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

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

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.

Harmonic analysis and statistics of the first Galois cohomology group

We utilize harmonic analytic tools to count the number of elements of the Galois cohomology group $$f\in H^1(K,T)$$ f ∈ H 1 ( K , T ) with discriminant-like invariant $$\text {inv}(f)\le X$$ inv ( f ) ≤ X as $$X\rightarrow \infty $$ X → ∞ . Specifically, Poisson summation produces a canonical decomposition for the corresponding generating series as a sum of Euler products for a very general counting problem. This type of decomposition is exactly what is needed to compute asymptotic growth rates using a Tauberian theorem. These new techniques allow for the removal of certain obstructions to known results and answer some outstanding questions on the generalized version of Malle’s conjecture for the first Galois cohomology group.

On a Tauberian theorem of Ingham and Euler–Maclaurin summation

We discuss two theorems in analytic number theory and combinatory analysis that have seen increased use in recent years. A corollary to a Tauberian theorem of Ingham allows one to quickly prove asymptotic formulas for arithmetic sequences, so long as the corresponding generating function exhibits exponential growth of a certain form near its radius of convergence. Two common methods for proving the required analytic behavior are modular transformations and Euler–Maclaurin summation. However, these results are sometimes stated without certain technical conditions that are necessary for the complex analytic techniques that appear in Ingham’s proof. We carefully examine the precise statements and proofs of these results, and find that in practice, the technical conditions are satisfied for those cases appearing in recent applications. We also generalize the classical approach of Euler–Maclaurin summation in order to prove asymptotic expansions for series with complex values, simple poles, or multi-dimensional summation indices.

A Mean Ergodic Theorem for Affine Nonexpansive Mappings in Nonpositive Curvature Metric Spaces

In this paper, we consider the orbits of an affine nonexpansive mapping in Hadamard (nonpositive curvature metric) spaces and prove an ergodic theorem for the inductive mean, which extends the von Neumann linear ergodic theorem. The main result shows that the sequence given by the inductive means of iterations of an affine nonexpansive mapping with a nonempty fixed point set converges strongly to a fixed point of the mapping. A Tauberian theorem is also proved in order to ensure convergence of the iterations.

Results on Tauberian Theorem for Cesàro Summable Double Sequences of Fuzzy Numbers

The paper aims to establish new results on Tauberian theorem for Cesàro summability of double sequences of fuzzy numbers, and thus to extend and unify several results in the available literature. Further, a number of special cases, corollaries and illustrative example in support of the investigation of this paper are also presented.