Chebyshev upper estimates for Beurling's generalized prime numbers

Let N be the counting function of a Beurling generalized number system and let pi be the counting function of its primes. We show that the L-1-condition integral(infinity)(1)vertical bar N(x) - ax/x vertical bar dx/x < infinity and the asymptotic behavior N(x) = ax + O (x/log x), for some a > 0, suffice for a Chebyshev upper estimate pi(x) log x/x <= B < infinity

The structure of quasiasymptotics of Schwartz distributions

In this article complete characterizations of quasiasymptotic behaviors of Schwartz distributions are presented by means of structural theorems. The cases at infinity and the origin are both analyzed. Special attention is paid to the quasiasymptotic of degree -1 and it is shown how the structural theorem can be used to study Ces\`{a}ro and Abel summability of trigonometric series and integrals. Further properties of quasiasymptotics at infinity are discussed, the author presents a condition over test functions which allows one to evaluate them at the quasiasymptotic, these test functions are in bigger spaces than $\mathcal{S}$. An extension of the structural theorems for quasiasymptotics is given, the author studies a structural characterization of the behavior $f(\lambda x)=O(\rho(\lambda))$ in $\mathcal{D'}$, where $\rho$ is a regularly varying function

Structural theorems for quasiasymptotics of distributions at infinity

Complete structural theorems for quasiasymptotics of distributions are presented in this article. For this, asymptotically homogeneous functions and associate asymptotically homogeneous functions at infinity with respect to a slowly varying function are employed. The proposed analysis, based on the concept of asymptotically and associate asymptotically homogeneous functions, allows to obtain easier proofs of the structural theorems for quasiasymptotics at infinity in the so far only known case: when the degree of the quasiasymptotic is not a negative integer. Furthermore, new structural theorems for the case of negative integral degrees are obtained by this method

Structural theorems for quasiasymptotics of ultradistributions

We provide complete structural theorems for the so-called quasiasymptotic behavior of non-quasianalytic ultradistributions. As an application of these results, we obtain descriptions of quasiasymptotic properties of regularizations at the origin of ultradistributions and discuss connections with Gelfand-Shilov type spaces

A generalization of the Banach-Steinhaus theorem for finite part limits

It is well known, as follows from the Banach-Steinhaus theorem, that if a sequence $\left\{ y_{n}\right\} _{n=1}^{\infty}$ of linear continuous functionals in a Fr\'{e}chet space converges pointwise to a linear functional $Y,$ $Y\left( x\right) =\lim_{n\rightarrow\infty}\left\langle y_{n} ,x\right\rangle$ for all $x,$ then $Y$ is actually continuous. In this article we prove that in a Fr\'{e}chet space\ the continuity of $Y$ still holds if $Y$ is the \emph{finite part} of the limit of $\left\langle y_{n},x\right\rangle$ as $n\rightarrow\infty.$ We also show that the continuity of finite part limits holds for other classes of topological vector spaces, such as \textsl{LF}-spaces, \textsl{DFS}-spaces, and \textsl{DFS} $^{\ast}$-spaces,\ and give examples where it does not hold

An elementary approach to asymptotic behavior in the Cesaro sense and applications to Stieltjes and Laplace transforms

We present an elementary approach to asymptotic behavior of generalized functions in the Cesaro sense. Our approach is based on Yosida's subspace of Mikusinski operators. Applications to Laplace and Stieltjes transforms are given

Multidimensional Tauberian theorems for vector-valued distributions

We prove several Tauberian theorems for regularizing transforms of vector-valued distributions. The regularizing transform of f is given by the integral transform M-phi(f)(x, y) = (f * phi(y))(x), (x, y) is an element of R-n x R+, with kernel phi(y) (t) = y(-n)phi(t/y). We apply our results to the analysis of asymptotic stability for a class of Cauchy problems, Tauberian theorems for the Laplace transform, the comparison of quasiasymptotics in distribution spaces, and we give a necessary and sufficient condition for the existence of the trace of a distribution on {x(0)} x R-m. In addition, we present a new proof of Littlewood's Tauberian theorem

Rotation invariant ultradistributions

We prove that an ultradistribution is rotation invariant if and only if it coincides with its spherical mean. For it, we study the problem of spherical representations of ultradistributions on $\mathbb{R}^{n}$. Our results apply to both the quasianalytic and the non-quasianalytic case

On Romanovski's lemma

Romanovski introduced a procedure, Romanovski's lemma, to construct the Denjoy integral without the use of transfinite induction. Here we give two versions of Romanovski's lemma which hold in general topological spaces. We provide several applications in various areas of mathematics