Pringsheim's Sense Research Articles

Some Tauberian theorems for the weighted mean method of summability of double sequence

UDC 517.5 Let p = ( p j ) and q = ( q k ) be real sequences of nonnegative numbers with the property that P m = ∑ j = 0 m p j ≠ 0 and Q n = ∑ k = 0 n q k ≠ 0 for all m and n . Let ( P m ) and ( Q n ) be regulary varying positive indices. Assume that ( u m n ) is a double sequence of complex (real) numbers, which is ( N ¯ , p , q ; α , β ) summable with a finite limit, where ( α , β ) = ( 1,1 ) , ( 1,0 ) , or ( 0,1 ) . We present some conditions imposed on the weights under which ( u m n ) converges in Pringsheim's sense. These results generalize and extend the results obtained by authors in [Comput. Math. Appl., <strong>62</strong>, No. 6, 2609–2615 (2011)].

Multidimensional Statistical Convergence in Credibility Theory

The purpose of this article is to present the concepts of double statistical convergence in credibility and [Formula: see text]-double statistical convergence in credibility in pringsheim sense. By using these definitions we present a natural multidimensional extension of Credibility theory via Summability methods.

IDEAL CONVERGENCE OF DOUBLE SEQUENCES OF CLOSED SETS

In the present paper, we introduce the concepts of ideal inner and ideal outer limits which always exist even if empty sets for double sequences of closed sets in Pringsheim's sense. Next, we give some formulas for finding ideal inner and outer limits in a metric space. After then, we define Kuratowski ideal convergence of double sequences of closed sets by means of the ideal inner and ideal outer limits of a double sequence of closed sets. Additionally, we give some examples that our result is more general than the results obtained before.

Rectangular Summation of Multiple Fourier Series and Multi-parametric Capacity

We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichlet space functions. As a by-product of the methods of proof, the results also hold in the vector-valued setting.

Tauberian Theorems for the Product of Weighted and Cesàro Summability Methods for Double Sequences

In this paper, we obtain necessary and sufficient conditions, under which convergence of a double sequence in Pringsheim's sense follows from its weighted-Cesaro summability. These Tauberian conditions are one-sided or two-sided if it is a sequence of real or complex numbers, respectively.

Uniform convergence of double sine transforms of general monotone functions

We consider different classes of functions of two variables that satisfy general monotonicity conditions, and obtain necessary and sufficient conditions for the uniform convergence in the regular sense and in the sense of Pringsheim of double sine transforms of functions of such classes.

On statistical acceleration convergence of double sequences

In this article the notion of statistical acceleration convergence of double sequences in Pringsheim's sense has been introduced. We prove the decompostion theorems for statistical acceleration convergence of double sequences and some theorems related to that concept have been established using the four dimensional matrix transformations. We provided some examples, where the results of acceleration convergence fails to hold for the statistical cases.

A Tauberian theorem for $(C,1,1)$ summable double sequences of fuzzy numbers

In this paper, we determine necessary and sufficient Tauberian conditions under which convergence in Pringsheim's sense of a double sequence of fuzzy numbers follows from its $(C,1,1)$ summability. These conditions are satisfied if the double sequence of fuzzy numbers is slowly oscillating in different senses. We also construct some interesting double sequences of fuzzy numbers.

A net convergence for Schauder double bases

The set of all pairs of natural numbers is considered as a directed set under the direction: [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. This directed set is used to study convergence of a double series in a sense of Pringsheim and to introduce double bases in topological vector spaces. An introductory study on double bases is presented.

Pringsheim Convergence and the Dirichlet Function

Double sequences have some unexpected properties which derive from the possibility of commuting limit operations. For example, may be defined so that the iterated limits and exist and are equal for all x, and yet the Pringsheim limit does not exist. The sequence is a classic example used to show that the iterated limit of a double sequence of continuous functions may exist, but result in an everywhere discontinuous limit. We explore whether the limit of this sequence in the Pringsheim sense equals the iterated result and derive an interesting property of cosines as a byproduct.

Domain of difference matrix of order one in some spaces of double sequences

In this study, we define the spaces Mu(Δ), Cp(Δ), C0p(Δ), Cr(Δ) and ℒq(Δ) of double sequences whose difference transforms are bounded , convergent in the Pringsheim's sense, null in the Pringsheim's sense, both convergent in the Pringsheim's sense and bounded, regularly convergent and absolutely q-summable, respectively, and also examine some inclusion relations related to those sequence spaces. Furthermore, we show that these sequence spaces are Banach spaces. We determine the alpha-dual of the space Mu(Δ) and the β(v)-dual of the space Cη(Δ) of double sequences, where v, η ∈ {p,bp,r}. Finally, we characterize the classes (μ : Cv(Δ)) for v ∈ {p,bp,r} of four dimensional matrix transformations, where μ is any given space of double sequences.

Some classical Tauberian theorems for (C,1,1,1) summable triple sequences

Abstract Let ( u m n s ) ${(u_{mns})}$ be a (C,1,1,1) summable triple sequence of real numbers. We give one-sided Tauberian conditions of Landau and Hardy type under which ( u m n s ) ${(u_{mns})}$ converges in Pringsheim's sense. We prove that ( u m n s ) ${(u_{mns})}$ converges in Pringsheim's sense if ( u m n s ) ${(u_{mns})}$ is slowly oscillating in certain senses. Moreover, we extend a Tauberian theorem given by Móricz [Studia Math. 110 (1994), 83–96] for double sequences to triple sequences.

Inverse continuous wavelet transform in Pringsheim's sense on Wiener amalgam spaces

The inversion formula for the continuous wavelet transform is usually considered in the weak sense. In the present note we trace back the inverse wavelet transform to summability means of Fourier transforms and obtain norm and almost everywhere convergence of the inversion formula for functions from the Lp and Wiener amalgam spaces.

Summation of double sequences and selection principles

This paper proves some selection properties of a class of positive real double sequences which converge to 0 in the sense of Pringsheim (see, e.g. [10]).

Almost everywhere $${(C, \alpha, \beta > 0)}$$ ( C , α , β > 0 ) -summability of the Fourier series of functions on the 2-adic additive group

We investigate Cesaro summability of bivariate integrable functions on the 2-adic group. We prove the a.e. convergence of 2-adic Cesaro means \({\sigma^{\alpha, \beta}_{n, m} f \rightarrow f}\) as \({n, m \rightarrow \infty}\) for functions \({f \in\,L\,{\rm log}^{+}\,L(\mathbb{I}^2)}\) and \({\alpha, \beta > 0}\). Then it is shown that this convergence result can not be improved in the Pringsheim sense, that is, \({L\,{\rm log}^{+}\,L}\) is the maximal convergence space for \({\sigma^{1, 1}_{n, m}}\) when there are no conditions for the indices except that they tend to infinity. We prove that for all measurable functions \({\delta : [o, \infty) \rightarrow [o, \infty)}\) for which \({{\rm lim}_{t \rightarrow \infty} \delta(t) = 0}\) there is a function \({f \in\,L\,\log^{+}\,L\, \delta(L)}\) with lim sup \({|\sigma_{2^{n_{1}}, 2^{n_{2}}} f(x)| = +\infty}\) a.e. as \({{\rm min}\,\{n_{1}, n_{2}\} \rightarrow \infty}\).

Four dimensional matrix characterization P-convergence fields of summability methods

In 1945 Agnew presented two dimensional matrix characterization of convergent fields. The goal of this paper is to present four dimensional matrix characterization of P-convergent field. This will be accomplished by the presentation of the following multiple dimensional analogues of Agnew’s theorem. If A is a four-dimensional multiplicative with multiplier 0, then there are sequences 0=σ0<σ1<σ2<⋯ and 0=ρ0<ρ1<ρ2<⋯ of integers such that each bounded double sequence {xk,l} oscillating so slowly thatP-limm,nmaxσm⩽k,r⩽σm+1;ρn⩽l,s⩽ρn+1xk,l-xr,s=0is A summable to 0 in the Pringsheim sense. In addition, natural implications and variations are also presented.

On the Pringsheim convergence of double series

Several aspects of the convergence of a double series in the sense of Pringsheim are considered in analogy with some well-known results for single series. They include various tests for absolute convergence and also criteria for convergence of the Cauchy product. Some errors in the works of earlier authors are corrected.

The best convergence of multiple trigonometric series

We consider the best convergence of multiple trigonometric series. We indicate essential distinction of the behavior (in this sense) of multiple series from that of simple ones. In particular, the well-known result obtained by S. N. Bernshtein on the best convergence of a series with an odd ratio of frequencies does not hold for a multiple series in the case of the approximation by polynomials with harmonics from rectangles (in the sense of Pringsheim), but it is true for “angular” approximations.

Uniform convergence of Cesàro means of negative order of double trigonometric Fourier series

In this paper we prove that if f ∊ C([−π, π]2) and the function f is bounded partial p-variation for some p ∊ [1,+ ∞), then the double trigonometric Fourier series of a function f is uniformly (C;−α,−β) summable (α+β> 1/p, α, β> 0) in the sense of Pringsheim. If α + β ≥ 1/p, then there exists a continuous function f0 of bounded partial p-variation on [−π, π]2 such that the Cesàro (C;−α,−β) means σ−α,−βn,m (f0;0,0) of the double trigonometric Fourier series of f0 diverge over cubes.

On Cesáro summability of fourier series with respect to double complete orthonormal systems

Let {ϕn(x), n = 1, 2,...} be an arbitrary complete orthonormal system on the interval I:= [0, 1]which consists of a.e. bounded functions. Suppose that E0 ⊂ I2 is any Lebesgue measurable set such that μ2E0 > 0, and φ, φ(0) = 0, is an increasing continuous function on [0, ∞) with φ(u) = o(u ln u) as u → ∞. Then there exist a function f ∈ L1(I2) and a set E0′, ⊂ E0, μ2E0′ > 0, such that $$\int_{I^2 } {\phi (|f(x,y)|)dxdy < \infty } $$ and the sequence of double Cesaro means of Fourier series of f with respect to the system {ϕn(x)ϕm(y): n,m = 1, 2,...} is unbounded in the sense of Pringsheim (by rectangles) on E0′. This statement gives critical integrability conditions for the Cesaro summability a.e. of Fourier series in the class of all complete orthonormal systems of the type {ϕ n(x)ϕm(y): n,m = 1, 2,...}.