Applications of q-Calculus in Operator Theory by Ali Aral

By Ali Aral

The approximation of services through linear confident operators is a vital learn subject quite often arithmetic and it additionally presents strong instruments to software parts equivalent to computer-aided geometric layout, numerical research, and ideas of differential equations. q-Calculus is a generalization of many matters, corresponding to hypergeometric sequence, complicated research, and particle physics. ​​This monograph is an creation to combining approximation thought and q-Calculus with purposes, through the use of good- identified operators. The presentation is systematic and the authors contain a short precis of the notations and simple definitions of q-calculus earlier than delving into extra complex fabric. the various purposes of q-calculus within the conception of approximation, specially on a number of operators, such as convergence of operators to capabilities in genuine and complicated area​ varieties the gist of the booklet.

This booklet is acceptable for researchers and scholars in arithmetic, physics and engineering, and for pros who may get pleasure from exploring the host of mathematical options and ideas which are accrued and mentioned within the book.

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36) we get ∞ Bn, q ( f ; x) =∑f x k=1 [k]q qk−1 [n]q n+k−1 k q k(k−1) 2 xk−1 (−x, q)−1 n+k + q f (0) (−x, q)−1 n . 4, then we have Dq Bn, q ( f ; x) x = ∞ ∑f k=2 [k]q k−1 q [n] q n+k−1 k q q k(k−1) 2 [k − 1]qxk−2 (−x, q)−1 n+k 44 2 q-Discrete Operators and Their Results ∞ [k]q k−1 q [n] −∑ f k=1 n+k−1 k q q k(k−1) 2 [n + k]qqk−1 xk−1 (−x, q)−1 n+k+1 q f (0) (−x, q)−1 . 6), we get f (0) (−x, q)−1 n x Dq =− f (0) f (0) (−x, q)−1 (−x, q)−1 n − [n]q n+1 2 qx x Therefore, Bn, q ( f ; x) x Dq ∞ ∑f = k=1 ∞ −∑ f k=1 − [k + 1]q qk [n]q n+k k+1 [k]q qk−1 [n] q k(k−1) 2 q n+k−1 k q qk [k]q xk−1 (−x, q)−1 n+k+1 q k(k−1) 2 [n + k]q qk−1 xk−1 (−x, q)−1 n+k+1 q f (0) f (0) (−x, q)−1 (−x, q)−1 n − [n]q n+1 qx2 x Using the identities n+k−1 k [n]q = n+k k [k + 1]q q [n + k]q = n+k k q n+k k+1 q q [n]q , we have Dq Bn, q ( f ; x) x = ∞ ∑ k=1 f − Since f (x) ≥ 0 and f (x) x is n+k k q k(k−1) 2 xk−1 (−x, q)−1 n+k+1 q [k + 1]q qk [n]q qk [n]q [k + 1]q −f [k]q qk−1 [n]q f (0) f (0) (−x, q)−1 (−x, q)−1 n − [n]q n+1 .

For each integer r > 0 Drq (Snq ( f ; x − [n]q q bn x)) = Eq ∞ [n]q j=0 bn ∑ r r Δrq f j [n]q x j [ j]q ! (bn ) j . 34) Proof. The proof is by induction on r. 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 !

7) that q Bn (x; x) = x. 4), 2 [2]q 2 Δ2q f0 = − [n]q 1 Thus [1]q q [n]q Bn (x2 ; x) = [n]q 2 [1]q − [n]q 2 +q 1 [n]2q [0]q [n]q x+ [0]q [n]q 2 = 2 = 1 [n]2q (1 + q)2 − (1 + q) [n]2q [n]q [n − 1]q q(1 + q) [2]q [n]2q = q(1 + q) [n]2q . x2 and, since [2]q = 1 + q and q [n − 1]q = [n]q − 1, we obtain Bn (x2 ; x) = x2 + x(1 − x) . 11) differs only in having [n]q in place of n. 3 Convergence In the classical case, the uniform convergence of Bn ( f ; x) to f (x) on [0, 1] for each f ∈ C [0, 1] is assured by the following two properties: 1.

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