By Ali Aral
The approximation of features by way of linear optimistic operators is a crucial learn subject generally arithmetic and it additionally offers strong instruments to software components similar to computer-aided geometric layout, numerical research, and strategies of differential equations. q-Calculus is a generalization of many matters, similar to hypergeometric sequence, complicated research, and particle physics. This monograph is an creation to combining approximation idea and q-Calculus with purposes, through the use of good- identified operators. The presentation is systematic and the authors comprise a short precis of the notations and simple definitions of q-calculus ahead of delving into extra complex fabric. the numerous functions of q-calculus within the thought of approximation, in particular on a variety of operators, such as convergence of operators to features in genuine and complicated area kinds the gist of the ebook.
This booklet is appropriate for researchers and scholars in arithmetic, physics and engineering, and for execs who may take pleasure in exploring the host of mathematical options and ideas which are accrued and mentioned within the book.
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Additional info for Applications of q-Calculus in Operator Theory
R Δ f (0) [n − 1]q! q for r ≥ 1. By using the above equality in q-Taylor formula given in , we get Bn,q ( f , x) = ∞ [n + r − 1]q! r xr Δq f (0) . [r]q ! r=0 [n − 1]q! 5, we have the following corollary. 6. The q-Baskakov operators can be represented as Bn,q ( f , x) = ∞ [r]q [n + r − 1]q! − r(r−1) 1 q xr . , , . . , r−1 q 2 f 0, [n]q q[n]q q [n]q r=0 [n − 1]q ! ∑ We are now in a position to give the moments of the first and second orders of the operators Bn,q . 5. For Bn,q (t m , x), m = 0, 1, 2, one has Bn,q (1, x) = 1.
4. , x. [n] [n] Proof. 35), the proof is obvious. 3 q-Baskakov Operators In this section we propose a generalization of the Baskakov operators, based on q-integers. We also estimate the rate of convergence in the weighted norm. We also study some shape-preserving and monotonicity properties of the q-Baskakov operators and also different generalizations of classical Baskakov operators based on q-integers defined in [30, 136]. First, we recall classical Baskakov operators , which for f ∈ C [0, ∞) are defined as ∞ k n+k−1 k Bn ( f , x) = ∑ x (1 + x)−n−k f k n k=0 This section is based on .
4) we set Dq Eq − [n] bxn x = − [n] bn Eq − [n] q bn . 3) we find q Dq Sn ( f ; x) =− + [n]q bn [n]q bn = Eq = Eq Eq Eq ∞ x − [n]q q bn ∑f j=0 ∞ x − [n]q q bn x − [n]q q bn x − [n]q q bn ∑f [n]q j=0 bn ∞ [n]q j=0 bn ∑ ∑ [n]q x [n]q [ j]q ! (bn ) j f Δ1q f j j [ j + 1]q bn [n]q x [n]q [ j]q ! (bn ) j j=0 ∞ j [ j]q bn [ j + 1]q bn [n]q [n]q x −f j [ j]q bn [n]q x [n]q [ j]q ! (bn ) j j [ j]q ! (bn ) j . Similarly, D2q (Snq ( f ; x)) = Dq (Dq (Snq ( f ; x))) = −q + = Eq 2 [n]q Eq bn [n]q bn 2 Eq x − [n]q q bn x − [n]q q bn 2 ∞ x − [n]q q2 bn 2 ∞ [n]q j=0 bn ∑ ∑ f [ j + 1]q bn [n]q j=0 ∞ ∑ f j=0 2 Δ2q f j [ j + 2]q bn [n]q [n]q x j [ j]q !
Applications of q-Calculus in Operator Theory by Ali Aral