On Tauber's second Tauberian theorem

We study Tauberian conditions for the existence of Cesàro limits in terms of the Laplace transform. We also analyze Tauberian theorems for the existence of distributional point values in terms of analytic representations. The development of these theorems is parallel to Tauber's second theorem on the converse of Abel's theorem. For Schwartz distributions, we obtain extensions of many classical Tauberians for Cesàro and Abel summability of functions and measures. We give general Tauberian conditions in order to guarantee $(\mathrm{C},beta)$ summability for a given order $\beta$. The results are directly applicable to series and Stieltjes integrals, and we therefore recover the classical cases and provide new Tauberians for the converse of Abel's theorem where the conclusion is Cesàro summability rather than convergence. We also apply our results to give new quick proofs of some theorems of Hardy-Littlewood and Szàsz for Dirichlet series

A Tauberian theorem for Cesàro summability of integrals

AbstractIn this paper we give a proof of the generalized Littlewood Tauberian theorem for Cesàro summability of improper integrals

Comparison theorems for summability methods of sequences of fuzzy numbers

In this study we compare Ces\`{a}ro and Euler weighted mean methods of summability of sequences of fuzzy numbers with Abel and Borel power series methods of summability of sequences of fuzzy numbers. Also some results dealing with series of fuzzy numbers are obtained.Comment: publication information is added, typos correcte

Indexed Absolute Summability Factor of Improper Integrals

In this paper, we have defined the summability for improper integrals and established a theorem on indexed absolute Cesaro summability factors of improper integral under sufficient conditions. Some auxiliary results (well known) have also been deduced from the main result under suitable conditions

Necessary and sufficient conditions under which convergence follows from summability by weighted means

We prove necessary and sufficient Tauberian conditions for sequences summable by weighted mean methods. The main results of this paper apply to all weighted mean methods and unify the results known in the literature for particular methods. Among others, the conditions in our theorems are easy consequences of the slowly decreasing condition for real numbers, or slowly oscillating condition for complex numbers. Therefore, practically all classical (one-sided as well as two-sided) Tauberian conditions for weighted mean methods are corollaries of our two main theorems

Wiener amalgams and summability of Fourier series

ome recent results on a general summability method, on the so-called µ-summability is summarized. New spaces, such as Wiener amalgams, Feichtinger’s algebra and modulation spaces are investigated in summability theory. Sufficient and necessary conditions are given for the norm and a.e. convergence of the µ-means. Key Words: Wiener amalgam spaces, Feichtinger’s algebra, homogeneous Banach spaces, Besov-, Sobolev-, fractional Sobolev spaces, modulation spaces, Herz spaces, Hardy-Littlewood maximal function, µ-summability of Fourier series, Lebesgue points. AMS Classification Number: Primary 42B08, 46E30, Secondary 42B30, 42A3