Banach Actions Preserving Unconditional Convergence

Let A,X,Y be Banach spaces and A×X→Y, (a,x)↦ax be a continuous bilinear function, called a Banach action. We say that this action preserves unconditional convergence if for every bounded sequence (an)n∈ω in A and unconditionally convergent series ∑n∈ωxn in X, the series ∑n∈ωanxn is unconditionally convergent in Y. We prove that a Banach action A×X→Y preserves unconditional convergence if and only if for any linear functional y*∈Y* the operator Dy*:X→A*, Dy*(x)(a)=y*(ax) is absolutely summing. Combining this characterization with the famous Grothendieck theorem on the absolute summability of operators from ℓ1 to ℓ2, we prove that a Banach action A×X→Y preserves unconditional convergence if A is a Hilbert space possessing an orthonormal basis (en)n∈ω such that for every x∈X, the series ∑n∈ωenx is weakly absolutely convergent. Applying known results of Garling on the absolute summability of diagonal operators between sequence spaces, we prove that for (finite or infinite) numbers p,q,r∈[1,∞] with 1r≤1p+1q, the coordinatewise multiplication ℓp×ℓq→ℓr preserves unconditional convergence if and only if one of the following conditions holds: (i) p≤2 and q≤r, (ii) 2<p<q≤r, (iii) 2<p=q<r, (iv) r=∞, (v) 2≤q<p≤r, (vi) q<2<p and 1p+1q≥1r+12.

Absolute summability of power $p$ of series, associated with conjugate Fourier series, by triangular matrix methods

We establish conditions of absolute summability of powers of series that are associated with conjugate Fourier series, by triangular matrix methods, and provide the application of the theorems proved to Voronoi-Nerlund method.

Tauberian theorems in the case of absolute summability in degree $p$ of double series

We establish a Tauberian theorem in the case of absolute summability in degree $p$ of double series by matrix methods, give its application to Abel methods.

Absolute summability of conjugate Fourier integrals by Voronoi-Nerlund method

We establish conditions of absolute summability of Fourier integrals and conjugate Fourier integrals by Voronoi-Nerlund method.

On the Sample Autocorrelation Function’s Absolute Summability

The sample ACF is the most common basic tool in analyzing time-series data. This paper provides a theoretical proof that, under some regularity conditions, sample ACF of a given stationary time series is not absolutely summable. Furthermore, it shows that under some mild conditions, the number of positive and negative sample ACFs and their absolute summation tend to infinity as the length of time series increases. The theoretical results are supported by practical evidence from a simulation study.

On absolute summability of integrals with multipliers

We establish sufficient conditions for summability of integrals with multiplier by $|\overline{W}, p(y)|_k$, $k \geqslant 1$, method.

APPLICATIONS OF MATRIX TRANSFORMATIONS TO ABSOLUTE SUMMABILITY

Rhoades and Sava¸s [6],[11] established necessary for inclusions of the absolute matrix summabilities under additional conditions. In this paper we determine necessary or su¢ cient conditions for some classes of in…nite matrices, and using this get necessary or su¢ cient conditions for more general absolute summabilities applied to all matrices.

Absolute Summability for N-Tupled Triangle Matrices on Sequence Space

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Summability of Orthogonal Series by Absolute Matrix Method: قابلية جمع المتسلسلات المتعامدة بالطريقة المصفوفية المطلقة

At the beginning of the 20th century, the theory of perpendicular series orthogonal, which treated sentences of orthogonal functions as a natural generalization of series theory. The perpendicular mass idyllability many methods, such as the Reimann, Norlund, Cesaro, Holder, and generalized Norlund methods, has been studied. In this research, we studied the summability of simple orthogonal series in different methods called absolute methods that rely heavily on differences. This is done by relying on some research, articles and scientific books in the field of summability of series. The absolute matrix methods are one of the most important methods of absolute summability. Most absolute summability methods are special cases of absolute matrix method. We finally recommend a double case study.

Bounds for uniform resolvent conditions

For a bounded linear operator on a Banach space, the uniform resolvent condition implies the absolute summability of the powers of the operator. In this paper, we study the bounds for the absolute sum of the powers of an operator that satisfies the uniform resolvent condition. Some known bounds on general Banach spaces as well as on finite-dimensional Banach spaces are improved.