Degree
Doctor of Philosophy (PhD)
Department
Mathematics
Document Type
Dissertation
Abstract
Each chapter in this dissertation focuses on some of the most fundamental objects a student may encounter in analysis: sequences, series, and integrals. The common thread that runs through each of these chapters is the connection that these fundamental objects have to \textit{Distributions} understood in the sense of generalized functions.
Chapter \ref{1} introduces the notion of $\nu-$separated increasing sequences $\left\{ x_{n}\right\} _{n=1}^{\infty}$. We establish that interpolation problems of the kind $\varphi\left( x_{n}\right) =z_{n}$ have solutions $\varphi \in\mathcal{S}\left( \mathbb{R}\right) $ for all sequences $\left\{ z_{n}\right\} $ of rapid decay in the sense that $z_{n}=o\left( x_{n}^{-\alpha}\right) $ for all $\alpha>0$ if and only if $\left\{ x_{n}\right\} $ is $\nu-$separated for some $\nu.$ We also give several generalizations of this result.
In chapter \ref{2}, we consider questions related to the behavior of moments $M_{m}\left( \left\{ z_{j}\right\} \right) $. We introduce the notion of symmetrical series of order $n$, for $n\geq 2$. We prove that when $\left\{ z_{j}\right\} \in l^{p}$ for some $p$ then several results characterizing the sequence from its moments hold. We then construct examples of sequences whose moments vanish with required density. Lastly, we construct counterexamples of several of the results valid in the $l^{p}$ case if we allow the moment series to all be \emph{conditionally convergent. }We show that for each \emph{arbitrary} sequence of real numbers $\left\{ \mu _{m}\right\} _{m=0}^{\infty}$ there are real sequences $\left\{ u_{j}\right\} _{j=0}^{\infty}$ such that% \[ \sum_{j=0}^{\infty}u_{j}^{2m+1}=\mu_{m}\,,\ \ \ m\geq0\,. \]
In chapters \ref{3} and \ref{4}, we consider Frullani's integral formula. In chapter 3, the existence of Frullani integral in the distributional sense is proven to be equivalent to the existence of the distributional point value at zero and Ces\`{a}ro limit at infinity. We draw connections to finite parts and Ces\`{a}ro summability culminating in applications. In Chapter 4, we discuss the different, albeit equivalent conditions provided by Iyengar and Ostrowski for the existence of Frullani Integral. We then identify other equivalent conditions and show that these conditions are solutions to a family of linear differential equations of the first order. We study the limiting behavior of these solutions at zero and infinity, providing applications of our results towards the end.
Date
5-22-2025
Recommended Citation
Ahuja, Jiten Amar, "Topics in Series, Integrals, and Distributions" (2025). LSU Doctoral Dissertations. 6792.
https://repository.lsu.edu/gradschool_dissertations/6792
Committee Chair
Estrada, Ricardo