Gumrah Uysal

Gumrah Uysal

Ph.D. in Mathematics
Independent researcher...

About

42
Publications
3,017
Reads
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91
Citations
Introduction
Interested in Korovkin - type approximation, Linear positive operators, nonlinear analysis, theory of real and complex functions, numerical analysis, Fourier analysis and approximation theory.
Education
February 2011 - May 2016
Ankara University
Field of study
  • Mathematics
September 2008 - June 2010
Sakarya University
Field of study
  • Mathematics
September 2006 - June 2007
Dokuz Eylul University
Field of study
  • Mathematics

Publications

Publications (42)
Article
Full-text available
In this paper, we give some pointwise convergence and Fatou type convergence theorems for a family of nonlinear bivariate m1,m2− singular integral operators in the following form: Tωm1,m2f;x,y=∬R2Kωt,s,∑v1=1m1∑v2=1m2(−1)(v1+v2)m1v1m2v2fx+v1t,y+v2sdsdt, where m1,m2 ≥ 1 are fixed natural numbers, x,y∈R2 and ω ∈ Ω, Ω denotes a nonempty set of indices...
Article
In this paper, we present some theorems on pointwise convergence and the rate of pointwise convergence for the family of nonlinear bivariate singular integral operators of the following form: (EQUATION PRESENTED) where f is a real valued and integrable function on a bounded arbitrary closed, semi-closed or open region D = 〈a, b〉 × 〈c, d〉 in ℝ² or D...
Article
In the present paper, we consider a general class of operators enriched with some properties in order to act on \begin{document}$ C^{\ast }( \mathbb{R} _{0}^{+}) $\end{document}. We establish uniform convergence of the operators for every function in \begin{document}$ C^{\ast }( \mathbb{R} _{0}^{+}) $\end{document} on \begin{document}$ \mathbb{R} _...
Article
Full-text available
We propose a modification for moment-type operators in order to preserve the exponential function $e^{2cx}$ with $c>0$ on real axis. First, we present moment identities. Then, we prove two weighted convergence theorems. Finally, we present a Voronovskaya-type theorem for the new operators.
Conference Paper
Full-text available
In this work, we will prove some theorems for nonlinear multivariate convolution operators in order to approximate one-sided partial derivatives of functions of multivariables by using extended definition of the notion of one-sided derivative in univariate case.
Article
Full-text available
In this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.
Article
Full-text available
We propose two modifications for Gauss-Weierstrass operators and moment-type operators which fix e ax and e 2ax with a > 0. First, we present moment identities for new operators. Then, we discuss weighted approximation and prove Voronovskaya-type theorems for them in exponentially weighted spaces. Using modulus of continuity in exponentially weight...
Chapter
This chapter consists of five sections. First section is devoted to introduction part in which the description of the problem is presented and theoretical background is given. In the second section, the preliminary concepts which are utilized in the sequel are introduced. Then, the conditions under which double singular integral operators involving...
Conference Paper
Full-text available
In this study, unlike the previous studies, a theorem concerning exponentially weighted pointwise approximation by nonlinear integral operators with non-isotropic kernels at continuity points of integrable functions is proved.
Conference Paper
Full-text available
In the current manuscript, modified Picard operators defined by Agratini et al. [Positivity 21 (2017), no. 3, 1189-1199] are incorporated by m-singular integral operators defined by Mamedov [Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 287-304] and a pointwise approximation result is proved as a continuation of the work by Aral et al. [Positivity (201...
Conference Paper
Full-text available
In the current study, normwise approximation results inspired by the work of Aral et al. [Positivity (2019), 1-13, doi.org/10.1007/s11117-019-00687-z] are proved for m-singular modified Picard operators which are obtained by blending modified Picard operators defined by Agratini et al. [Positivity 21 (2017), no. 3, 1189-1199] with m-singular settin...
Chapter
Let Λ be a non-empty index set consisting of σ indices and σ0 is allowed to be either accumulation point of Λ or infinity. We assume that the function Kσ, Kσ:R×R→R, has finite Lebesgue integral value on R for all values of its second variable and for any σ∈Λ and satisfies some conditions. The main purpose of this work is to investigate the conditio...
Chapter
In this chapter, m-singularity notion is discussed for double singular integral operators. In this direction, several results concerning pointwise convergence of nonlinear double m-singular integral operators are presented. This chapter is divided into six sections. In the first section, the reasons giving birth to m-singularity notion are explaine...
Article
Full-text available
In this paper, we present some theorems concerning weighted pointwise convergence of nonlinear singular integral operators of the form: (T f)(x) = b Z a K (t x; f (t)) dt; x 2 (a; b) ; 2 ; where (a; b) is a certain, …nite interval in R; is a non-empty set of indices and f is measurable function on (a; b) in the sense of Lebesgue.
Article
In this study, we investigate the conditions under which weighted pointwise convergence exists for the singular integral operators involving power nonlinearity, that is the operators under the spotlights contain the term f m, where m ≥ 1 is a natural number.
Article
Full-text available
The paper is devoted to the study of pointwise approximation of functions f in L1;phi(D) by double singular integral operators with radial kernels at mu-generalized Lebesgue points. Here, phi : R2 -> R+ is a weight function satisfying some sharp conditions including almost everywhere differentiability on its domain, and L1;phi' (D) is the collectio...
Article
Full-text available
In this paper, we present some theorems on weighted approximation by two dimensional nonlinear singular integral operators in the following form: Tλ(f;x,y)=∬R2Kλ(t−x,s−y,f(t,s))dsdt,(x,y)∈R2,λ∈Λ,$${T_\lambda }(f;x,y) = \iint\limits_{{\mathbb{R}^2}}K_\lambda {(t - x,s - y,f(t,s))dsdt,\;(x,y) \in {\mathbb{R}^2},\lambda \in \Lambda ,}$$ where Λ is a s...
Article
In this paper, we present some general results on the pointwise convergence of the non-convolution type nonlinear singular integral operators in the following form: $$T_{\lambda}(f;x)
Article
Full-text available
In this study, we present some results on the weighted pointwise convergence of a family of singular integral operators with radial kernels given in the following form: $$\begin{aligned} L_{\lambda }\left( f;x,y\right) =\underset{ \mathbb {R} ^{2}}{\iint }f\left( t,s\right) H_{\lambda }\left( t-x,s-y\right) \mathrm{d}s\,\mathrm{d}t,\quad \left( x,y...
Conference Paper
Full-text available
In this paper we present some theorems concerning existence and Fatou type weighted pointwise convergence of nonlinear singular integral operators of the form: (Tλf)(x)=∫RKλ(t−x; f(t))dt, x∈R, λ∈Λ$({T_\lambda }f)(x) = \int\limits_R {{K_\lambda }} (t - x;{\rm{ }}f(t))dt,{\rm{ x}} \in R,{\rm{ }}\lambda \in \Lambda $ where Λ ≠ ∅ is a set of non-negat...
Article
Full-text available
The paper is devoted to the study of pointwise approximation of functions f ∈ L p,ϕ (D) by double singular integral operators with radial kernels at p−generalized Lebesgue points. Here, ϕ > 0 is a weight function satisfying some sharp conditions and L p,ϕ (D) is the collection of all measurable and non-integrable functions for which f ϕ p is integr...
Conference Paper
In this work, we present extra results on the weighted pointwise convergence for a family of singular integral operators with radial kernels. Also, we present a theorem concerning the rate of pointwise convergence.
Article
In the present paper, we study the pointwise approximation of multidimensional singular integral operators with radial kernel such that Hλ (t − x) = Kλ (|t − x|) of the form: Lλ (f; x) = Z D f (t) Kλ (|t − x|) dt, x ∈ D where D = n Π i=1 < ai , bi > is open, semi-open or closed multidimensional arbitrary bounded box in Rn or D = Rn, λ ∈ Λ ⊂ R + 0 ,...
Article
Full-text available
In this paper, the pointwise approximation of nonlinear singular integral operators of the form:% \begin{equation*} T_{\lambda }\left( f;x\right) =\underset{D}{\int }K_{\lambda }\left( t-x;f(t)\right) dt,\text{ }x\in D,\text{ }\lambda \in \Lambda \end{equation*}% where$\ D=$ is open, semi-open or closed arbitrary bounded interval in $% %TCIMACRO{\U...
Article
In this paper we study the pointwise approximation of functions which are non-integrable on D =< -π,π > x < -π,π > where 〈-π,π〉 x 〈-π,π〉 is a closed, semi-closed or open region, by double singular integral operators with radial kernels of the form: Lλ= (f;x,y)= ∫Df (s,t)Kλ(√(s - x)2+(t - y)2)dsdtat generalized Lebesgue point. Also we investigate th...
Article
Full-text available
In the present work we prove the pointwise convergence and the rate of pointwise convergence for a family of singular integral operators with radial kernel in two-dimensional setting in the following form: Lx(f;x,y) = integral integral(D)f(t,s)H-lambda(t-x,s-y)dt ds, (x,y) is an element of D, where D = < a, b > x < c,d > is an arbitrary closed, sem...
Article
In this paper we prove the pointwise convergence and the rate of pointwise convergence for a family of singular integral operators with radial kernel in two-dimensional setting in the following form: L-lambda(f; x, y) = integral integral(D) f (s, t) H-lambda (s-x, t-y) ds dt, (x, y) is an element of D, where D = < a, b > x < c, d > (< a, b > x < c,...

Questions

Questions (3)
Question
I am exacty searching for the case p=infinity and q=1. That inequality contains supremum.
Question
I mean what happens if v1'y1+v2'y2=0 is changed with an another condition? Is this certain other than the comfort of the getting desired solution?
Question
Does proof by contradiction method make you convinced ?
Explanation: In mathematics, the proof by contradiction method supposes the converse of the hypothesis is true then arrives that the converse is contradicting with the first assumption. Hence the hypothesis is true. This method based on the idea such that if the converse is false then the straight is true. However in this method i think we miss some other cases. Personally i believe in constructive proofs.

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