Applications of q-Calculus in Operator Theory by Ali Aral

By Ali Aral

The approximation of services through linear optimistic operators is a crucial study subject ordinarily arithmetic and it additionally offers strong instruments to software components comparable to computer-aided geometric layout, numerical research, and suggestions of differential equations. q-Calculus is a generalization of many matters, resembling hypergeometric sequence, complicated research, and particle physics. ​​This monograph is an creation to combining approximation concept and q-Calculus with purposes, through the use of good- identified operators. The presentation is systematic and the authors contain a quick precis of the notations and simple definitions of q-calculus sooner than delving into extra complex fabric. the numerous purposes of q-calculus within the thought of approximation, specifically on quite a few operators, consisting of convergence of operators to capabilities in genuine and complicated area​ varieties the gist of the booklet.

This publication is appropriate for researchers and scholars in arithmetic, physics and engineering, and for execs who may take pleasure in exploring the host of mathematical recommendations and concepts which are accumulated and mentioned within the book.

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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, . .. 55) for all f ∈ C˜B [0, ∞) and n = 3, 4, . .. 49). 16. 56) for all f ∈ C˜B [0, ∞), n = 1, 2, . . and q ∈ (0, 1). 3 Proofs The q-forward differences lead us to the moments of the first and second orders of Bn,q .

268]). From this equality, it is obvious that qdifferences of monomial t m of order greater than m are zero. q j( j−1) 2 ∑ (−1)i q i(i−1) 2 i=0 j [ j − i]m q i are the Stirling polynomials of the second kind satisfying the equality Sq (m + 1, j) = Sq (m, j − 1) + [ j]q Sq (m, j) , for m ≥ 0 and j ≥ 1 with Sq (0, 0) = 1, Sq (m, 0) = 0 for m > 0. Also Sq (m, j) = 0 for j > m. Thus the proof is completed. 25) Snq (e2 ; x) = qx2 + bn x. 1, b). 1. 21), we will assume q = qn as a sequence such that qn → 1 as n → ∞ for 0 < qn < 1 and so that [n]qn → ∞ as n → ∞.

Note that these representations are not obtained using classical derivatives and forward differences. 2. 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 !

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