Approximation by Bivariate Bernstein-Durrmeyer Operators on a Triangle

The purpose of the present paper is to define the GBS (Generalized Boolean Sum) operators associated with the two‐dimensional Bernstein‐Durrmeyer operators introduced by Zhou 1992 and study its approximation properties. Furthermore, we show the convergence and comparison of convergence with the GBS of the Bernstein‐Kantorovich operators proposed by Deshwal et al 2017 by numerical examples and illustrations.

“…Lemma (Goyal et al) For

e_{i j} = s^{i} t^{j}, false(i, j false) \in N^{0} \times N^{0}, N^{0} = double-struckN \cup false{0 false}

, we have (i)

V_{n} false(e_{00}; x, y false) = 1

; (ii)

V_{n} false(e_{10}; x, y false) = \frac{1 + n x}{n + 3}

; (iii)

V_{n} false(e_{01}; x, y false) = \frac{1 + n y}{n + 3}

; (iv)

V_{n} false(e_{20}; x, y false) = \frac{n false(n - 1 false) x^{2} + 4 n x + 2}{false(n + 3 false) false(n + 4 false)}

; (v)

V_{n} false(e_{02}; x, y false) = \frac{n false(n - 1 false) y^{2} + 4 n y + 2}{false(n + 3 false) false(n + 4 false)}

; (vi)

V_{n} false(e_{11}; x, y false) = \frac{n false(n - 1 false) x y + n false(x + y false) + 1}{false(n + 3 false) false(n + 4 false)}

; (vii)

V_{n} false(e_{40}; x, y false) = \frac{1}{false(n + 3 false) false(n + 4 false) false(n + 5 false) false(<...>}

…”

Section: Preliminary Resultsunclassified

“…Lemma (Goyal et al) For the bivariate operators

V_{n} false(f; x, y false)

, we have (i)

\lim_{n \to \infty} n V_{n} false(false(s - x false); x, y false) = 1 - 3 x

; (ii)

\lim_{n \to \infty} n V_{n} false(false(t - y false); x, y false) = 1 - 3 y

; (iii)

\lim_{n \to \infty} n V_{n} false({false(s - x false)}^{2}; x, y false) = 2 x false(1 - x false)

; (iv)

\lim_{n \to \infty} n V_{n} false({false(t - y false)}^{2}; x, y false) = 2 y false(1 - y false)

; (v)

\lim_{n \to \infty} n V_{n} false(false(s - x false) false(t - y false); x, y false) = - 2 x y

; (vi)

\lim_{n \to \infty} n^{2} V_{n} false({false(s - x false)}^{4}; x, y false) = 12 x^{2} {false(x - 1 false)}^{2}

; (vii)

\lim_{n \to \infty} n^{2} V_{n} false({false(t - y false)}^{4}; x, y false) = 12 y^{2} {false(y - 1 false)}^{2}

. …”

Section: Preliminary Resultsunclassified

“…Theorem (Goyal et al) Let f ∈ C 2 ( S ). Then, we have

\lim_{n \to \infty} ({scriptV}_{n} (f; x, y) - f (x, y)) = f_{x}^{'} (x, y) (1 - 3 x) + f_{y}^{'} (x, y) (1 - 3 y) + f_{x x}^{''} (x, y) x (1 - x) - 2 f_{x y}^{''} (x, y) x y + f_{y y}^{''} (x, y) y (1 - y),

uniformly in ( x , y )∈ S .…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…Deo and Bhardwaj established some direct theorems and the inverse theorem of the operators

V_{n}

V_{n}

.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

GBS operators of bivariate Bernstein‐Durrmeyer–type on a triangle

Ruchi

Baxhaku

Agrawal

2018

“…Pop and Fărcaş introduced the GBS operators associated with the bivariate operators on the triangle. Goyal et al investigated the rate of convergence and simultaneous approximation for first order partial derivatives of these operators. Acu and Muraru proposed two dimensional Bernstein‐Schurer‐Kantorovich operators involving q ‐integers and established some approximation theorems of these operators.…”

Section: Introductionmentioning

confidence: 99%

Generalized Bernstein‐Kantorovich–type operators on a triangle

Kajla

2019

Self Cite

In this note, we construct generalized Bernstein‐Kantorovich–type operators on a triangle. The concern of this note is to present a Voronovskaja‐type and Grüss Voronovskaja‐type asymptotic theorems, and some estimates of the rate of approximation with the help of K‐functional, first and second order modulus of continuity. We also obtain Korovkin‐ and Voronovskaja‐type statistical approximation theorems via weighted mean matrix method. Lastly, we show that the numerical results which explain the validity of the theoretical results and the effectiveness of the constructed operators.

Quantitative estimations of bivariate summation‐integral–type operators

Yadav

Meher

Mishra

2019

In this paper, we study the approximation properties of bivariate summationintegral-type operators with two parameters m, n ∈ N. The present work deals within the polynomial weight space. The rate of convergence is obtained while the function belonging to the set of all continuous and bounded function defined on ([0], ∞)(×[0], ∞) and function belonging to the polynomial weight space with two parameters, also convergence properties, are studied. To know the asymptotic behavior of the proposed bivariate operators, we prove the Voronovskaya type theorem and show the graphical representation for the convergence of the bivariate operators, which is illustrated by graphics using Mathematica. Also with the help of Mathematica, we discuss the comparison by means of the convergence of the proposed bivariate summation-integral-type operators and Szász-Mirakjan-Kantorovich operators for function of two variables with two parameters to the function. In the same direction, we compute the absolute numerical error for the bivariate operators by using Mathematica and is illustrated by tables and also the comparison takes place of the proposed bivariate operators with the bivariate Szász-Mirakjan operators in the sense of absolute error, which is represented by table. At last, we study the simultaneous approximation for the first-order partial derivative of the function. KEYWORDS polynomial weight space, rate of convergence, Szász-Mirakjan-Kantorovich operators MSC CLASSIFICATION 41A25; 41A35; 41A36 INTRODUCTIONFrom 1941 to 1950, there were three authors 1-3 who independently introduced operators, so-called Szász-Mirakjan operators, and studied many properties related to those operators for function belonging to the set of continuous function defined on [0, ∞] (extension of the Bernstein's operators, which is defined on [0, 1]). To the generalization and for the study of approximation properties, some researchers, professors, presented their papers related to Szász-Mirakjan operators concerning some modifications (including extension). Some are as those in the previous studies. [4][5][6][7][8][9][10][11][12][13][14] But still the above mentioned papers and monographs represent only extension or modification of the Szász-Mirakjan operators for functions (f ∈ C[0], ∞), in case if function is integrable then in that condition, modification took place in 1954. First of all, Butzer 15 introduced an integral modification of the Szász-Mirakjan operators (in general, we represent by S n (f; x)) by considering Lebesgue integrable function and were named in Totik 16 as Szász-Mirakjan-Kantorovich operators.