arXiv subjects
Publications and source records attributed to Jasson Vindas.
Let $N$ be the counting function of a Beurling generalized number system and let $π$ be the counting function of its primes. We show that the $L^{1}$-condition $$ \int_{1}^{\infty}|\frac{N(x)-ax}{x}|\frac{\mathrm{d}x}{x}<\infty $$ and the asymptotic behavior $$N(x)=ax+O(\frac{x}{\log x}),$$ for some $a>0$, suffice for a Chebyshev upper estimate $$ \frac{π(x)\log x}{x}\leq B<\infty. $$
We characterize Schwartz distributions having a value at a single point in the sense introduced by means of nonstandard analysis by A. Robinson. They appear to be distributions continuous in a neighborhood of the point.
We give necessary and sufficient criteria for a distribution to be smooth or uniformly Hölder continuous in terms of approximation sequences by smooth functions; in particular, in terms of those arising as regularizations $(T\astϕ_{n})$.
We provide several general versions of Littlewood's Tauberian theorem. These versions are applicable to Laplace transforms of Schwartz distributions. We apply these Tauberian results to deduce a number of Tauberian theorems for power series where Cesàro summability follows from Abel summability. We also use our general results to give a new simple proof of the classical Littlewood one-sided Tauberian theorem for power series.
We define an integral, the distributional integral of functions of one real variable, that is more general than the Lebesgue and the Denjoy-Perron-Henstock-Kurzweil integrals, and which allows the integration of functions with distributional values everywhere or nearly everywhere. Our integral has the property that if $f$ is locally distributionally integrable over the real line and $ψ\in\mathcal{D}(\mathbb{R}%) $ is a test function, then $fψ$ is distributionally integrable, and the formula% [<\mathsf{f},ψ> =(\mathfrak{dist}) \int_{-\infty}^{\infty}f(x) ψ(x) \,\mathrm{d}% x\,,] defines a distribution $\mathsf{f}\in\mathcal{D}^{\prime}(\mathbb{R}) $ that has distributional point values almost everywhere and actually $\mathsf{f}(x) =f(x) $ almost everywhere. The indefinite distributional integral $F(x) =(\mathfrak{dist}) \int_{a}^{x}f(t) \,\mathrm{d}t$ corresponds to a distribution with point values everywhere and whose distributional derivative has point values almost everywhere equal to $f(x).$ The distributional integral is more general than the standard integrals, but it still has many of the useful properties of those standard ones, including integration by parts formulas, substitution formulas, even for infinite intervals --in the Cesàro sense--, mean value theorems, and convergence theorems. The distributional integral satisfies a version of Hake's theorem. Unlike general distributions, locally distributionally integrable functions can be restricted to closed sets and can be multiplied by power functions with real positive exponents.
We study several Tauberian properties of regularizing transforms of tempered distributions with values in Banach spaces, that is, transforms of the form $M^{\mathbf{f}}_ϕ(x,y)=(\mathbf{f}\astϕ_{y})(x)$, where the kernel $ϕ$ is a test function and $ϕ_{y}(\cdot)=y^{-n}ϕ(\cdot/y)$. If the zeroth moment of $ϕ$ vanishes, it is a wavelet type transform; otherwise, we say it is a non-wavelet type transform. The first aim of this work is to show that the scaling (weak) asymptotic properties of distributions are \emph{completely} determined by boundary asymptotics of the regularizing transform plus natural Tauberian hypotheses. Our second goal is to characterize the spaces of Banach space-valued tempered distributions in terms of the transform $M^{\mathbf{f}}_ϕ(x,y)$. We investigate conditions which ensure that a distribution that a priori takes values in locally convex space actually takes values in a narrower Banach space. Special attention is paid to find the \emph{optimal} class of kernels $ϕ$ for which these Tauberian results hold. We give various applications of our Tauberian theory in the pointwise and (micro-)local regularity analysis of Banach space-valued distributions, and develop a number of techniques which are specially useful when applied to scalar-valued functions and distributions. Among such applications, we obtain the full weak-asymptotic series expansion of the family of Riemann-type distributions $R_β(x)=\sum_{n=1}^{\infty}e^{iπxn^{2}}/n^{2β}$, $β\in\mathbb{C}$, at every rational point. We also apply the results to regularity theory within generalized function algebras, to the stabilization of solutions for a class of Cauchy problems, and to Tauberian theorems for the Laplace transform.