The Multiple K -Riemann Integral

The aim of this paper is to extend the notion of K -Riemann integrability of functions defined over a , b to functions defined over a rectangular box of ℝ n . As a generalization of step functions, we introduce a notion of K -step functions which allows us to give an equivalent definition of the K -Riemann integrable functions.

General Tauberian theorems for the Cesàro integrability of functions

For a locally integrable function f on {[0,\infty)}, we defineF(t)=\int_{0}^{t}f(u)\mathop{}\!du\quad\text{and}\quad\sigma_{\alpha}(t)=\int_% {0}^{t}\biggl{(}1-\frac{u}{t}\biggr{)}^{\alpha}f(u)\mathop{}\!dufor {t>0}. The improper integral {\int_{0}^{\infty}f(u)\mathop{}\!du} is said to be {(C,\alpha)} integrable to L for some {\alpha>-1} if the limit {\lim_{x\to\infty}\sigma_{\alpha}(t)=L} exists. It is known that {\lim_{t\to\infty}F(t)=\ell} implies {\lim_{t\to\infty}\sigma_{\alpha}(t)=\ell} for {\alpha>-1}, but the converse of this implication is not true in general. In this paper, we introduce the concept of the general control modulo of non-integer order for functions and obtain some Tauberian conditions in terms of this concept for the {(C,\alpha)} integrability method in order that the converse implication hold true. Our results extend the main theorems in [Ü. Totur and İ. Çanak, Tauberian conditions for the (C,\alpha) integrability of functions, Positivity 21 2017, 1, 73–83].

On weighted integrability of functions defined by trigonometric series with p-bounded variation coefficients

In this paper we introduce new classes of p-bounded variation sequences and give a sufficient and necessary condition for weighted integrability of trigonometric series with coefficients belonging to these classes. This is a generalization of the results obtained by the first author [J. Inequal. Appl. 2010:1–19, 2010] and Dyachenko and Tikhonov [Stud. Math. 193(3):285–306, 2009].

The (C, α) integrability of functions by weighted mean methods

Let p(x) be a nondecreasing continuous function on [0, ?) such that p(0) = 0 and p(t) ? ? as t ? ?. For a continuous function f (x) on [0, ?), we define s(t)= ?0t f(u)du and ?? (t) =?0t? (1- p(u)/p(t))? f(u)du. We say that a continuous function f (x) on [0, ?) is (C, ?) integrable to a by the weighted mean method determined by the function p(x) for some ? > ?1 if the limit limt?? ?? (t) = a exists. We prove that if the limit limt?? ?? (t) = a exists for some ? > ?1, then the limit limt?? ??+h (t) = a exists for all h > 0. Next, we prove that if the limit limt?? ?? (t) = a exists for some ? > 0 and p(t)/p?(t) f(t)= O(1), t ? ?, then the limit limt?? ???1 (t) = a exists.

Sharp Exponential Integrability for Traces of Monotone Sobolev Functions

We answer a question posed in [12] on exponential integrability of functions of restricted n-energy. We use geometric methods to obtain a sharp exponential integrability result for boundary traces of monotone Sobolev functions defined on the unit ball.