By Ali Aral
The approximation of services through linear optimistic operators is a vital learn subject usually arithmetic and it additionally offers robust instruments to software parts comparable to computer-aided geometric layout, numerical research, and options of differential equations. q-Calculus is a generalization of many matters, reminiscent of hypergeometric sequence, advanced research, and particle physics. This monograph is an advent to combining approximation concept and q-Calculus with functions, through the use of good- recognized operators. The presentation is systematic and the authors comprise a short precis of the notations and simple definitions of q-calculus sooner than delving into extra complicated fabric. the numerous functions of q-calculus within the thought of approximation, particularly on a variety of operators, which include convergence of operators to services in actual and intricate area varieties the gist of the publication.
This publication is acceptable for researchers and scholars in arithmetic, physics and engineering, and for pros who could get pleasure from exploring the host of mathematical recommendations and ideas which are accrued and mentioned within the book.
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X j+r+1 − f [x j , . . , x j+r ] x j+r+1 − x j q j+r [r + 1]q =q =q q r(2 j+r+1) 2 qr Δrq f x j+1 − Δrq f (x j ) [r + 1]q ! r(2 j+r−1) + j+r 2 (r+1)(2 j+r) 2 r Δrq f x j+1 r(2 j+r−1) Δq f (x j ) −q 2 [r]q ! [r]q ! Δr+1 q f (x j ) . [r + 1]q ! 4. 38) Proof. First, we prove that Dq (−x, q)n = [n]q (−qx, q)n−1 . 5) we have Dq (−x, q)n = 1 (q − 1)x n−1 ∏ j=0 n−1 1 + q j+1x − ∏ 1 + q j x j=0 = n−2 1 1 + q j+1x ((1 + qnx) − (1 + x)) (q − 1)x ∏ j=0 = qn − 1 n−2 1 + q j+1x q−1 ∏ j=0 = [n]q (−qx, q)n−1 .
By the q-Taylor formula  for f , we have f (t) = f (x) + Dq f (x) (t − x) + 1 2 D f (x) (t − x)2q + Φq (x; t) (t − x)2q q q for 0 < q < 1 where (t − x)2q = (t − x) (t − qx). 2 q-Sz´asz Operators 27 where n−1 (t − x)nq = ∏ t − qk x k=0 (see ). By applying again of q-L’Hopital’s Rule, there exist q2 ∈ (0, 1) (q1 < q2 ) such that for all q ∈ (q2 , 1) D2q f (t) − D2q f (x) = 0. 27) By assumption the function Φq (t) := Φq (t; x) is a bounded function for all q ∈ (q2 , 1). 29) and thus lim n→∞ [n]qn bn (Snqn ( f ; x) − f (x)) [n]qn q 1 S n Φqn (t) (t − x)2qn ; x = x lim D2qn f (x) + lim n→∞ 2 qn →1 bn n 28 2 q-Discrete Operators and Their Results Now, the last term on the right-hand side can be estimated in the following way.
Let q = (qn ) satisfy 0 < qn < 1 and let qn → 1 as n → ∞. Then, if f ∈ C [0, 1], Bn ( f ; x) = n ∑ fr r=0 n r xr qn n−r−1 ∏ (1 − qsnx) s=0 converges uniformly to f on [0, 1]. Proof. This is a special case of the Bohman–Korovkin theorem. 11) above. Given a function f defined on [0, 1], let w (δ ) = sup |x1 −x2 |<δ | f (x1 ) − f (x2 ) |, the usual modulus of continuity, where the supremum is taken over all x1 , x2 ∈ [0, 1] such that | x1 − x2 |≤ δ . 4. 5), then f − Bn f ∞≤ 1 3 w( 1/2 ). 12) q Proof.
Applications of q-Calculus in Operator Theory by Ali Aral