New PDF release: Applications of q-Calculus in Operator Theory

By Ali Aral

ISBN-10: 1461469457

ISBN-13: 9781461469452

ISBN-10: 1461469465

ISBN-13: 9781461469469

The approximation of capabilities via linear optimistic operators is a crucial examine subject usually arithmetic and it additionally presents strong instruments to software parts reminiscent of computer-aided geometric layout, numerical research, and ideas of differential equations. q-Calculus is a generalization of many topics, resembling hypergeometric sequence, complicated research, and particle physics. ​​This monograph is an creation to combining approximation conception and q-Calculus with purposes, by utilizing good- recognized operators. The presentation is systematic and the authors contain 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 idea of approximation, specifically on quite a few operators, along with convergence of operators to capabilities in actual and complicated area​ types the gist of the ebook.

This e-book is acceptable for researchers and scholars in arithmetic, physics and engineering, and for execs who could get pleasure from exploring the host of mathematical suggestions and concepts which are gathered and mentioned within the book.

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Extra resources for Applications of q-Calculus in Operator Theory

Example text

1 − qk )−1 = ∏ (1 + xqs), s=0 where x ∈ [0, ∞) and q ∈ (0, 1). 54) k=0 for q ∈ (0, 1) and x ∈ [0, ∞). , B∞,q f ≤ f for f ∈ CB [0, ∞). This means that the limit q-Baskakov operator is well defined. 49)). 2 Main Results We introduce the space C˜B [0, ∞) = { f ∈ CB [0, ∞) : there exists limx→∞ f (x) = 0}. Obviously C˜B [0, ∞) ⊂ CB [0, ∞) and C˜B [0, ∞) is also a Banach space. The following theorems were studied in [63]. 15. Let C0 ∈ (0, 1) be an absolute constant with the property that q = 1/n q(n) ∈ (C0 , 1) for every n = 1, 2, .

The function f is star-shaped if and only if xDq ( f ) (x) ≥ f (x) for each q ∈ (0, 1) and x ∈ [0, a]. 12. If f is star-shaped, then Bn,q ( f ) is star-shape. Proof. 3 q-Baskakov Operators ∞ [k]q k−1 q [n] −∑ f k=1 ∞ 43 n+k k = [n] ∑ k=0 q q k(k−1) 2 q k(k−1) 2 xk−1 (−x, q)−1 n+k q qk xk (−x, q)−1 n+k+1 q [k + 1]q qk [n]q f n+k−1 k [k]q k−1 q [n] −f − q 1 f [k + 1]q [k + 1]q qk [n]q . [k]q qk−1 [n]q . 48) Since f is star-shaped, we have q[k]q f [k + 1]q [k + 1]q qk [n]q ≥f [k]q k−1 q [n] q . 48), we have the desired result.

In the lemma below, we are concerned with the dependence of S6 on [n]q . 1 q-Bernstein Operators 21 a1 = 1 − (4 + 10q + 10q2 + 5q3 + q4 )x + (6 + 20q + 35q2 + 39q3 + 29q4 + 15q5 + 5q6 + q7)x2 − (4 + 15q + 31q2 + 46q3 + 51q4 + 44q5 + 29q6 + 14q7 + 5q8 + q9)x3 + (1 + 4q + 9q2 + 15q3 + 20q4 + 22q5 + 20q6 + 15q7 + 9q8 + 4q9 + q10 )x4 , a2 = (−1 + 10q + 10q2 + 5q3 + q4 ) + (3 − 4q − 36q2 − 52q3 − 40q4 − 19q5 − 6q6 − q7 )x (−3 − 3q + 12q2 + 46q3 + 72q4 + 67q5 + 43q6 + 19q7 + 6q8 + q9)x2 + (1 + 2q − 7q3 − 20q4 − 30q5 − 32q6 − 25q7 − 13q8 − 5q9 − q10)x3 , and a3 = (1 − 16q + q2 + 13q3 + 11q4 + 4q5 + q6 ) + (−2 + 6q + 30q2 + 5q3 − 22q4 − 26q5 − 15q6 − 5q7 − q8)x (1 − 6q2 − 14q3 − 6q4 + 9q5 + 15q6 + 11q7 + 4q8 + q9)x2 .

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Applications of q-Calculus in Operator Theory by Ali Aral


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