A Dunkl-Gamma Type Operators in Terms of Two-Variable Hermite Polynomials
aa r X i v : . [ m a t h . C A ] J a n A Dunkl-Gamma Type Operators in Terms of Two-VariableHermite Polynomials
Bayram C¸ ekim, Rabia Akta¸s, and Fatma Ta¸sdelen
Abstract.
The goal of this paper is to present a Dunkl-Gamma type operatorwith the help of two-variable Hermite polynomials and to derive its approxi-mating properties via the classical modulus of continuity, second modulus ofcontinuity and Peetre’s K -functional.
1. Introduction
By now, several research workers have investigated linear positive operators andtheir approximation properties, see for instance [ ], [ ], [ ], [ ? ], [ ], [ ], [ ] andreferences so on. Furthermore, many authors have studied linear positive operatorscontaining generating functions and given some approximation properties of theseoperators. To see such operators, we give the references such as Altin et.al [ ],Dogru et.al [ ], Krech [ ], Olgun et.al [ ], Sucu et.al [ ], Tasdelen et.al [ ],Varma et.al [
20, 21 ].Latterly, with the help of Dunkl exponential function, several authors havedefined some linear positive operators. First of them is a Dunkl analogue of Sz´aszoperators given in [ ] as follows: S ∗ n ( g ; x ) = 1 e ν ( nx ) ∞ X k =0 ( nx ) k γ ν ( k ) g (cid:18) k + 2 νθ k n (cid:19) ; n ∈ N , ν, x ∈ [0 , ∞ ) (1.1)for g ∈ C [0 , ∞ ) . Here the Dunkl exponential function is defined by e ν ( x ) = ∞ X k =0 x k γ ν ( k ) (1.2)for ν > − and the coefficients γ ν are given by γ ν (2 k ) = 2 k k !Γ ( k + ν + 1 / ν + 1 /
2) and γ ν (2 k + 1) = 2 k +1 k !Γ ( k + ν + 3 / ν + 1 /
2) (1.3)Also, for the coefficients γ ν , the following recursion relation holds γ ν ( k + 1) γ ν ( k ) = (2 νθ k +1 + k + 1) , k ∈ N , (1.4) Mathematics Subject Classification.
Primary 41A25, 41A36; Secondary 33C45.
Key words and phrases.
Dunkl exponential, Hermite polynomial, Gamma function, modulusof continuity, Peetre’s K-functional. where θ k is defined by θ k = (cid:26) , if k = 2 p , if k = 2 p + 1 (1.5)for p ∈ N in [ ]. Then, ˙I¸c¨oz et.al has given a Stancu-type generalization of Sz´asz-Kantorovich operators and q − Sz´asz operators with the help of he Dunkl exponentialfunction in [
9, 10 ].Next, Wafi et.al [ ] has introduced Sz´asz–Gamma operators based on Dunklanalogue as D fn ( x ) = 1 e µ ( nx ) ∞ X k =0 ( nx ) k γ µ ( k ) n k +2 µθ k + λ +1 Γ( k + 2 µθ k + λ + 1) R ∞ t k +2 µθ k + λ e − nt f ( t ) dt, (1.6)where λ ≥ x ) = R ∞ t x − e − t dt for x > . (1.7)Finally, Akta¸s et.al [ ] introduce the operator T n ( f ; x ) , n ∈ N T n ( f ; x ) := 1 e αx e µ ( nx ) ∞ X k =0 h µk ( n, α ) γ µ ( k ) x k f (cid:18) k + 2 µθ k n (cid:19) where α ≥ , µ ≥ x ∈ [0 , ∞ ) via the Dunkl generalization of two-variableHermite polynomials h µn ( ξ, α ) defined as follows ∞ X n =0 h µn ( ξ, α ) γ µ ( n ) t n = e αt e µ ( ξt ) . (1.8)Here, h µn ( ξ, α ) = γ µ ( n ) H µn ( ξ, α ) n !and H µn ( ξ, α ) has the following explicit representation H µn ( ξ, α ) = n ! [ n ] X k =0 α k ξ n − k k ! γ µ ( n − k ) . We note that H µn ( ξ, α ) reduces to the two-variable Hermite polynomials as µ = 0 seedetail [ ]. In the case of µ = 0 , the operator T n ( f ; x ) gives the operator G αn ( f ; x )defined by Krech [ ].The paper is organized as follows. In the next section, we introduce a Dunkl-Gamma type operator consisting of the two-variable Hermite polynomials. In thethird section, the rates of convergence of the operator are obtained by means of theclassical modulus of continuity, second modulus of continuity, Peetre’s K -functionaland the Lipschitz class Lip M ( α ) .
2. The Dunkl-Gamma Type Operators
Firstly, before we introduce our operator, let us give some features and resultsrelated to the Dunkl generalization of two-variable Hermite polynomials h µn ( ξ, α )generated by (1.8).We first recall the following definition and lemma in [ ]. DUNKL-GAMMA TYPE OPERATORS IN TERMS OF TWO-VARIABLE HERMITE POLYNOMIALS3
Definition . [ ] Assume that µ ∈ C ( C := C \ (cid:8) − , − , ... (cid:9) , x ∈ C and ϕ is a entire function. On all entire functions ϕ on C , Rosenblum defines the linearoperator D µ as follows: D µ,x ( ϕ ( x )) = ( D µ ϕ ) ( x ) = ϕ ′ ( x ) + µx ( ϕ ( x ) − ϕ ( − x )) , x ∈ C . (2.1) Lemma . [ ] Assume that ϕ, ψ are entire functions. With the help of thelinear operator D µ , the following relations are satisfied: i ) D jµ : x n → γ µ ( n ) γ µ ( n − j ) x n − j , j = 0 , , , ..., n ( n ∈ N ); D jµ : 1 → ,ii ) D µ ( ϕψ ) = D µ ( ϕ ) ψ + ϕ D µ ( ψ ) , if ϕ is an even function, iii ) D µ : e µ ( λx ) → λe µ ( λx ) . By using this definition and lemma 1, the results in the next lemma hold true (seedetail [ ] ). Lemma . [ ] A new generalization of two-variable Hermite polynomials h µn ( ξ, α ) has the following results ( i ) ∞ P n =0 h µn +1 ( ξ,α ) γ µ ( n ) t n = ( ξ + 2 αt ) e αt e µ ( ξt ) , ( ii ) ∞ P n =0 h µn +2 ( ξ,α ) γ µ ( n ) t n = ( ξ + 4 ξαt + 4 α t + 2 α ) e αt e µ ( ξt ) + 4 αµe αt e µ ( − ξt ) , Now we can define our operator as follows:
Definition . Via a new generalization of two variable Hermite polynomials h µn ( ξ, α ) given in (1.8), we consider the operator S n ( f ; x ) , n ∈ N given by S n ( f ; x ) := 1 e αx e µ ( nx ) ∞ X k =0 h µk ( n, α ) γ µ ( k ) x k n k +2 µθ k + λ +1 Γ( k + 2 µθ k + λ + 1) R ∞ t k +2 µθ k + λ e − nt f ( t ) dt (2.2) where f ∈ C [0 , ∞ ) , α ≥ , µ ≥ , λ ≥ and x ∈ [0 , ∞ ) . We note that theoperator in (2.2) is positive and linear. For α = 0 , it reduces to D fn ( x ) given by(1.6). Lemma . The following equations can be derived from the definition of theoperator S n ( f ; x ) : ( i ) S n (1; x ) = 1 , ( ii ) S n ( t ; x ) = x + αx n + λ +1 n , ( iii ) S n ( t ; x ) = x n n n + 4 nαx + 4 α x + 2 α + 4 αµ e µ ( − nx ) e µ ( nx ) o + µxn ( n − αx ) e µ ( − nx ) e µ ( nx ) + λ +2) n ( n + 2 αx ) x + ( λ +1)( λ +2) n . Proof.
From the definition of Gamma function in (1.7), we have R ∞ t k +2 µθ k + λ e − nt dt = Γ( k + 2 µθ k + λ + 1) n k +2 µθ k + λ +1 . By using the above equation and the generating function in (1.8), we get the relation( i ). Using the definition of Gamma function again, we have R ∞ t k +2 µθ k + λ +1 e − nt dt = Γ( k + 2 µθ k + λ + 2) n k +2 µθ k + λ +2 . BAYRAM C¸ EKIM, RABIA AKTAS¸, AND FATMA TAS¸DELEN
Thus we get the relations S n ( t ; x ) = 1 e αx e µ ( nx ) ∞ X k =0 h µk ( n, α ) γ µ ( k ) x k n k +2 µθ k + λ +1 Γ( k + 2 µθ k + λ + 1) Γ( k + 2 µθ k + λ + 2) n k +2 µθ k + λ +2 = 1 ne αx e µ ( nx ) ∞ X k =0 ( k + 2 µθ k + λ + 1) h µk ( n, α ) γ µ ( k ) x k = 1 ne αx e µ ( nx ) ∞ X k =0 ( k + 2 µθ k ) h µk ( n, α ) γ µ ( k ) x k + ( λ + 1) ne αx e µ ( nx ) ∞ X k =0 h µk ( n, α ) γ µ ( k ) x k . The second series in right hand side of the above equation from the generatingfunction in (1.8) is ( λ +1) n . Also, if we use the recursion relation in (1.4) for the firstterm, we get S n ( t ; x ) = 1 ne αx e µ ( nx ) ∞ X k =1 h µk ( n, α ) γ µ ( k − x k + ( λ + 1) n . While we are substituting k by k + 1 and using Lemma 2 ( i ), we arrive at therelation ( ii ). From the definition of Gamma function in (1.7) again, the followingequality holds R ∞ t k +2 µθ k + λ +2 e − nt dt = Γ( k + 2 µθ k + λ + 3) n k +2 µθ k + λ +3 , from which, it follows S n ( t ; x ) = 1 e αx e µ ( nx ) ∞ X k =0 h µk ( n, α ) γ µ ( k ) x k n k +2 µθ k + λ +1 Γ( k + 2 µθ k + λ + 1) Γ( k + 2 µθ k + λ + 3) n k +2 µθ k + λ +3 = 1 n e αx e µ ( nx ) ∞ X k =0 ( k + 2 µθ k + λ + 1)( k + 2 µθ k ) h µk ( n, α ) γ µ ( k ) x k + ( λ + 2) n e αx e µ ( nx ) ∞ X k =0 ( k + 2 µθ k ) h µk ( n, α ) γ µ ( k ) x k + ( λ + 2)( λ + 1) n e αx e µ ( nx ) ∞ X k =0 h µk ( n, α ) γ µ ( k ) x k . The third term in right hand side of the above equation from the generating functionin (1.8) is ( λ +2)( λ +1) n . Also by taking into account the recursion relation in (1.4) thefor the first and second series, we obtain S n ( t ; x ) = xn e αx e µ ( nx ) ∞ X k =0 ( k + 2 µθ k +1 + λ + 2) h µk +1 ( n, α ) γ µ ( k ) x k + ( λ + 2) xn e αx e µ ( nx ) ∞ X k =0 h µk +1 ( n, α ) γ µ ( k ) x k + ( λ + 2)( λ + 1) n . Using the equation θ k +1 = θ k + ( − k , (2.3) DUNKL-GAMMA TYPE OPERATORS IN TERMS OF TWO-VARIABLE HERMITE POLYNOMIALS5 it yields S n ( t ; x ) = xn e αx e µ ( nx ) ∞ X k =0 ( k + 2 µθ k ) h µk +1 ( n, α ) γ µ ( k ) x k + 2 µxn e αx e µ ( nx ) ∞ X k =0 h µk +1 ( n, α ) γ µ ( k ) ( − x ) k + 2( λ + 2) xn e αx e µ ( nx ) ∞ X k =0 h µk +1 ( n, α ) γ µ ( k ) x k + ( λ + 2)( λ + 1) n . Finally using the recursion relation in (1.4) in the first series, from Lemma 2 ( i ) forthe second and third series and Lemma 2 ( ii ) for the first series, we complete theproof of ( iii ). (cid:3) Remark . In case of α = 0 , the results of Lemma 3 reduce to the results inthe paper of Wafi and Rao in [ ] . Lemma . From the results of Lemma 3 and the linearity of the operator, wecan obtain the next results for S n operator Λ = S n ( t − x ; x ) = 2 αx + λ + 1 n , Λ = S n (( t − x ) ; x )= 1 n (cid:20) x n (cid:0) x α + 4 λα + 10 α (cid:1) + 2 x (cid:16) µ e µ ( − nx ) e µ ( nx ) + 1 (cid:17) + ( λ +1)( λ +2) n (cid:21) . (2.4) Theorem . Assume that the function g on the interval [0 , ∞ ) is uniformlycontinuous bounded function. For each function g on [0 , ∞ ) , we can give S n ( g ; x ) uniformly ⇒ g ( x ) on each compact set A ⊂ [0 , ∞ ) when n → ∞ .
3. The Convergence Rates of Operator S n In this part, we obtain some rates of convergence of the operator S n . Theorem . If h ∈ Lip M ( α ) , which satisfies the inequality | h ( s ) − h ( t ) | ≤ M | s − t | α where s, t ∈ [0 , ∞ ) , < α ≤ and M > , we have |S n ( h ; x ) − h ( x ) | ≤ M (Λ ) α/ where Λ is given in Lemma 4. Proof.
From the linearity of operator and h ∈ Lip M ( α ) , we get |S n ( h ; x ) − h ( x ) | ≤ S n ( | h ( t ) − h ( x ) | ; x ) ≤ M S n ( | t − x | α ; x ) . Under favour of H¨older’s inequality and Lemma 4, we can give the following requiredinequality |S n ( h ; x ) − h ( x ) | ≤ M [Λ ] α . (cid:3) BAYRAM C¸ EKIM, RABIA AKTAS¸, AND FATMA TAS¸DELEN
Theorem . The operator S n in (2.2) satisfies the inequality |S n ( g ; x ) − g ( x ) | ≤ r x n (4 x α + 4 λα + 10 α ) + 2 x (cid:16) µ e µ ( − nx ) e µ ( nx ) + 1 (cid:17) + ( λ +1)( λ +2) n ! ω (cid:18) g ; 1 √ n (cid:19) , where g ∈ e C [0 , ∞ ) , which is the space of uniformly continuous functions on [0 , ∞ ) , andthe modulus of continuity is defined by ω ( g ; δ ) := sup s,t ∈ [0 , ∞ ) | s − t |≤ δ | g ( s ) − g ( t ) | (3.1) for g ∈ e C [0 , ∞ ) . Proof.
Firstly we note that the modulus of continuity verifies the followinginequality | g ( t ) − g ( x ) | ≤ w ( g ; δ ) (cid:18) | t − x | δ + 1 (cid:19) . (3.2)Under favour of the linearity of operator, Cauchy-Schwarz’s inequality, and Lemma4, respectively, it follows |S n ( g ; x ) − g ( x ) | ≤ S n ( | g ( t ) − g ( x ) | ; x ) ≤ (cid:18) δ S n ( | t − x | ; x ) (cid:19) ω ( g ; δ ) ≤ (cid:18) δ p Λ (cid:19) ω ( g ; δ ) . By choosing δ = √ n , we complete the proof. (cid:3) Lemma . For h ∈ e C B [0 , ∞ ) , which is denoted by e C B [0 , ∞ ) = { g ∈ e C B [0 , ∞ ) : g ′ , g ′′ ∈ e C B [0 , ∞ ) } (3.3) with the norm k h k e C B [0 , ∞ ) = k h k e C B [0 , ∞ ) + k h ′ k e C B [0 , ∞ ) + k h ′′ k e C B [0 , ∞ ) where e C B [0 , ∞ ) is the space of uniformly continuous and bounded functions on [0 , ∞ ) and k h k = sup {| h ( x ) | : x ∈ [0 , ∞ ) } , the following inequality holds true |S n ( h ; x ) − h ( x ) | ≤ [Λ + Λ ] k h k e C B [0 , ∞ ) , (3.4) where Λ and Λ are given by in Lemma 4. Proof.
With the help of the Taylor’s series of the function h , it follows that h ( s ) = h ( x ) + ( s − x ) h ′ ( x ) + ( s − x ) h ′′ ( ς ) , ς ∈ ( x, s ) . Then, by applying S n to this equality and using the linearity of the operator, weget S n ( h ; x ) − h ( x ) = h ′ ( x ) Λ + h ′′ ( ς )2 Λ . DUNKL-GAMMA TYPE OPERATORS IN TERMS OF TWO-VARIABLE HERMITE POLYNOMIALS7
Using Lemma 4, the following inequality is satisfied |S n ( h ; x ) − h ( x ) | ≤ (cid:16) αx n + ( λ +1) n (cid:17) k h ′ k e C B [0 , ∞ ) + n h x n (cid:0) x α + 4 λα + 10 α (cid:1) + 2 x (cid:16) µ e µ ( − nx ) e µ ( nx ) + 1 (cid:17) + ( λ +1)( λ +2) n i k h ′′ k e C B [0 , ∞ ) ≤ [Λ + Λ ] k h k e C B [0 , ∞ ) . (cid:3) We recall that C B [0 , ∞ ) is the space of real valued continuous and boundedfunctions with the norm k f k = sup | f ( x ) | ≤ x< ∞ . For any f ∈ C B [0 , ∞ ) and δ > , Peetre’s K-functional is given by K ( f ; δ ) := inf (cid:8) k f − g k + δ k g ′′ k : g ∈ C B [0 , ∞ ) (cid:9) , where C B [0 , ∞ ) = { g ∈ C B [0 , ∞ ) : g ′ , g ′′ ∈ C B [0 , ∞ ) } and the second modulus ofcontinuity ω ( f ; δ ) is defined as ω (cid:16) f ; √ δ (cid:17) := sup , (3.6)between Peetre’s K-functional and second modulus of continuity ω (see [ ]) Lemma . For g ∈ C B [0 , ∞ ) and x ≥ , α, λ ≥ , we get (cid:12)(cid:12)(cid:12) e S n ( g ; x ) − g ( x ) (cid:12)(cid:12)(cid:12) ≤ Υ( n, x ) (cid:13)(cid:13)(cid:13) g ′′ (cid:13)(cid:13)(cid:13) where e S n ( g ; x ) = S n ( g ; x ) + g ( x ) − g (cid:16) x + αx + λ +1 n (cid:17) and Υ( n, x ) = Λ + Λ . Proof.
Let us define e S n ( g ; x ) = S n ( g ; x ) + g ( x ) − g (cid:16) x + αx + λ +1 n (cid:17) which islinear operator and e S n ( t − x ; x ) = 0 , t ∈ [0 , ∞ ). For g ∈ C B [0 , ∞ ), the Taylor’sexpression is g ( t ) = g ( x ) + ( t − x ) g ′ ( x ) + R tx ( t − v ) g ′′ ( v ) dv, t ∈ [0 , ∞ ) . If we apply e S n to the last equality and then use e S n ( t − x ; x ) = 0 , we have e S n ( g ; x ) = g ( x ) + e S n (cid:16)R tx ( t − v ) g ′′ ( v ) dv ; x (cid:17) , BAYRAM C¸ EKIM, RABIA AKTAS¸, AND FATMA TAS¸DELEN from which, it follows (cid:12)(cid:12)(cid:12) e S n ( g ; x ) − g ( x ) (cid:12)(cid:12)(cid:12) ≤ e S n (cid:16)(cid:12)(cid:12)(cid:12)R tx ( t − v ) g ′′ ( v ) dv (cid:12)(cid:12)(cid:12) ; x (cid:17) ≤ S n (cid:0) ( t − x ) ; x (cid:1) (cid:13)(cid:13)(cid:13) g ′′ (cid:13)(cid:13)(cid:13) + (cid:12)(cid:12)(cid:12)(cid:12)R x + 2 αx + λ +1 nx ( x + αx + λ +1 n − v ) g ′′ ( v ) dv (cid:12)(cid:12)(cid:12)(cid:12) ≤ S n (cid:0) ( t − x ) ; x (cid:1) (cid:13)(cid:13)(cid:13) g ′′ (cid:13)(cid:13)(cid:13) + (cid:16) αx + λ +1 n (cid:17) (cid:13)(cid:13)(cid:13) g ′′ (cid:13)(cid:13)(cid:13) ≤ Υ( n, x ) (cid:13)(cid:13)(cid:13) g ′′ (cid:13)(cid:13)(cid:13) , where Υ( n, x ) = Λ + Λ . (cid:3) Theorem . Let f ∈ C B [0 , ∞ ) and c > . The following inequality holds |S n ( f ; x ) − f ( x ) | ≤ cω (cid:18) f ; 12 p Υ( n, x ) (cid:19) + ω ( f ; Λ ) . Proof.
For f ∈ C B [0 , ∞ ) and g ∈ C B [0 , ∞ ) , we get e S n ( f − g ; x ) = S n ( f − g ; x ) + ( f − g )( x ) − ( f − g ) (cid:16) x + αx + λ +1 n (cid:17) = S n ( f ; x ) + f ( x ) − f (cid:16) x + αx + λ +1 n (cid:17) − e S n ( g ; x ) . On the other hand, we give S n ( f ; x ) − f ( x ) = e S n ( f − g ; x ) + e S n ( g ; x ) − g ( x ) + g ( x ) + f (cid:16) x + αx + λ +1 n (cid:17) − f ( x ) − f ( x )= e S n ( f − g ; x ) − ( f ( x ) − g ( x )) + e S n ( g ; x ) − g ( x ) + f (cid:16) x + αx + λ +1 n (cid:17) − f ( x ) . Thus we have |S n ( f ; x ) − f ( x ) | ≤ (cid:12)(cid:12)(cid:12) e S n ( f − g ; x ) (cid:12)(cid:12)(cid:12) + | f ( x ) − g ( x ) | + (cid:12)(cid:12)(cid:12) e S n ( g ; x ) − g ( x ) (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) f (cid:16) x + αx + λ +1 n (cid:17) − f ( x ) (cid:12)(cid:12)(cid:12) ≤ k f − g k + Υ( n, x ) (cid:13)(cid:13)(cid:13) g ′′ (cid:13)(cid:13)(cid:13) + ω ( f ; Λ ) . From the inequality (3.6) between Peetre’s K -functional and the second modulusof continuity ω , we have the desired result. (cid:3) References [1] Altın, A., Do˘gru, O. and Ta¸sdelen, F., The generalization of Meyer-K¨onig and Zeller operatorsby generating functions, J. Math. Anal. Appl., 312 (1) (2005) , 181-194.[2] Appell, P. and Kampe de Feriet, J., Hypergeometriques et Hyperspheriques: Polynomesd’Hermite, Gauthier-Villars, Paris, 1926.[3] Atakut, C¸ . and ˙Ispir, N., Approximation by modified Sz´asz–Mirakjan operators on weightedspaces, Proc. Indian Acad. Sci. Math. 112 (2002) , 571–578[4] Atakut, C¸ . and B¨uy¨ukyazici, ˙I., Stancu type generalization of the Favard Sz´asz operators,Appl. Math. Lett., 23 (12) (2010) , 1479-1482.[5] Ciupa, A., A class of integral Favard–Sz´asz type operators. Stud. Univ. Babes-Bolyai Math.40 (1) (1995) , 39–47.[6] DeVore, R.A. and Lorentz, G.G., Construtive Approximation, Springer, Berlin, .[7] Do˘gru, O., ¨Ozarslan, M.A. and Ta¸sdelen, F., On positive operators involving a certain classof generating functions, Studia Sci. Math. Hungar., 41 (4) (2004) , 415-429.[8] Gupta, V., Vasishtha, V. and Gupta, M.K., Rate of convergence of the Sz´asz–Kantorovich–Bezier operators for bounded variation functions, Publ. Inst. Math. (Beograd) (N.S.), 72 (2006) , 137–143.
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