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These investigations are related to symmetric and supersymmetric polynomials and function calculus in algebras of analytic functions. 10.3390/axioms11100511 3 Vasylyshyn T., Zhyhallo K. Entire Symmetric Functions on the Space of Essentially Bounded Integrable Functions on the Union of Lebesgue-Rohlin Spaces. The theory of approximation of functions is one of the central branches of mathematical analysis. [Extracted from the article]

The main object of this article is to introduce the concepts of (f, ρ)− statistical convergence of order α and strong (f, ρ)− summability of order α of sequences of real numbers and give some inclusion relations between these spaces. [ABSTRACT FROM AUTHOR]

In this study, by using definition of ρ-statistical convergence which was defined by Çakallı [5], we introduce the concepts of pointwise wρα(f)-summability, pointwise ρ-statistical convergence of order α and uniform ρ-statistical convergence of order α sequences of real-valued functions. Also, we give some inclusion relations between these concepts. [ABSTRACT FROM AUTHOR]

We generalize the classical Lebesgue's theorem to multi-dimensional functions. We prove that the Cesàro means of the Fourier series of the multi-dimensional function f ∈ L1(log L) d-1(Td) ⊃ Lp(Td) (1 < p ≤ ∞) converge to f at each strong Lebesgue point. [ABSTRACT FROM AUTHOR]

In this paper, we introduce the concept of (H, 1)(E, 1) product transform and obtained two quite new theorems on (H, 1)(E, 1) product transform of Fourier series and its conjugate series. Our result extends several known result on single summability methods. [ABSTRACT FROM AUTHOR]

A moderated theorem was established in this study to derive an infinite series as an absolute Riesz n k |N̄,pnα; δ; γ|k summable by applying an almost increasing sequence. Suitable conditions were applied for obtaining some well-known applications of the presented theorem. It was observed from the investigation that absolute summability could be used for Bounded Input Bounded Output stability and to bind the output of the signals as absolute summable, it was a necessary and sufficient condition for BIBO stability. [ABSTRACT FROM AUTHOR]

In this paper, we have generalized a known theorem on |N, pn|k summability factors of infinite series to the φ - |A, pn|k summability by using an almost increasing sequence. This new theorem also includes several new results. [ABSTRACT FROM AUTHOR]

Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence M = ( M p ) p ∈ N 0 , have been put forward by A. Lastra, S. Malek and the second author [10] , and their validity depends on the possibility of associating to M a nonzero proximate order. We provide several characterizations of this and other related properties, in which the concept of regular variation for functions and sequences plays a prominent role. In particular, we show how to construct well-behaved strongly regular sequences from nonzero proximate orders. [ABSTRACT FROM AUTHOR]

We show a picture of the relations among different types of summability of series in the space [ABSTRACT FROM AUTHOR]

For a given (real analytic) slow–fast system { x ˙ = ε f ( x , y , ε ) y ˙ = g ( x , y , ε ) , that admits a slow–fast saddle and that satisfies some mild assumptions, the Borel-summability properties of the saddle separatrix tangent in the direction of the critical curve are investigated: 1-summability is shown. It is also shown that slow–fast saddle connections of canard type have summability properties, in contrast to the typical lack of Borel-summability for canard solutions of general equations. [ABSTRACT FROM AUTHOR]

We prove a general theorem dealing with the generalized absolute Cesàro summability factors of infinite series. This theorem also includes some new and known results. [ABSTRACT FROM AUTHOR]

In this paper, we investigate summability methods of logarithmic averages of the numerical sequences and its applications such as Tauberian type theorems. [ABSTRACT FROM AUTHOR]

We prove some inequalities related to the concepts of N (st)-conservative matrices, N (st)-lim sup and N (st)-lim inf which are natural analogues of (c, st ∩ l∞ )-matrices, st-lim sup and st-lim inf respectively. [ABSTRACT FROM AUTHOR]

In this study, using the generalized difference operator Δm, we introduce the concepts of (Δm; I)-lacunary statistical convergence of order α and lacunary strong Δmp-summability of order α of sequences and give some relations about these concepts. [ABSTRACT FROM AUTHOR]

We prove and discuss some new $H_{p}$ - $L_{p}$ type inequalities of weighted maximal operators of Vilenkin-Nörlund means with monotone coefficients. It is also proved that these inequalities are the best possible in a special sense. We also apply these results to prove strong summability for such Vilenkin-Nörlund means. As applications, both some well-known and new results are pointed out. [ABSTRACT FROM AUTHOR]

Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence M = ( M p ) p ∈ N 0 , have been put forward by A. Lastra, S. Malek and the second author [28] . We study several open questions related to the existence of kernels of summability constructed by means of analytic proximate orders. In particular, we give a simple condition that allows us to associate a proximate order with a strongly regular sequence. Under this assumption, and through the characterization of strongly regular sequences in terms of so-called regular variation, we show that the growth index γ ( M ) defined by V. Thilliez [54] and the order of quasianalyticity ω ( M ) introduced by the second author [50] are the same. [ABSTRACT FROM AUTHOR]

We study the Mathias-Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias-Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias-Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of $${\omega}$$ -hitting and $${\omega}$$ -splitting families which retain their property in the extension by a Laver type forcing associated with a coideal. [ABSTRACT FROM AUTHOR]

In the present paper, a general theorem dealing with |A; pn; δ|k summability factors of infinite series has been proved by using almost increasing sequence. This theorem also includes some known and new results. [ABSTRACT FROM AUTHOR]

In this paper we give necessary and suficient conditions for ... and vise versa, and ... and vise versa for the case ... is absolute factorable summability. So we also complete some open problems in the paper of Sarıgöl [10]. [ABSTRACT FROM AUTHOR]

In this paper we give necessary and suficient conditions for ... and vise versa, and ... and vise versa for the case ... is absolute factorable summability. So we also complete some open problems in the paper of Sarıgöl [10]. [ABSTRACT FROM AUTHOR]

In this paper we continue in the line of recent investigation of order summability of nets using ideals by Boccuto et al. (Czechoslovak Math. J. 62(137):1073-1083 ; J. Appl. Anal. 20(1), ) where they had introduced the notions of $$\mathcal {I}$$ and $$\mathcal {I}^*$$ order convergence, $$\mathcal {I}$$ and $$\mathcal {I}^*$$ divergence of nets and its further extensions, namely the notions of $$\mathcal {I}^{\mathcal {K}}$$ -order convergence and $$\mathcal {I}^{\mathcal {K}}$$ -divergence of nets in a $$(\ell )$$ -group and investigate the relation between $$\mathcal {I}$$ and $$\mathcal {I}^{\mathcal {K}}$$ -concepts where a special class of ideals called $$\Lambda P(\mathcal {I},\mathcal {K})-$$ ideals plays very important role. We also introduce, for the first time, the notion of $$\mathcal {I}^{\mathcal {K}}$$ -order Cauchy condition and $$\mathcal {I}$$ -order cluster points of nets in ( $$\ell $$ ) groups and examine some of its characterizations and its consequences. In particular the role of $$\mathcal {I}$$ -order cluster points in making the above mentioned Cauchy nets convergent is studied. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce a new generalized absolute summability method and some new definitions on classes of pairs of sequences. By using these new definitions, we generalize the result of Bor (Math Commun 13:21-25, ) dealing with $$\left| C,\alpha ,\gamma ;\beta \right| _{k}$$ summability factors of infinite series. [ABSTRACT FROM AUTHOR]

The paper deals with the spherical Fourier sums $$S_r (f,x) = \sum\nolimits_{\left\| k \right\| \leqslant r} {\hat f} (k)e^{ik \cdot x}$$ of a periodic function f in m variables and the strong integral means of these sums ((∫ | S( f, x)| dr)/ R for p ≥ 1. We establish the exact growth order as R → +∞ of the corresponding operators, i.e., the growth order of the quantities sup((∫| S( f, 0)| dr)/ R). The upper and lower bounds differ by their coefficients, which depend only on the dimension m. A sufficient condition on the function ensuring the uniform strong p-summability of its Fourier series is given. [ABSTRACT FROM AUTHOR]

F. Schipp in 1969 proved the almost everywhere p-strong summability of Walsh-Fourier series and showed that if λ( n) →∞, then there exists a function f ∈ L[0, 1) for which the Walsh partial sums S( x, f) satisfy the divergence condition almost everywhere on [0, 1). In the present paper, we show that this condition holds everywhere. [ABSTRACT FROM AUTHOR]

Abstract: Schoenberg (1959) [23] proved that if a sequence is bounded and statistically convergent to L then it is statistically summable to L. However the converse is not true. In this paper we prove necessary and sufficient Tauberian conditions based on generator sequences under which statistical convergence follows from statistical summability . [Copyright &y& Elsevier]

Abstract: We present criteria for deciding whether a bivariate rational function in two variables can be written as a sum of two (q-)differences of bivariate rational functions. Using these criteria, we show how certain double sums can be evaluated, first, in terms of single sums and, finally, in terms of values of special functions. [Copyright &y& Elsevier]

In this paper, making use of a known summation formulae for bilateral series, an attempt has been made to establish certain interesting results involving continued fractions. [ABSTRACT FROM AUTHOR]

Let s denote the q-ary sum-of-digits function and let $${P_1(X), P_2(X) \in \mathbb{Z}[X]}$$ with $${P_1(\mathbb{N}), P_2(\mathbb{N}) \subset \mathbb{N}}$$ be polynomials of degree $${h, l \geqq 1, h \neq l}$$ , respectively. In this note we show that ( $${s_q(P_1(n))/s_q(P_2(n)))_{n \geqq 1}}$$ is dense in $${\mathbb{R}^+}$$ . This extends work by Stolarsky [] and Hare, Laishram and Stoll []. [ABSTRACT FROM AUTHOR]

In normed linear space settings, modifying the sequential definition of continuity of an operator by replacing the usual limit "lim" with arbitrary linear regular summability methods G we consider the notion of a generalized continuity ((G1,G2)-continuity) and examine some of its consequences in respect of usual continuity and linearity of the operators between two normed linear spaces. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove two theorems on , summability of orthogonal series. Also, several known and new results are deduced as corollaries of the main results. [Copyright &y& Elsevier]

Abstract: It is shown that if then Lebesgue summability, summability ( ), Abel summability, Riemann summability, and summability ( ) of the series are all equivalent to one another. [Copyright &y& Elsevier]

We prove a new Tauberian-like theorem. For a real sequence u = ( u), on the basis of the weighted mean summability of its generator sequence ( V(∆ u)) and some other conditions, this theorem establishes the property of slow oscillation of the indicated sequence. [ABSTRACT FROM AUTHOR]

We address an open problem posed recently by Almeida and Hästö. They defined the spaces ℓq(·)(Lp(·)) of variable integrability and summability and showed that ∥ · ∣ℓq(·)(Lp(·))∥ is a norm if q = 1 is constant almost every-where or if 1/p(x)+1/q(x) = 1 for almost every x ∈ Rn. Nevertheless, the natural conjecture (expressed also by Almeida and Hästö) is that the expression is a norm if p(x), q(x) = 1 almost everywhere. We show that ∥ · ∣ℓq(·)(Lp(·))∥ is a norm if 1 = q(x) = p(x) for almost every x ∈ Rn. Furthermore, we construct an example of p(x) and q(x) with min(p(x), q(x)) = 1 for every x ∈ Rn such that the triangle inequality does not hold for ∥ · ∣ℓq(·)(Lp(·))∥. [ABSTRACT FROM AUTHOR]

A complete convergence result is obtained for weighted sums of identically distributed ρ-mixing random variables with E| X| log(1 + | X|) < ∞ for some 0 < α ≤ 2. This result partially extends the result of Sung (Stat Papers 52: 447-454, ) for negatively associated random variables to ρ-mixing random variables. It also settles the open problem posed by Zhou et al. (J Inequal Appl, , doi:). [ABSTRACT FROM AUTHOR]

In this paper, we continue our investigations in line of our recent papers, Savas and Das [16] and Das, Savas and Ghosal [5]. We introduce the notion of AI-statistical convergence which includes the new summability methods studied in [16] and [5] as special cases and make certain observations on this new and more general summability method. [ABSTRACT FROM AUTHOR]

In the present paper, by using a new defined -|C, ασ, αn |fc summability method and some classes of pairs of sequences, we generalize a result of Bor [5] dealing with < -- |C, ασ,a; β|k summability factors. [ABSTRACT FROM AUTHOR]

Abstract: We introduce a new class of matrices and establish a general absolute summability factor theorem by applying a recently defined absolute summability, summability, and the class . As applications of the main result, we further verify that many well-known classes of matrices such as Cesàro’s matrices, Rhaly’s generalized Cesàro matrices, the -Cesàro matrices, and Riesz matrices are all in for some certain sequences and , and, therefore, many new factor theorems can be obtained under suitable conditions. [Copyright &y& Elsevier]

We prove for the (C, 1) summability method several Tauberian remainder theorems using the general control modulo of the oscillatory behavior. [ABSTRACT FROM PUBLISHER]

We introduce the concept of statistical convergence of order α of difference sequences, and we give some relations between the set of statistical convergence of order α of difference sequences and strong wαγ(△m,p)-summability of order α. Furthermore some relations between the spacewαγ(△m and Sαγ are examined. [ABSTRACT FROM AUTHOR]

Abstract: Text: We prove congruences, modulo a power of a prime p, for certain finite sums involving central binomial coefficients , partly motivated by analogies with the well-known power series for and . The right-hand sides of those congruences involve values of the finite polylogarithms . Exploiting the available functional equations for the latter we compute those values, modulo the required powers of p, in terms of familiar quantities such as Fermat quotients and Bernoulli numbers. Video: For a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=W54Ad0YFr8A. [Copyright &y& Elsevier]

Abstract: We give a generator of the space spanned by double zeta values of level 2 with odd weight by using explicit formulas for double Euler sums. [Copyright &y& Elsevier]

Abstract: The function was introduced by Hardy and Littlewood (1936) in their study of Lambert summability, and since then it has attracted attention of many researchers. In particular, this function has made a surprising appearance in the recent disproof by Alzer, Berg and Koumandos (2005) of a conjecture by Clark and Ismail (2003) . More precisely, Alzer et al. have shown that the Clark and Ismail conjecture is true if and only if for all . It is known that is unbounded in the domain from above and below, which disproves the Clark and Ismail conjecture, and at the same time raises a natural question of whether we can exhibit at least one point for which . This turns out to be a surprisingly hard problem, which leads to an interesting and non-trivial question of how to approximate for very large values of . In this paper we continue the work started by Gautschi (2005) in and develop several approximations to for large values of . We use these approximations to find an explicit value of for which . [Copyright &y& Elsevier]

Abstract: The notation means the finite sum of all numbers as ranges over the integers that divide including and . The purpose of this paper is to define a theory of summability suggested by this type of sum and an associated theory of convergence. This paper studies five sequence spaces that are related to the arithmetical summability thus defined. [Copyright &y& Elsevier]

Abstract: This paper is concerned with a uniform approach–the -coefficient method–to basic hypergeometric series. By virtue of this method, some new and elementary proofs for four partial theta function identities due to Alladi and Berkovich (2004) , Andrews and Warnaar (2007) , Berkovich  and Warnaar (2003) , as well as for Ramanujan’s summation formula are obtained. [Copyright &y& Elsevier]

Abstract: The exterior nonstationary problem is studied for the 3D Navier–Stokes equations, for which the associated total net force to the boundary may not vanish. The time-decay properties of the strong solution including the first and second derivatives are shown in and weighted spaces. In particular, the relation of (weighted) -summability for smooth solutions is discussed in details between the time decay and the total net force exerted by the fluid to the body. The conclusions in this article improve and extend results in: [H. Bae, B. Jin, Asymptotic behavior for the Navier–Stokes equations in 2D exterior domains, J. Funct. Anal. 240 (2006) 508–529; H. Bae, B. Jin, Temporal and spatial decay rates of Navier–Stokes solutions in exterior domains, Bull. Korean Math. Soc. 44 (2007) 547–567] and in: [C. He, T. Miyakawa, On -summability and asymptotic profiles for smooth solutions to Navier–Stokes equations in a 3D exterior domain, Math. Z. 245 (2003) 387–417; C. He, T. Miyakawa, On weighted-norm estimates for nonstationary incompressible Navier–Stokes flows in a 3D exterior domain, J. Differential Equations 246 (2009) 2355–2386], respectively. [Copyright &y& Elsevier]

Abstract: The energy of a graph is equal to the sum of the absolute values of its eigenvalues. Line graphs play an important role in the study of graph theory. Generalized line graphs extend the ideas of both line graphs and cocktail party graphs. In this paper, we establish relations between the energy of the generalized line graph of a graph G and the Laplacian and signless Laplacian energies of G. We give upper and lower bounds for the energy of generalized line graphs. Finally, we present upper and lower bounds for some special graphs. [Copyright &y& Elsevier]