arXiv subjects

Ryoichi Kunisada

Publications and source records attributed to Ryoichi Kunisada.

9 recordsLinked to original sources

arXiv2023

On almost convergence on locally compact abelian groups

We study a summability method called almost convergence for bounded measurable functions defined on a locally compact abelian group. We define almost convergence using topologically invariant means and exhibit two different kinds of necessary and sufficient conditions, one is analytic and the other is functional analytic, for a given function to be almost convergent. As an application, we show complex Tauberian theorems for almost convergence on the integers and the real numbers. In particular, the latter one can be viewed as an analogue of the Wiener-Ikehara theorem.

math.FA↗

arXiv2023

On Almost convergence on the real line and its application to bounded analytic functions

We address the study of topologically invariant means and almost convergence on the real numbers $\mathbb{R}$. Here, the former is a certain class of invariant means on $L^{\infty}(\mathbb{R})$ and the latter is a summability method defined by them. Almost convergence on $\mathbb{R}$ was firstly introduced by Raimi (1957) as a generalization of Lorentz's almost convergence for bounded sequences. We extensively generalize his result of analytic characterization of almost convergence and explore its application to the theory of Hardy space. Specifically, we establish the relation between the asymptotic behavior on the imaginary axis and that at infinity of bounded analytic functions defined on the right half plane.

math.FA↗

arXiv2020

Convolution invariant linear functionals and applications to summability methods

We study topologically invariant means on $L^{\infty}(\mathbb{R})$, the set of all essentially bounded functions on the real line, and prove that invariance with respect to a single convolution operator is sufficient for a mean to be topologically invariant. We also consider some applications of this result to summability methods. In particular, the notion of almost convergence is introduced for a function in $L^{\infty}(\mathbb{R})$, and a Tauberian theorem concerning almost convergence and a summability method defined by a Wiener kernel is obtained. Further, for the $C_{\infty}$ summability method, which is defined by the limit of Hölder summability methods, we provide a necessary and sufficient condition for a given function to be $C_{\infty}$ summable.

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arXiv2019

On additive property of finitely additive measures

By the additive property, we mean a condition under which $L^p$ spaces over finitely additive measures are complete. Basile and Rao gives a necessary and sufficient condition that a finite sum of finitely additive measures has the additive property. We generalize this result to the case of a countable sum of finitely additive meaures. An application of this result to density measures are also presented.

math.FA↗

arXiv2017

Summability methods and Fourier analysis on $\mathbb{R}^{\times}$

We introduce a certain class of summability methods on $L^{\infty}([1, \infty))$ which is defined by convolution in the group algebra $L^1(\mathbb{R}^{\times})$. This class contains an integral version of Cesàro summability method and we particularly give a necessary and sufficient condition for a summability method in the class to equivalent to this one. Also, we consider another class of summability methods concerned with convolution in the group algebra $L^1(\mathbb{R})$ and give similar results.

math.FA↗

arXiv2017

Invariant linear functionals on $L^{\infty}(\mathbb{R}_+)$

We consider a continuous version of the classical notion of Banach limits, namely, positive linear functionals on $L^{\infty}(\mathbb{R}_+)$ invariant under translations $f(x) \mapsto f(x+s)$ of $L^{\infty}(\mathbb{R}_+)$ for every $s \ge 0$. We give its characterization in terms of the invariance under the operation of a certain linear mapping on $L^{\infty}(\mathbb{R}_+)$. Applications to summability methods are provided in the last section.

math.FA↗

arXiv2017

Functional analytic approach to Cesàro mean

We study a certain class $\mathcal{P}$ of positive linear functionals $φ$ on $L^{\infty}([1,\infty))$ for which $φ(f) = α$ if $\lim_{x \to \infty} \frac{1}{x} \int_1^x f(t)dt = α$. It turns out that translations $f(x) \mapsto f(rx)$ on $L^{\infty}([1, \infty))$, where $r \in [1, \infty)$, which are induced by the action of the multiplicative semigroup $[1, \infty)$ on itself, plays an intrinsic role in the study of $\mathcal{P}$. We also deal with an analogue $\mathcal{K}$ of $\mathcal{P}$ of positive linear functionals on $L^{\infty}([0, \infty))$ partaining to the action of the additive semigroup $[0, \infty)$ on itself. In particular, we give some expressions of maximal possible values of $\mathcal{P}$ and $\mathcal{K}$ for a given function respectively.

math.FA↗

arXiv2016

Density measures and additive property

We deal with finitely additive measures defined on all subsets of natural numbers which extend the asymptotic density (density measures). We consider a class of density measures which are constructed from free ultrafilters on natural numbers and study a certain additivity property of such density measures.

math.NT↗

arXiv2016

Density measures on a certain flow

We study finitely additive extensions of the asymptotic density to all the subsets of natural numbers. Such measures are called density measures. We consider a class of density measures constructed from free ultrafilters on $\mathbb{N}$ and investigate absolute continuity and singularity for those density measures. In particular, for any pair of such density measures we prove necessary and sufficient conditions that one is absolutely continuous with respect to the other and that they are singular. Also we prove the same results for weak absolute continuity and strong singularity.

math.NT↗