $$\ell ^2 $$ ℓ 2 , extending a result of E. Ricard.

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