By Ali Aral

ISBN-10: 1461469457

ISBN-13: 9781461469452

ISBN-10: 1461469465

ISBN-13: 9781461469469

The approximation of features by means of linear optimistic operators is a vital examine subject more often than not arithmetic and it additionally presents strong instruments to software parts reminiscent of computer-aided geometric layout, numerical research, and strategies of differential equations. q-Calculus is a generalization of many matters, akin to hypergeometric sequence, advanced research, and particle physics. This monograph is an creation to combining approximation idea and q-Calculus with functions, by utilizing good- recognized operators. The presentation is systematic and the authors comprise a quick precis of the notations and uncomplicated definitions of q-calculus ahead of delving into extra complicated fabric. the various functions of q-calculus within the idea of approximation, specially on quite a few operators, together with convergence of operators to services in actual and complicated area varieties the gist of the ebook.

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

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**Extra resources for Applications of q-Calculus in Operator Theory**

**Sample text**

15) that (1 − x) is a factor of Sm (x), for m > 0. 16) r=1 where ar is a polynomial in x and q. 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 .

8) and the Cauchy rule for multiplication of two series [21, p. 21) we have the representation Snq ( f ; x) = ∞ j ∑ ∑ (−1)i q i(i−1) 2 f [n]q x [n]q [i]q ! bnj j=0 i=0 = ∞ [n]q j=0 bn ∑ j j [ j − i]q bn xj , [ j]q ! 4). 4) for f (x) = xm , m = 0, 1, 2, . . Δqj (t m ) (0) = j ∑ (−1) q i i(i−1) 2 i=0 m bn [n]q j i [ j − i]m q for j ≥ 0. Also we know that the connection with q-differences Δqj f0 and jth derivative f ( j) is the following: Δqj f0 f ( j) (ξ ) , = j! q j( j−1) [ j]q ! where ξ ∈ (0, [ j]) (see [134, p.

Depending on the selection of q, these operators are more flexible than the classical Sz´asz–Mirakyan operators while retaining their approximation properties. For these operators, we give a Voronovskaya-type theorem related to q-derivatives. Furthermore, we obtain convergence properties for functions belonging to particular subspaces of C [0, ∞) and give some representation formulae of q-Sz´asz–Mirakyan operators and their rth qderivatives. This section is based on [25, 29]. 1 Introduction In this section, as Phillips has done for Bernstein operators, we introduce a similar modification of the Sz´asz–Mirakyan operators [148] that we call q-Sz´asz–Mirakyan operators and examine the main properties of this new approximation process.

### Applications of q-Calculus in Operator Theory by Ali Aral

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