The Experts below are selected from a list of 34638 Experts worldwide ranked by ideXlab platform
Uğur Kadak - One of the best experts on this subject based on the ideXlab platform.
-
Relative weighted almost convergence based on fractional-order Difference Operator in multivariate modular function spaces
Revista de la Real Academia de Ciencias Exactas Físicas y Naturales. Serie A. Matemáticas, 2018Co-Authors: Uğur KadakAbstract:In the present paper, we introduce the concept of relative weighted almost convergence and its weighted statistical extensions in multivariate modular function spaces based on a new type fractional-order double Difference Operator $$\nabla ^{a,b,c}_h$$ . We first define the concepts of weighted almost $$\nabla $$ -statistical convergence and statistical weighted almost $$\nabla $$ -convergence of double sequences. By using the notion of relative uniform convergence involving a scale function $$\sigma $$ , we introduce modular relative weighted almost statistical convergence and modular relative statistical weighted almost convergence of double sequences. We then obtain some inclusion relations between these proposed methods and provide some counterexamples that show that these are non-trivial and proper extensions of the existing literature on this topic. Moreover, we apply the relative statistical weighted almost convergence of a double sequence of positive linear Operators to prove some Korovkin-type theorems in multivariate modular spaces by considering several kinds of test functions. Finally, we present a non-trivial application to generalized Boolean sum (GBS) Operators of bivariate generalized Bernstein–Durrmeyer Operators.
-
generalized statistically almost convergence based on the Difference Operator which includes the p q gamma function and related approximation theorems
Results in Mathematics, 2018Co-Authors: Uğur Kadak, S A MohiuddineAbstract:This paper is devoted to extend the notion of almost convergence and its statistical forms with respect to the Difference Operator involving (p, q)-gamma function and an increasing sequence $$(\lambda _n)$$ of positive numbers. We firstly introduce some new concepts of almost $${\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )$$ -statistical convergence, statistical almost $${\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )$$ -convergence and strong almost $$[{\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )]_r$$ -convergence. Moreover, we present some inclusion relations between these newly proposed methods and give some counterexamples to show that these are non-trivial generalizations of existing literature on this topic. We then prove a Korovkin type approximation theorem for functions of two variables through statistically almost $${\Delta }^{[a,b,c]}_{h, \alpha , \beta }(\lambda )$$ -convergence and also present an illustrative example via bivariate non-tensor type Meyer–Konig and Zeller generalization of Bernstein power series. Furthermore, we estimate the rate of almost convergence of approximating linear Operators by means of the modulus of continuity and derive some Voronovskaja type results by using the generalized Meyer–Konig and Zeller Operators. Finally, some computational and geometrical interpretations for the convergence of Operators to a function are presented.
-
generalized weighted invariant mean based on fractional Difference Operator with applications to approximation theorems for functions of two variables
Results in Mathematics, 2017Co-Authors: Uğur KadakAbstract:In the present article, following a very recent and new approach of Baliarsingh (Alexandria Eng. J., 55(2):1811–1816, 2016), we introduce the notions of statistically \(\sigma \)-convergence and \(\sigma \)-statistically convergence by the weighted method with respect to the Difference Operator \(\varDelta ^{\alpha ,\beta ,\gamma }_h\). Some inclusion relations between proposed methods are examined. As an application, we prove a Korovkin type approximation theorem for functions of two variables. Also, by using generalized Meyer–Konig and Zeller Operator, we present an example such that our proposed method works but its classical and statistical versions do not work. Finally, we estimate the rate of statistically weighted \(\sigma (\varDelta ^{\alpha ,\beta ,\gamma }_h)\)-convergence of approximating positive linear Operators and give a Voronovskaja-type theorem.
-
weighted statistical convergence based on generalized Difference Operator involving p q gamma function and its applications to approximation theorems
Journal of Mathematical Analysis and Applications, 2017Co-Authors: Uğur KadakAbstract:Abstract In this work, following a new approach of Baliarsingh (2016) [6] , we introduce the concepts of statistically weighted Ψ Δ p , q -summability, weighted Ψ Δ p , q -statistical convergence and weighted strongly Ψ Δ p , q -summability with respect to the Difference Operator Δ h , p , q α , β , γ including ( p , q ) -analogue of Gamma function. Some inclusion relations between newly proposed methods are examined. We then prove a Korovkin type approximation theorem for functions of two variables and also present an example via ( p , q ) -analogue of modified Bernstein–Schurer Operators to show that our proposed method is stronger than its classical and weighted statistical versions. Furthermore, we compute the rate of convergence of approximating positive linear Operators through the modulus of continuity. Finally, we present computational and geometrical approaches to illustrate some of our results in this paper.
Mohamed Ali Mourou - One of the best experts on this subject based on the ideXlab platform.
-
taylor series associated with a differential Difference Operator on the real line
Journal of Computational and Applied Mathematics, 2003Co-Authors: Mohamed Ali MourouAbstract:We extend the classical theory of Taylor series to a first-order differential-Difference Operator ? on the real line which includes as a particular case the Dunkl Operator associated with the reflection group Z2 on R. More precisely, we establish first a generalized Taylor formula with integral remainder, and then specify sufficient conditions for a function on R to be expanded as a generalized Taylor series. Moreover, we provide a criterion of analyticity for functions on R involving the differential-Difference Operator ?.
-
transmutation Operators and paley wiener theorem associated with a singular differential Difference Operator on the real line
Analysis and Applications, 2003Co-Authors: Mohamed Ali Mourou, Khalifa TrimecheAbstract:We consider a singular differential-Difference Operator Λ on the real line which includes, as particular case, the Dunkl Operator associated with the reflection group Z2 on R. We exhibit a Laplace integral representation for the eigenfunctions of the Operator Λ. From this representation, we construct a pair of integral transforms which turn out to be transmutation Operators of Λ into the first derivative Operator d/dx. We exploit these transmutation Operators to develop a new commutative harmonic analysis on the real line corresponding to the Operator Λ. In particular, we establish a Paley–Wiener theorem and a Plancherel theorem for the Fourier transform associated to Λ.
-
transmutation Operators associated with a dunkl type differential Difference Operator on the real line and certain of their applications
Integral Transforms and Special Functions, 2001Co-Authors: Mohamed Ali MourouAbstract:We consider a singular differential-Difference Operator Λ on the real line which generalizes the Dunkl Operator associated with the reflection group Z2 on R. We construct transmutation Operators between Λ and the first derivative Operator d/dx. We exploit these transmutation Operators, firstly to determine the elementary solution of certain classes of singular differential-Difference Operators on a product of Euclidean spaces, and secondly to introduce a generalized translation on the real line corresponding to the Operator Λ
P. Baliarsingh - One of the best experts on this subject based on the ideXlab platform.
-
Compactness of binomial Difference Operator of fractional order and sequence spaces
Rendiconti del Circolo Matematico di Palermo Series 2, 2019Co-Authors: Taja Yaying, Bipan Hazarika, P. BaliarsinghAbstract:In this article we introduce binomial Difference sequence spaces of fractional order $$\alpha ,$$ α , $$b_0^{r,s}\left( \Delta ^{(\alpha )}\right) ,$$ b 0 r , s Δ ( α ) , $$b_c^{r,s}\left( \Delta ^{(\alpha )}\right) $$ b c r , s Δ ( α ) and $$b_{\infty }^{r,s}\left( \Delta ^{(\alpha )}\right) $$ b ∞ r , s Δ ( α ) by employing fractional Difference Operator $$\Delta ^{(\alpha )},$$ Δ ( α ) , defined by $$\Delta ^{(\alpha )}x_k=\sum \limits _{i=0}^{\infty }(-1)^i\frac{\Gamma (\alpha +1)}{i!\Gamma (\alpha -i+1)}x_{k-i}.$$ Δ ( α ) x k = ∑ i = 0 ∞ ( - 1 ) i Γ ( α + 1 ) i ! Γ ( α - i + 1 ) x k - i . We give some topological properties, obtain the Schauder basis and determine the $$\alpha -,$$ α - , $$\beta -$$ β - and $$\gamma -$$ γ - duals of the spaces. We characterize the matrix classes $$(b_c^{r,s}(\Delta ^{(\alpha )}),\ell _p),$$ ( b c r , s ( Δ ( α ) ) , ℓ p ) , $$(b_c^{r,s}(\Delta ^{(\alpha )}),\ell _{\infty })$$ ( b c r , s ( Δ ( α ) ) , ℓ ∞ ) and $$(b_c^{r,s}(\Delta ^{(\alpha )}),c).$$ ( b c r , s ( Δ ( α ) ) , c ) . We characterize certain classes of compact Operators on the space $$b_c^{r,s}(\Delta ^{(\alpha )})$$ b c r , s ( Δ ( α ) ) using Hausdorff measure of non-compactness. Finally, we present the graphical interpretation of the Operator $$B^{r,s}\left( \Delta ^{(\alpha )}\right) $$ B r , s Δ ( α ) .
-
On certain Toeplitz matrices via Difference Operator and their applications
Afrika Matematika, 2016Co-Authors: P. Baliarsingh, S. DuttaAbstract:In this paper, we introduce a new Difference Operator $$B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})$$ B ( r ~ , s ~ , t ~ , u ~ ) , defined by $$(B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})x)_k=r_kx_k+s_{k-1}x_{k-1}+t_{k-2}x_{k-2}+u_{k-3}x_{k-3},$$ ( B ( r ~ , s ~ , t ~ , u ~ ) x ) k = r k x k + s k - 1 x k - 1 + t k - 2 x k - 2 + u k - 3 x k - 3 , where $$\tilde{r},\tilde{s},\tilde{t},\tilde{u}$$ r ~ , s ~ , t ~ , u ~ are convergent sequences of real numbers satisfying certain conditions and any term with negative subscript is equal to zero. In fact, as the generalizations of most of the Difference Operators, the Operator $$B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})$$ B ( r ~ , s ~ , t ~ , u ~ ) includes the Difference Operators $$\Delta ^{1}, \Delta ^{2}, \Delta ^{3}, \Delta _\nu ,\Delta _{uv},B(r,s)$$ Δ 1 , Δ 2 , Δ 3 , Δ ν , Δ u v , B ( r , s ) and B ( r , s , t ). We determine certain Toeplitz matrices such as Fibonacci and other summable matrices which can be obtained immediately by taking the inverse of the Difference Operator $$B(\tilde{r},\tilde{s},\tilde{t},\tilde{u})$$ B ( r ~ , s ~ , t ~ , u ~ ) under some limiting conditions. Moreover, their spectral and other basic properties have been studied.
-
On a fractional Difference Operator
Alexandria Engineering Journal, 2016Co-Authors: P. BaliarsinghAbstract:Abstract In the present article, a set of new Difference sequence spaces of fractional order has been introduced and subsequently, an application of these spaces, the notion of the derivatives and the integrals of a function to the case of non-integer order have been generalized. Certain results involving the unusual and non-uniform behavior of the corresponding Difference Operator have been investigated and also been verified by using some counter examples. We also verify these unusual and non-uniform behaviors by studying the geometry of fractional calculus.
-
On the Spectral Properties of the Weighted Mean Difference Operator over the Sequence Space
International Journal of Analysis, 2014Co-Authors: P. Baliarsingh, S. DuttaAbstract:In the present work the generalized weighted mean Difference Operator has been introduced by combining the generalized weighted mean and Difference Operator under certain special cases of sequences and . For any two sequences and of either constant or strictly decreasing real numbers satisfying certain conditions the Difference Operator is defined by with for all . Furthermore, we compute the spectrum and the fine spectrum of the Operator over the sequence space . In fact, we determine the spectrum, the point spectrum, the residual spectrum, and the continuous spectrum of this Operator on the sequence space .
-
on the spectrum of 2 nd order generalized Difference Operator delta 2 over the sequence space c_0
Boletim da Sociedade Paranaense de Matemática, 2013Co-Authors: S. Dutta, P. BaliarsinghAbstract:The main purpose of this article is to determine the spectrum and the fine spectrum of second order Difference Operator � 2 over the sequence space c0. For any sequence (xk) 1 in c0, the generalized second order Difference Operator � 2 over c0 is defined by � 2 (x) = P2=0 ( 1) i 2 � xk i = xk 2xk 1 + xk 2, with
Khalifa Trimeche - One of the best experts on this subject based on the ideXlab platform.
-
transmutation Operators and paley wiener theorem associated with a singular differential Difference Operator on the real line
Analysis and Applications, 2003Co-Authors: Mohamed Ali Mourou, Khalifa TrimecheAbstract:We consider a singular differential-Difference Operator Λ on the real line which includes, as particular case, the Dunkl Operator associated with the reflection group Z2 on R. We exhibit a Laplace integral representation for the eigenfunctions of the Operator Λ. From this representation, we construct a pair of integral transforms which turn out to be transmutation Operators of Λ into the first derivative Operator d/dx. We exploit these transmutation Operators to develop a new commutative harmonic analysis on the real line corresponding to the Operator Λ. In particular, we establish a Paley–Wiener theorem and a Plancherel theorem for the Fourier transform associated to Λ.
Yu. B. Melnikov - One of the best experts on this subject based on the ideXlab platform.
-
On Spectral Analysis of an Integral-Difference Operator
Letters in Mathematical Physics, 1997Co-Authors: Yu. B. MelnikovAbstract:A complete spectral analysis of an integral-Difference Operator arising as a collision Operator in some nonequilibrium statistical physics models is presented. Eigenfunctions of both discrete and continuous spectrum are constructed.
