RATE OF CONVERGENCE FOR CERTAIN BASKAKOV. DURRMEYER TYPE OPERATORS. VIJAY GUPTA AND P. N. AGRAWAL. Abstract. In the present paper ...
Analele Universit˘ a¸tii Oradea Fasc. Matematica, Tom XIV (2007), 33–39
RATE OF CONVERGENCE FOR CERTAIN BASKAKOV DURRMEYER TYPE OPERATORS VIJAY GUPTA AND P. N. AGRAWAL Abstract. In the present paper, we study a certain integral modification of the well known Baskakov operators with the weight function of Beta basis function. We establish rate of convergence for these operators for functions having derivatives of bounded variation.
1. Introduction The integral modification of Baskakov operators, considered in this paper are defined by Z ∞ ∞ X bn,k (t)f (t)dt + (1 + x)−n f (0) Bn (f, x) = pn,k (x) 0
k=1 ∞
Z =
Wn (x, t)f (t)dt,
(1.1)
0
where n+k−1 k
pn,k (x) = bn,v (t) = and Wn (x, t) =
!
xv ; (1 + x)n+k
1 tk−1 , B(n + 1, k) (1 + t)n+k+1
∞ X
pn,k (x)bn,k (t) + (1 + x)−n δ(t),
k=1
2000 Mathematics Subject Classification. 41A25; 41A30. Key words and phrases. Linear positive operators, Baskakov operators, Beta basis function, rate of convergence. 33
34
VIJAY GUPTA AND P. N. AGRAWAL
δ(t) being the Dirac delta function. The operators (1.1) were introduced by Finta [2], these operators are different from the usual Baskakov Durrmeyer operators. Actually these operators reproduce not only the constant ones, but also the linear functions. Very recently Gupta et al. [3] also estimated some direct and inverse results in simultaneous approximation for these operators (1.1). Rt We denote βn (x, t) = 0 Wn (x, s)ds, then in particular, we have Z ∞ Wn (x, s)ds = 1. βn (x, ∞) = 0
By DBr (0, ∞), r ≥ 0 we mean the class of absolutely continuous functions f defined on the interval (0, ∞) such that: (i)f (t) = O(tr ), t → ∞, (ii) having a derivative f 0 on the interval (0, ∞) coinciding a.e. with a function which is of bounded variation on every finite subinterval of (0, ∞). It can be observed that all functions f ∈ BDr (0, ∞) possess for each c > 0 a representation Z x f (x) = f (c) + ψ(t)dt, x ≥ c. c
Bojanic and Cheng [1] and Gupta et al. [4] obtained interesting results on the rate of convergence of Bernstein polynomials and some summationintegral type operators respectively, for functions having derivatives of bounded variation. We now extend the study for these new BaskakovBeta type operators (1.1) and estimate the rate of convergence for functions having derivatives of bounded variation. Our main result is stated as follows: Theorem. Let f ∈ DBr (0, ∞),r ∈ N , and x ∈ (0, ∞). Then for n sufficiently large, we have √
√
[ n] x+x/k x+x/ n 2(1 + x) X _ x _ 0 | Bn (f, x) − f (x) | ≤ ( ((f )x ) + √ ((f 0 )x )) n − 1 k=1 n √ x−x/k
x−x/ n
CONVERGENCE OF CERTAIN BASKAKOV DURRMEYER TYPE OPERATORS35
2(1 + x) (| f (2x) − f (x) − xf 0 (x+ ) | + | f (x) |) (n − 1)x p + 2x(1 + x)/n − 1(M 2r O(n−r/2 )+ | f 0 (x+ ) |) 1p 2x(1 + x)/(n − 1) | f (0 x+ ) | − | f 0 (x− ) |, + 2 where the auxiliary function fx is given by − f (t) − f (x ) , 0 ≤ t < x; fx (t) = 0 , t = x; f (t) − f (x+ ) , x < t < ∞. Wb a f (x) denotes the total variation of fx on [a, b]. +
2. Basic Results We shall use the following lemmas to prove our main theorem. Lemma 2.1. Let the function Tn,m (x), m ∈ N ∪ {0}, be defined as Tn,m (x) = Bn (t − x)m x Z ∞ ∞ X bn,k (t)(t − x)m dt + (1 + x)−n (−x)m = pn,k (x) v=1
0
then Tn,0 (x) = 1, Tn,1 = 0, Tn,2 (x) = recurrence relation
2x(1 + x) and also there holds the n−1
(1) (n−m)Tn,m+1 (x) = x(1+x)bTn,m (x)+2mTn,m−1 (x)c+m(1+2x)Tn,m (x).
Consequently, for each x ∈ [0, ∞) we get Tn,m (x) = O(n−[(m+1)/2] ). Remark 2.1. From Lemma 2.1, using Cauchy-Schwarz inequality, it follows that p 1/2 Bn (| t − x |, x) ≤ [Bn ((t − x)2 , x)] ≤ 2x(1 + x)/(n − 1).
36
VIJAY GUPTA AND P. N. AGRAWAL
Lemma 2.2. Let x ∈ (0, ∞) and Wn (x, t) be the kernel defined in (1.1). Then for n sufficiently large, we have Z y 2x(1 + x) Wn (x, t)dt ≤ (i) βn (x, y) = , 0≤y
