General estimates for the difference of operators
Abstract
This paper deals with the quantitative estimates for the difference of operators having the same basis functions. The results essentially improve the earlier known results. In the end, some applications for the differences of Szász-Mirakyan operators with its different variants are presented.
1 DIFFERENCES OF OPERATORS
, some results have been estimated.7, 8 Let
be an interval and
, with C (I ) containing the polynomials and CB(I ) the space of all f ∈ C(I ) such that
be a positive linear functional such that F (e0) = 1. Denote
. For the functional F, the following basic results were obtained.4
Lemma 1. (See the work of Acu and Raşa[4])Let f ∈ C(I ) with
∈ CB(I ). Then,
Lemma 2. (See the work of Acu and Raşa[4])Let f ∈ C(I ) with f (iv) ∈ CB(I). Then,
. Consider the positive linear functionals
such that Fn,k(e0) = Gn,k(e0) = 1 and denote D(I ) as the set of all f ∈ C(I ) for which
Theorem 1.If f ∈ D(I ) with
∈ CB(I), then
.
Remark 1.We may remark here that in theorem 3 in the work of Acu and Raşa4, the authors considered
, which is true in general only when summation index K is finite, ie,
, but in case
, the value of δ(x) becomes infinite. Second, for λ > 0, we know ω( f,λδ) ≤ (λ + 1)ω( f,δ). Thus, the improved conclusion of theorem 3 of Acu and Raşa4 must be considered as the following theorem.
Theorem 2.If f ∈ D(I ) with
∈ CB(I), then
Corollary 1.If f ∈ D(I ) with
∈ CB(I ) and both the operators Un and Vn preserve linear functions, then we have
Theorem 3.If f ∈ D(I ) with
∈ CB(I ), then for each x ∈ I, we have
, the theorem follows.
. Thus, the above theorem may be utilized for a wide range of operators.2 SOME APPLICATIONS
In this section, we provide the applications of Theorem 1 and estimate the quantitative difference estimates between Szász-Mirakyan operators and its variants.
(1)
Lemma 3.The following recurrence relation holds for moments
Remark 3.We have
, we have
2.1 Difference with Szász-Mirakyan-Baskakov operators
(2)
Remark 4.By definition of beta function, we have
Below, we present the quantitative estimate for the difference of Szász-Mirakyan-Baskakov and Szász-Mirakyan operators.
Proposition 1.Let f (s) ∈ CB[0,∞),s ∈ {0,1,2} and x ∈ [0,∞). Then, for
, we have
Proof.Using Remarks 3 and 4 and Lemma 3, we have
Proposition 2.Let f (s) ∈ CB[0,∞),s ∈ {0,1,2}, and x ∈ [0,∞). Then, for
, we have
Proposition 3.Let f (s) ∈ CB[0,∞),s ∈ {0,1,2,3,4}. Then for each x ∈ [0,∞), we have
2.2 Difference with Pǎltǎnea operators
(3)
Remark 5.By definition of gamma function, we have
We present below the quantitative estimates for the difference of Pǎltǎnea and Szász-Mirakyan operators.
Proposition 4.Let f (s) ∈ CB[0,∞),s ∈ {0,1,2}, and x ∈ [0,∞). Then for
, we have
Proof.As both the operators here preserve constant and linear functions, using Corollary 1 and applying Lemma 3 and Remarks 3 and 5, we have
Proposition 5.Let f (s) ∈ CB[0,∞),s ∈ {0,1,2,3,4}. Then for each x ∈ [0,∞), we have
, we get δ(x) = 0; thus, the proposition follows from Theorem 2.3 Difference with Szász-Mirakyan-Kantorovich operators
(4)
Remark 6.By simple computation, we have
Below, we present the quantitative estimate for the difference of Szász-Mirakyan-Kantorovich and Szász-Mirakyan operators.
Proposition 6.Let f (s) ∈ CB[0,∞),s ∈ {0,1,2} and x ∈ [0,∞), then for
, we have
Proof.Using Remark 3, Remark 6 and Lemma 3, we have
Proposition 7.Let f (s) ∈ CB[0,∞),s ∈ {0,1,2,3,4}, then for each x ∈ [0,∞), we have
2.4 Difference with Szász-Mirakyan-Durrmeyer operators
(5)
Remark 7.By simple computation, we have
We present below the quantitative estimates for difference of Szász-Mirakyan-Durrmeyer and Szász-Mirakyan operators.
Proposition 8.Let f (s) ∈ CB[0,∞),s ∈ {0,1,2}, and x ∈ [0,∞). Then for
, we have
Proof.Using Remarks 3 and 7 and Lemma 3, we have
Proposition 9.Let f (s) ∈ CB[0,∞),s ∈ {0,1,2,3,4}. Then for each x ∈ [0,∞), we have
ACKNOWLEDGEMENT
The author would like to thank the referees for their valuable comments, leading to overall improvements of the paper.
Biography
Vijay Gupta PhD, is a professor at the Department of Mathematics, Netaji Subhas University of Technology (formerly, Netaji Subhas Institute of Technology), New Delhi, India. He received the PhD degree from the University of Roorkee (currently Indian Institute of Technology, Roorkee, India. His area of research is approximation theory especially on linear positive operators. He is the author of several books and book chapters, and he has over 300 research papers to his credit. He visited many universities globally on academic invitations. His present h-index is 36 (as per Google Scholar records).
