Vo lu m e 6 5 , N u m b e r 1 , P a g e s 1 –9 ( 2 0 1 6 ) D O I: 1 0 .1 5 0 1 / C o m m u a 1 _ 0 0 0 0 0 0 0 7 3 9 IS S N 1 3 0 3 –5 9 9 1
MODIFIED q BASKAKOV OPERATORS
DILEK SÖYLEMEZ
Abstract. In the present paper, a generalization of the sequences of q Baskakov operators, which are based on a function having continuously di¤erentiable on [0; 1) with (0) = 0; inf 0(x) 1;has been considered.
Uniform approximation of such a sequence has been studied and degree of approximation has been obtained. Moreover, monotonicity properties of the sequence of operators are investigated.
1. Introduction In [7], Baskakov operator was introduced as
Bn(f ) (x) = 1 (1 + x)n 1 X k=0 n + k 1 k x 1 + x k f k n
for n 2 N, x 2 [0; 1) and f 2 C [0; 1) where C[0; 1) denote the space of all continuous and real valued functions de…ned on [0; 1). This operator and its various extentions have been intensively studied. Some are in [1], [6], [8], [15], [16].
Let us recall some notations on q analysis ([10], [17]). The q integer, [n] and the q factorial, [n]! are de…ned by
[n] := [n]q = 1 qn 1 q ; q 6= 1 n; q = 1 for n 2 N [0] = 0; and [n]! := [1]q[2]q::: [n]q; n = 1; 2; ::: 1 ; n = 0 ; for n 2 N and [0]! = 1; Received by the editors: Oct. 25, 2015, Accepted: Dec. 23, 2015.
2010 Mathematics Subject Classi…cation. Primary ; 41A36 Secondary 41A25.
Key words and phrases. Modi…ed q Baskakov operators, weighted Korovkin theorem, monotonicity properties.
c 2 0 1 6 A n ka ra U n ive rsity C o m m u n ic a tio n s d e la Fa c u lté d e s S c ie n c e s d e l’U n ive rs ité d ’A n ka ra . S é rie s A 1 . M a th e m a t ic s a n d S t a tis tic s .
respectively where q > 0. For integers n r 0 the q binomial coe¢ cient is de…ned as n r q = [n]q! [r]q! [n r]q!:
The q derivative of f (x) is denoted by Dqf (x) and de…ned as
Dqf (x) := f (qx) f (x) (q 1)x ; x 6= 0; Dqf (0) = f 0 (0) ; also D0qf := f; Dnqf := Dq(Dn 1q f ); n = 1; 2; :::
q Pochammer formula is given by
(x; q)0= 1; (x; q)n=
n 1Q k=0
1 qkx
with x 2 R ; n 2 N [ f1g. The q derivative of the product and quotient of two functions f and g are
Dq(f (x)g(x)) = g(x)Dq(f (x)) + f (qx)Dq(g(x)) and Dq( f (x) g(x)) = g(x)Dq(f (x)) f (x)Dq(g(x)) g(x)g(qx) ; respectively.
A generalization of the Baskakov operator based on q integers is de…ned by Aral and Gupta [4]. The authors constructed the q Baskakov operator as
Bn;q(f ; x) = 1 X k=0 n + k 1 k q k(k 1) 2 xk( x; q) 1 n+kf [k] qk 1[n] ; n 2 N; (1.1)
where x 0; q > 0 and f is a real valued continuous function on [0; 1). They established moments using q derivatives, expressed the operator in terms of divided di¤erences, studied the rate of convergence in a polynomial weighted norm and gave a theorem related to monotonic convergence of the sequence of operators with respect to n:
Finta and Gupta [11] obtained direct estimates for the operators (1:1), using the second order Ditzian-Totik modulus of smoothness. A Voronovskaja-type result for q derivative of q Baskakov operators is given in [2].
Yet, a di¤erent type of q Baskakov operator has also been introduced by Aral and Gupta in [3].
Recently, Cárdenas-Morales, Garrancho and Ra¸sa [9] introduced a new type generelization of Bernstein polynomials denoted by Bn and de…ned as
Bn(f ; x) : = Bn f 1; (x) (1.2) = n X k=0 n k k(x) (1 (x))n k (f 1)(k n);
where Bn is the n th Bernstein polynomial, f 2 C [0; 1], x 2 [0; 1] and is a
continuously di¤erentiable of in…nite order on [0; 1] such that (0) = 0; (1) = 1 and 0(x) > 0 for x 2 [0; 1] : Also, the authors studied some shape preserving and convergence properties concerning the generalized Bernstein operators Bn(f ; x) :
In [5], Aral, Inoan and Ra¸sa constructed sequences of Szasz-Mirakyan operators which are based on a function . They studied weighted approximation properties, Voronovskaja-type result for these operators. They also showed that the sequence of the generalized Szász-Mirakyan operators is monotonically nonincreasing under the convexity of the original function.
In the present paper, we consider a modi…cation of the q Baskakov opera-tors (1:1) in the sense of [5], we study some approximation and shape preserving properties of the new operators.
Motivated from [5] and [9], we de…ne a new generalization of q Baskakov oper-ators for f 2 C[0; 1) by Bn;q(f ; x) = 1 X k=0 f 1 [k] qk 1[n] n + k 1 k q k(k 1) 2 k(x) ( (x) ; q) 1 n+k (1.3) q > 0 and is a continuously di¤erentiable function on [0; 1) such that
(0) = 0; inf
x2[0;1)
0(x) 1:
An example of such a function is given in [5]. Note that, in the setting (1:3) we have
Bn;qf := Bn;q f 1 ;
where the operator Bn;q is de…ned by (1:1). If = e1;then Bn;q = Bn;q: We can
write the following equalities that are similar to the corresponding results for the q Baskakov operators (1:1) Bn;q(1; x) = 1; (1.4) Bn;q( ; x) = (x) (1.5) Bn;q 2; x = 2(x) + (x) [n] 1 + (x) q : (1.6)
Bn;q 3; x = 3(x) + 1 [n] 2(x) 1 + (x) q 2q + 1 q (1.7) 1 [n]2 (x) 1 + (x) q 1 + (x) q2 + 2(x) q 1 + (x) q :
The …rst purpose of the paper is to investigate uniform convergence of the operators (1.3) on weighted spaces which are de…ned using the function and obtain the degree of weighted convergence, using weighted modulus of continuity. Next, we study the monotonic convergence under convexity of the function.
Troughout the paper we will consider the following class of functions. Let ' (x) = 1 + 2(x)
B' R+ = f : R+! R; jf (x)j Mf' (x) ; x 0
where Mf is a constant depending on f .
C' R+ = f 2 B' R+ ; f is continuous on R+
C'k R+ = f 2 C' R+ ; limx!1
f (x) ' (x) = kf where kf is a constant depending on f .
U' R+ = f 2 C' R+ ;
f (x)
' (x) is uniformly continuous on R
+ :
These spaces are normed spaces with the norm kfk'= sup
x2R+
jf (x)j ' (x) :
Moreover, we shall use the following weighted modulus of continuity ! (f ; ) = sup
x;t2R+
j (t) (x)j
jf (t) f (x)j ' (t) + ' (x)
for each f 2 C'(R+) and for every > 0 [14]: We observe that ! (f ; 0) = 0 for
every f 2 C'(R+) and the function ! (f ; ) is nonnegative and nondecreasing
with respect to for f 2 C'(R+).
De…nition 1. A continuous, real valued function f is said to be convex in D [0; 1) ; if f m X i=1 ixi ! m X i=1 if (xi)
for every x1; x2; :::; xm2 D and for every nonnegative numbers 1; 2; :::; msuch
that 1+ 2+ ::: + m= 1:
In [9] Cárdenas-Morales, Garrancho and Ra¸sa introduced the following de…nition of convexity of a continuous function.
De…nition 2. A continuous, real valued function f is said to be convex in D; if f 1 is convex in the sense of De…nition 1.
2. Approximation Properties
In this section, we obtain the weighted uniform convergence of Bn;q to f and the degree of approximation with the aid of weighted modulus of continuity. Let us recall the weighted form of the Korovkin Theorem ([12], [13]).
Lemma 1. [12]The positive linear operators Ln; n 1; act from C'(R+) to
B'(R+) if and only if the inequality
jLn('; x)j Kn' (x) ; x 0
holds; where Kn is a positive constant.
Theorem 1. [12] Let the sequence of linear positive operators (Ln)n 1acting from
C'(R+) to B'(R+) satisfy the three conditions
lim
n!1kLn
v vk
'= 0; v = 0; 1; 2: (2.1)
Then for any function f 2 Ck
'(R+)
lim
n!1kLnf f k'= 0:
Now, we are ready to give the following theorem.
Theorem 2. Let Bn;q be the operator de…ned by (1.3) : Then for any f 2 Ck '(R+)
and q > 1, we have
lim
n!1 Bn;qf f '= 0:
Proof. By Lemma 1 Bn;q are linear operators acting from C'(R+) to B'(R+).
Indeed, from (1:4) and (1:6) we easily obtain that
Bn;q('; x) 1 + 2(x) q [n] + q + 1
q [n] :
On the other hand, using (1:4) ; (1:5) and (1:6) ; one can write Bn;q1 1 '= 0; Bn;q( ) '= 0; and Bn;q 2 2 '= sup x2R+ (x) 1 + (x)q [n] 1 + 2(x) 2 [n]: (2.2)
In [14], the following theorem is given.
Theorem 3. Let Ln: C'(R+) ! B'(R+) be a sequence of positive linear operators
with kLn(1) 1k'0 = an; kLn( ) k'1 2 = bn; Ln 2 2 '= cn; Ln 3 3 '3 2 = dn;
where an; bn; cn and dn tend to zero as n ! 1: Then
kLn(f ) f k'3
2 (7 + 4an+ 2cn) ! (f ; n) + kfk'an
for all f 2 C'(R+) ; where
n = 2
p
(an+ 2bn+ cn) (1 + an) + (an+ 3bn+ 3cn+ dn) :
Applying the above theorem, we obtain the degree of approximation. Theorem 4. For all f 2 C'(R+) and q > 1; we have
Bn;q(f ) f '32 7 + 4 [n] ! f ; 2p2 p [n]+ 18 [n] ! :
Proof. According to Theorem 3, we shall calculate the sequences an; bn; cn and dn.
From (1:4), (1:5), (2:2) and (1:7)we get
an= Bn;q(1) 1 '0 = 0; bn= Bn;q( ) '1 2 = 0; cn = Bn;q 2 2 '= sup x2R+ (x) 1 + q(x)n [n] (1 + 2(x)) 2 [n]; dn = Bn;q 3 3 '3 2 = sup x2R+ 8 < : 1 [n] 2 4 2(x) 1 + (x) q 2q+1 q (1 + 2(x))32 3 5 1 [n]2 2 4 (x) 1 + (x) q 1 + (x) q2 + 2(x) q 1 + (x) q (1 + 2(x))32 3 5 9 = ; 12 [n]:
3. Monotonicity Properties of Bn;q
Here, we study the monotonic convergence of the operators (1:3) under the convexity.
Theorem 5. Let f be a convex function on [0; 1) : Then we have
Bn;q(f ; x) Bn+1;q(f ; x)
for n 2 N:
Proof. From (1:3), one can write
Bn;q(f ; x) Bn+1;q(f ; x) = 1 X k=0 n + k 1 k q k(k 1) 2 k(x) ( (x) ; q)n+k f 1 [k] qk 1[n] 1 X k=0 n + k k q k(k 1) 2 k(x) ( (x) ; q)n+k+1 f 1 [k] qk 1[n + 1] = 1 X k=0 n + k 1 k q k(k 1) 2 k(x) ( (x) ; q)n+k f 1 [k] qk 1[n] 1 X k=0 n + k k q k(k 1) 2 k(x) ( (x) ; q)n+k f 1 [k] qk 1[n + 1] + 1 X k=0 n + k k q k(k 1) 2 qn+k k+1(x) ( (x) ; q)n+k+1 f 1 [k] qk 1[n + 1] = 1 X k=1 n + k 1 k q k(k 1) 2 k(x) ( (x) ; q)n+k f 1 [k] qk 1[n] 1 X k=1 n + k k q k(k 1) 2 k(x) ( (x) ; q)n+k f 1 [k] qk 1[n + 1] + 1 X k=0 n + k k q k(k 1) 2 qn+k k+1(x) ( (x) ; q)n+k+1 f 1 [k] qk 1[n + 1] :
Rearranging the above equality, we have Bn;q(f ; x) Bn+1;q(f ; x) = 1 X k=0 n + k k + 1 q k(k 1) 2 qk k+1(x) ( (x) ; q)n+k+1 f 1 [k + 1] qk[n] 1 X k=0 n + k + 1 k + 1 q k(k 1) 2 qk k+1(x) ( (x) ; q)n+k+1 f 1 [k + 1] qk[n + 1] + 1 X k=0 n + k k q k(k 1) 2 qn+k k+1(x) ( (x) ; q)n+k+1 f 1 [k] qk 1[n + 1] = 1 X k=0 qk(k21)qk k+1(x) ( (x) ; q)n+k+1 n + k k + 1 f 1 [k + 1] qk[n] n + k + 1 k + 1 f 1 [k + 1] qk[n + 1] + n + k k q n f 1 [k] qk 1[n + 1]
By the following equalities n + k + 1 k + 1 = [n + k + 1] [k + 1] n + k k n + k k + 1 = [n] [k + 1] n + k k ; we get Bn;q(f ; x) Bn+1;q(f ; x) = 1 X k=0 n + k k q k(k 1) 2 qk k+1(x) ( (x) ; q)n+k+1 [n + k + 1] [k + 1] [n] [n + k + 1] f 1 [k + 1] qk[n] f 1 [k + 1] qk[n + 1] + qn [k + 1] [n + k + 1] f 1 [k] qk 1[n + 1] : By taking, 1=[n+k+1][n] 0; 2= qn[n+k+1][k+1] 0; 1+ 2= 1 and x1= [k+1]qk[n]; x2= [k] qk 1[n+1] one has 1x1+ 2x2= [k + 1] qk[n + 1]:
Therefore, we obtain that
by convexity of f for x 2 [0; 1) and n 2 N: This proves the theorem. References
[1] Altomare, F. and Mangino, E. M. 1999. On a generalization of Baskakov operator, Rev. Roumaine Math. Pures Appl. 44, 683–705.
[2] Aral, A. Acar, T. 2012. Voronovskaja type result for q derivative of q Baskakov operators. Journal of Applied Functional Analysis, vol. 7(4), 321-331.
[3] Aral, A. Gupta, V. 2009. On q Baskakov type operators, Demonstratio Math. 42 (1), 109– 122.
[4] Aral, A. Gupta, V. 2011. Generalized q Baskakov operators, Math. Slovaca 61 (4), 619–634. [5] Aral, A., Inoan, D. and Ra¸sa, I. 2014. On the generalized Szász-Mirakyan operators. Resulr
in Mathematics. 65, 441-452.
[6] Atakut, Ç. 1997. On the approximation of functions together with derivatives by certain linear positive operators, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat. 46 (1 2) 57–65. [7] Baskakov, V. A. 1957. An example of a sequence of linear positive operators in the spaces of
continuous functions, Doklady Akademii Nauk SSSR 112, 249–251.
[8] Cao, F. , Ding, C. and Xu, Z. 2005. On multivariate Baskakov operator, J. Math. Anal. Appl. 307 no. 1, 274–291
[9] Cárdenas-Moreles, D. Garrancho, P., Ra¸sa, I.: 2011. Berstein-type operators which preserve polynomials. Compt. Math. Appl. 62, 158-163.
[10] Ernst, T. 2000.The history of q calculus and a new method, U.U.U.D.M Report 2000, 16, ISSN 1101-3591, Department of Mathematics, Upsala University.
[11] Finta, Z. Gupta, V. 2010. Approximation properties of q Baskakov operators, Cent. Eur. J. Math. 8(1), 199-211.
[12] Gadjiev, A. D. 1974. The convergence problem for a sequence of positive linear operators on unbounded sets and theorems analogues to that of P. P. Korovkin. Dokl. Akad. Nauk SSSR 218, 1001-1004. Also in Soviet Math. Dokl. 15, 1433-1436.
[13] Gadjiev, A. D. 1976. Theorems of the type of P. P. Korovkin’s theorems (in Russian), Math. Z. 205, 781-786, translated in Maths Notes, 20 (5-6), 995-998 (1977).
[14] Holho¸s, A. 2008. Quantitative estimates for positive linear operators in weighted space. Gen. Math. 16(4), 99-110.
[15] ·Ispir N., 2001. On Modi…ed Baskakov Operators on Weighted Spaces. Turkish Journal of Mathematics, 25, No:3 (355-365).
[16] Pethe, S. 1984. On the Baskakov operator, Indian J. Math. 26, No. 1–3, 43–48 (1985). [17] Phillips, G. M. 2003. Interpolation and approximation by polynomials. CMS Books in
Math-ematics/Ouvrages de Mathématiques de la SMC, 14. Springer-Verlag, New York.
Current address : Ankara University, Elmadag Vocational School, Department of Computer Programming, 06780, Ankara, Turkey
