TAUBERIAN THEOREMS FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY OF INTEGRALS

Let $q$ be a positive weight function on $\mathbf{R}_{+}:=[0, \infty)$ which is integrable in Lebesgue's sense over every finite interval $(0,x)$ for $00$, $Q(0)=0$ and $Q(x) \rightarrow \infty$ as $x \to \infty$.Given a real or complex-valued function $f \in L^{1}_{loc} (\mathbf{R}_{+})$, we define $s(x):=\int_{0}^{x}f(t)dt$ and$$\tau^{(0)}_q(x):=s(x), \tau^{(m)}_q(x):=\frac{1}{Q(x)}\int_0^x \tau^{(m-1)}_q(t) q(t)dt\,\,\, (x>0, m=1,2,...),$$provided that $Q(x)>0$. We say that $\int_{0}^{\infty}f(x)dx$ is summable to $L$ by the $m$-th iteration of weighted mean method determined by the function $q(x)$, or for short, $(\overline{N},q,m)$ integrable to a finite number $L$ if$$\lim_{x\to \infty}\tau^{(m)}_q(x)=L.$$In this case, we write $s(x)\rightarrow L(\overline{N},q,m)$. It is known thatif the limit $\lim _{x \to \infty} s(x)=L$ exists, then $\lim _{x \to \infty} \tau^{(m)}_q(x)=L$ also exists. However, the converse of this implicationis not always true. Some suitable conditions together with the existence of the limit $\lim _{x \to \infty} \tau^{(m)}_q(x)$, which is so called Tauberian conditions, may imply convergence of $\lim _{x \to \infty} s(x)$. In this paper, one- and two-sided Tauberian conditions in terms of the generating function and its generalizations for $(\overline{N},q,m)$ summable integrals of real- or complex-valued functions have been obtained. Some classical type Tauberian theorems given for Ces\`{a}ro summability $(C,1)$ and weighted mean method of summability $(\overline{N},q)$ have been extended and generalized. 

Tauberian theorems for the weighted mean method of summability of integrals

3rd International Conference of Mathematical Sciences (ICMS) -- SEP 04-08, 2019 -- Maltepe Univ, Istanbul, TURKEYWOS: 000505225800026Let q be a positive weight function on R+ := [0, infinity) which is integrable in Lebesgue's sense over every finite interval (0, x) for 0 0, Q(0) = 0 and Q(x) -> infinity as x -> infinity. Given a real or complex-valued function f is an element of L-loc(1)(R+), we define s(x) := integral(x)(0) f(t)dt and tau((0))(q)(x) := s(x), tau((m))(q)(x) := 1/Q(x) integral(x)(0) tau((m 1))(q)(t)q(t)dt (x > 0, m = 1, 2, ...), provided that Q(x) > 0. We say that integral(infinity)(0) f(x)dx is summable to L by the m-th iteration of weighted mean method determined by the function q(x), or for short, ((N) over bar, q, m) integrable to a finite number L if lim(x ->infinity) tau((m))(q)(x) = L. In this case, we write s(x) -> L((N) over bar, q, m). It is known that if the limit lim(x ->infinity) s(x) = L exists, then lim(x ->infinity) tau((m))(q)(x) = L also exists. However, the converse of this implication is not always true. Some suitable conditions together with the existence of the limit lim(x ->infinity) tau((m))(q)(x), which is so called Tauberian conditions, may imply convergence of lim(x ->infinity) s(x). In this paper, one- and two-sided Tauberian conditions in terms of the generating function and its generalizations for ((N) over bar, q, m) summable integrals of real- or complex-valued functions have been obtained. Some classical type Tauberian theorems given for Cesaro summability (C, 1) and weighted mean method of summability ((N) over bar, q) have been extended and generalized

On the generalized Mellin integral operators

In this study, we give a modification of Mellin convolution-type operators. In this way, we obtain the rate of convergence with the modulus of the continuity of the mmth-order Mellin derivative of function ff, but without the derivative of the operator. Then, we express the Taylor formula including Mellin derivatives with integral remainder. Later, a Voronovskaya-type theorem is proved. In the last part, we state order of approximation of the modified operators, and the obtained results are restated for the Mellin-Gauss-Weierstrass operator

Differentiated Bernstein Type Operators

ARAL, Ali/0000-0002-2024-8607WOS:000553032700001The present paper deals with the derivatives of Bernstein type operators preserving some exponential functions. We investigate the uniform convergence of the differentiated operators. The rate of convergence by means of a modulus of continuity is studied, an upper estimate theorem for the difference of new constructed differentiated Bernstein type operators is presented.TUBITAK (The Scientific and Technological Research Council of Turkey)Turkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK) [1002, 119F191]; TUBITAKTurkiye Bilimsel ve Teknolojik Arastirma Kurumu (TUBITAK)1. The second author has been supported within TUBITAK (The Scientific and Technological Research Council of Turkey) 1002 -Project 119F191 and the third author would like to thank to TUBITAK for their financial supports during his PhD studies