Some general statistical approximation results for λ -Bernstein operators
aa r X i v : . [ m a t h . C A ] F e b SOME GENERAL STATISTICAL APPROXIMATION RESULTS FOR λ -BERNSTEIN OPERATORS FARUK ¨OZGER
Department of Engineering Sciences, ˙Izmir Katip C¸ elebi University, 35620, ˙Izmir, [email protected]
Abstract.
In this article, we achieve some general statistical approximation results for λ -Bernstein operators in addition to some other approximation properties. We prove a statisticalVoronovskaja-type approximation theorem. We also construct bivariate λ -Bernstein operatorsand study their approximation properties. Keywords:
Rate of weighted A statistical convergence, λ -Bernstein operators, bivariate λ -Bernstein operators, statistical approximation properties, Gr¨ussVoronovskaja-type theorem,weighted A -statistical Voronovskaja-type theorem, weighted space MSC: Introduction
Bernstein used famous polynomials nowadays called Bernstein polynomials, in 1912, to obtainan alternative proof of Weierstrass’s fundamental theorem [2]. Approximation properties ofBernstein operators and their applications in Computer Aided Geometric Design and ComputerGraphics have been extensively studied in many articles.Bernstein basis of degree n on x ∈ [0 ,
1] is defined by b n,i ( x ) = (cid:18) ni (cid:19) x i (1 − x ) n − i i = 0 , . . . , n, and n th order Bernstein polynomial is given by B n ( f ; x ) = n X i =0 f (cid:18) in (cid:19) b n,i ( x ) (1.1)for any continuous function f ( x ) defined on [0 , λ Bernstein operators [4] B n,λ ( f ; x ) = n X i =0 f (cid:18) in (cid:19) ˜ b n,i ( λ ; x ) (1.2)with B´ezier bases ˜ b n,i ( λ ; x ) [15]: ˜ b n, ( λ ; x ) = b n, ( x ) − λn + 1 b n +1 , ( x ) , ˜ b n,i ( λ ; x ) = b n,i ( x ) + λ (cid:18) n − i + 1 n − b n +1 ,i ( x ) − n − i − n − b n +1 ,i +1 ( x ) (cid:19) , i = 1 , . . . , n − , ˜ b n,n ( λ ; x ) = b n,n ( x ) − λn + 1 b n +1 ,n ( x ) , (1.3)where shape parameters λ ∈ [ − , Preliminary Results
In this part, we obtain global approximation formula in terms of Ditzian-Totik uniform mod-ulus of smoothness of first and second order and give a local direct estimate of the rate ofconvergence by Lipschitz-type function involving two parameters for λ Bernstein operators. Wealso give a Gr¨uss-Voronovskaja and a quantitative Voronovskaja-type theoremResults in the following lemma were obtained for λ Bernstein operators in [4, Lemma 2.1].
Lemma 2.1.
We have following equalities for λ Bernstein operators: B n,λ (1; x ) = 1; (2.1) B n,λ ( t ; x ) = x + 1 − x + x n +1 − (1 − x ) n +1 n ( n − λ ; B n,λ ( t ; x ) = x + x (1 − x ) n + (cid:20) x − x + 2 x n +1 n ( n −
1) + x n +1 + (1 − x ) n +1 − n ( n − (cid:21) λ ; B n,λ ( t ; x ) = x + 3 x (1 − x ) n + 2 x − x + xn + (cid:20) x n +1 − x n + 3 x − x n +1 n ( n − x n +1 − x n ( n −
1) + 4 x n +1 − xn ( n −
1) + 1 − x n +1 + (1 − x ) n +1 n ( n − (cid:21) λ ; B n,λ ( t ; x ) = x + 6 x (1 − x ) n + 7 x − x + 11 x n + x − x + 12 x − x n + (cid:20) x − x − x + 4 x n +1 n + 17 x n +1 + 16 x − x − x n + x − x n +1 n + 7 x − x n +1 n ( n −
1) + x − x + 22 x n +1 n ( n −
1) + (1 − x ) n +1 + x − n ( n − (cid:21) λ. Global and local approximations.
First we obtain global approximation formula interms of Ditzian-Totik uniform modulus of smoothness of first and second order defined by ω ξ ( f, δ ) := sup < | h |≤ δ sup x,x + hξ ( x ) ∈ [0 , {| f ( x + hξ ( x )) − f ( x ) |} and ω φ ( f, δ ) := sup < | h |≤ δ sup x,x ± hφ ( x ) ∈ [0 , {| f ( x + hφ ( x )) − f ( x ) + f ( x − hφ ( x )) |} , respectively, where φ is an admissible step-weight function on [ a, b ], i.e. φ ( x ) = [( x − a )( b − x )] / if x ∈ [ a, b ], [5]. Corresponding K -functional is K ,φ ( x ) ( f, δ ) = inf g ∈ W ( φ ) (cid:8) || f − g || C [0 , + δ || φ g ′′ || C [0 , : g ∈ C [0 , (cid:9) , where δ > W ( φ ) = { g ∈ C [0 ,
1] : g ′ ∈ AC [0 , , φ g ′′ ∈ C [0 , } and C [0 ,
1] = { g ∈ C [0 ,
1] : g ′ , g ′′ ∈ C [0 , } . Here, g ′ ∈ AC [0 ,
1] means that g ′ is absolutely continuous on [0 , C >
0, such that C − ω φ ( f, √ δ ) ≤ K ,φ ( x ) ( f, δ ) ≤ Cω φ ( f, √ δ ) . (2.2) Theorem 2.2.
Let λ ∈ [ − , , f ∈ C [0 , and φ ( φ = 0) be an admissible step-weight functionof Ditzian-Totik modulus of smoothness such that φ is concave. Then we have | B n,λ ( f ; x ) − f ( x ) | ≤ Cω φ (cid:18) f, δ n ( x )2 φ ( x ) (cid:19) + ω ξ (cid:18) f, β n ( x ) ξ ( x ) (cid:19) for x ∈ [0 , and C > . -BERNSTEIN OPERATORS 3 Now we give a local direct estimate of the rate of convergence with the help of Lipschitz-typefunction involving two parameters for operators (1.2). We write
Lip ( k ,k ) M ( η ) := n f ∈ C [0 ,
1] : | f ( t ) − f ( x ) | ≤ M | t − x | η ( k x + k x + t ) η ; x ∈ (0 , , t ∈ [0 , o for k ≥ , k >
0, where η ∈ (0 ,
1] and M is a positive constant (see [11]). Theorem 2.3. If f ∈ Lip ( k ,k ) M ( η ) , then we have | B n,λ ( f ; x ) − f ( x ) | ≤ M α η n ( x )( k x + k x ) − η for all λ ∈ [ − , , x ∈ (0 , and η ∈ (0 , . Theorem 2.4.
The following inequality holds: | B n,λ ( f ; x ) − f ( x ) | ≤ | β n ( x ) | | f ′ ( x ) | + 2 p α n ( x ) w (cid:0) f ′ , p α n ( x ) (cid:1) for f ∈ C [0 , and x ∈ [0 , . Voronovskaja-type theorems.
In this part, we give a Gr¨ussVoronovskaja-type theoremand a quantitative Voronovskaja-type theorem for B n,λ ( f ; x ).We first obtain a quantitative Voronovskaja-type theorem for B n,λ ( f ; x ) using Ditzian-Totikmodulus of smoothness defined as ω φ ( f, δ ) := sup < | h |≤ δ (cid:26)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) x + hφ ( x )2 (cid:19) − f (cid:18) x − hφ ( x )2 (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , x ± hφ ( x )2 ∈ [0 , (cid:27) , where φ ( x ) = ( x (1 − x )) / and f ∈ C [0 , K -functional is definedby K φ ( f, δ ) = inf g ∈ W φ [0 , (cid:8) || f − g || + δ || φg ′ || : g ∈ C [0 , , δ > (cid:9) , where W φ [0 ,
1] = { g : g ∈ AC loc [0 , , k φg ′ k < ∞} and AC loc [0 ,
1] is the class of absolutelycontinuous functions defined on [ a, b ] ⊂ [0 , C > K φ ( f, δ ) ≤ C ω φ ( f, δ ) . Theorem 2.5.
Assume that f ∈ C [0 , such that f ′ , f ′′ ∈ C [0 , . Then, we have (cid:12)(cid:12)(cid:12)(cid:12) B n,λ ( f ; x ) − f ( x ) − β n f ′ ( x ) − (cid:16) α n + 12 (cid:17) f ′′ ( x ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cn φ ( x ) ω φ (cid:18) f ′′ , n − / (cid:19) for every x ∈ [0 , and sufficiently large n , where C is a positive constant, α n and β n are definedin Theorem 2.2. Statistical approximation properties by weighted mean matrix method
In this part, we study on statistical approximation properties and estimate rate of weighted A -statistical convergence. We also use statistical convergence to prove a Voronovskaja-typeapproximation theorem. Theorem 3.1.
Let A = ( a nk ) be a weighted non-negative regular summability matrix for n, k ∈ N and q = ( q n ) be a sequence of non-negative numbers such that q > and Q n = P nk =0 q k → ∞ as n → ∞ . For any f ∈ C [0 , , we have S e NA − lim n →∞ k B n,λ ( f ; x ) − f ( x ) k C [0 , = 0 . F. ¨OZGER
A Voronovskaja-type approximation theorem.
We prove a Voronovskaja-type ap-proximation theorem by ˚ B n,λ ( f ; x ) family of linear operators. Theorem 3.2.
Let A = ( a nk ) be a weighted non-negative regular summability matrix and let ( x n ) be a sequence of real numbers such that S e NA − lim x n = 0 . Also let ˚ B n,λ ( f ; x ) be a sequenceof positive linear operators acting from C B [0 , into C [0 , defined by ˚ B n,λ ( f ; x ) = (1 + x n ) B n,λ ( f ; x ) . Then for every f ∈ C B [0 , , and f ′ , f ′′ ∈ C B [0 , we have S e NA − lim n →∞ n (cid:8) ˚ B n,λ ( f ; x ) − f ( x ) (cid:9) = f ′′ ( x )2 x (1 − x ) . Approximation properties for bivariate case
In this part, we construct bivariate λ Bernstein operators and study their approximationproperties.Let I = I × I = [0 , × [0 ,
1] and ( x, y ) ∈ I , then we construct bivariate λ Bernstein operatorsas ¯ B n,m ( f ; x, y ; λ ) = n X k =0 m X k =0 f (cid:18) k n , k m (cid:19) ˜ b n,k ( λ ; x )˜ b m,k ( λ ; y )for f ∈ C ( I ), where B´ezier bases ˜ b n,k ( λ ; x ) , ˜ b m,k ( λ ; x ) ( k = 0 , , . . . , n ; k = 0 , , . . . , m ) aredefined in (1.3). Lemma 4.1.
For any natural number n ( n ≥ the following equalities hold: ¯ B n,m (1; x, y ; λ ) = 1; (4.1)¯ B n,m ( s ; x, y ; λ ) = x + 1 − x + x n +1 − (1 − x ) n +1 n ( n − λ ; (4.2)¯ B n,m ( t ; x, y ; λ ) = y + 1 − y + y m +1 − (1 − y ) m +1 m ( m − λ ; (4.3)¯ B n,m ( s ; x, y ; λ ) = x + x (1 − x ) n + (cid:20) x − x + 2 x n +1 n ( n −
1) + x n +1 + (1 − x ) n +1 − n ( n − (cid:21) λ ; (4.4)¯ B n,m ( t ; x, y ; λ ) = y + y (1 − y ) m + (cid:20) y − y + 2 y m +1 m ( m −
1) + y m +1 + (1 − y ) m +1 − m ( m − (cid:21) λ. (4.5) Theorem 4.2.
The sequence ¯ B n,m ( f ; x, y ; λ ) of operators convergences uniformly to f ( x, y ) on I for each f ∈ C ( I ) .Proof. It is enough to prove the following conditionlim n,m →∞ ¯ B n,m ( e ij ( x, y ); x, y ; λ ) = x i y j , ( i, j ) ∈ { (0 , , (1 , , (0 , } converges uniformly on I . We clearly havelim m,n →∞ ¯ B n,m ( e ( x, y ); x, y ; λ ) = 1 . We havelim n,m →∞ ¯ B n,m ( e ( x, y ); x, y ; λ ) = lim n →∞ (cid:20) x + 1 − x + x n +1 − (1 − x ) n +1 n ( n − λ (cid:21) = e ( x, y ) , lim n,m →∞ ¯ B n,m ( e ( x, y ); x, y ; λ ) = lim m →∞ (cid:20) y + 1 − y + y m +1 − (1 − y ) m +1 m ( m − λ (cid:21) = e ( x, y ) -BERNSTEIN OPERATORS 5 by Lemma 4.1, andlim n,m →∞ ¯ B n,m ( e ( x, y ) + e ( x, y ); x, y ; λ )= lim n,m →∞ (cid:26) x + x (1 − x ) n + (cid:20) x − x + 2 x n +1 n ( n −
1) + x n +1 + (1 − x ) n +1 − n ( n − (cid:21) λ + y + y (1 − y ) m + (cid:20) y − y + 2 y m +1 m ( m −
1) + y m +1 + (1 − y ) m +1 − m ( m − (cid:21) λ (cid:27) = e ( x, y ) + e ( x, y ) . Bearing in mind the above conditions and Korovkin type theorem established by Volkov [14]lim m,n →∞ ¯ B n,m ( e ij ( x, y ); x, y ; λ ) = x i y j converges uniformly. (cid:3) Now we compute the rates of convergence of operators ¯ B n,m ( f ; x, y ; λ ) to f ( x, y ) by meansof the modulus of continuity. We first give the needed definitions.Complete modulus of continuity for a bivariate case is defined as follows: ω ( f, δ ) = sup n | f ( s, t ) − f ( x, y ) | : p ( s − x ) + ( t − y ) ≤ δ o for f ∈ C ( I ab ) and for every ( s, t ) , ( x, y ) ∈ I ab = [0 , a ] × [0 , b ]. Partial moduli of continuity withrespect to x and y are defined as ω ( f, δ ) = sup {| f ( x , y ) − f ( x , y ) | : y ∈ [0 , a ] and | x − x | ≤ δ } ,ω ( f, δ ) = sup {| f ( x, y ) − f ( x, y ) | : x ∈ [0 , b ] and | y − y | ≤ δ } . Peetre’s K -functional is given by K ( f, δ ) = inf g ∈ C ( I ab ) (cid:8) k f − g k C ( I ab ) + δ k g k C ( I ab ) (cid:9) for δ >
0, where C ( I ab ) is the space of functions of f such that f , ∂ j f∂x j and ∂ j f∂y j ( j = 1 ,
2) in C ( I ab ) [12]. We now give an estimate of the rates of convergence of operators ¯ B n,m ( f ; x, y ; λ ). Theorem 4.3.
Let f ∈ C ( I ) , then we have (cid:12)(cid:12) ¯ B n,m ( f ; x, y ; λ ) − f ( x, y ) (cid:12)(cid:12) ≤ ω (cid:16) f ; p δ n ( x ) , p δ n ( y ) (cid:17) for all x ∈ I . Now we investigate convergence of the sequence of linear positive operators ¯ B n,m ( f ; x, y ; λ )to a function of two variables which defined on weighted space.Let ρ ( x, y ) = x + y + 1 and B ρ be the space of all functions defined on the real axis providedwith | f ( x, y ) | ≤ M f ρ ( x, y ) , where M f is a positive constant depending only on f . Theorem 4.4.