Applications of q-Calculus in Operator Theory

Abstract

In order to approximate integrable functions on the interval [0, 1], Kantorovich gave modified Bernstein polynomials. Later in the year 1967 Durrmeyer [58] considered a more general integral modification of the classical Bernstein polynomials, which were studied first by Derriennic [47]. Also some other generalizations of the Bernstein polynomials are available in the literature. The other most popular generalization as considered by Goodman and Sharma [82], namely, genuine Bernstein-Durrmeyer operators. In this chapter we discuss the q analogues of various integral modifications of Bernstein polynomials. The results were discussed in recent papers [45, 62, 86, 89, 92, 94, 121], etc. q-Bernstein-Kantorovich OperatorsRecently, Dalmanoglu [45] proposed the q-Kantorovich-Bernstein operators asIn case q = 1, the operators (4.1) reduce to well-known Bernstein-Kantorovich operators Direct ResultsFor the operators (4.1), Dalmanoglu [45] obtained the following theorems:Theorem 4.1. If the sequence (q n ) satisfies the conditions lim n→∞ q n = 1 and lim n→∞Proof. First, we haveAlso by definition of q-integral q-Bernstein-Kantorovich Operators 115Again by definition of q-integralTo estimate K n,q (t 2 , x), we haveTherefore using [k + 1] q = q[k] q + 1 and using the similar methods as above, we haveReplacing q by a sequence {q n } such that lim n→∞ q n = 1, it is easily seen that K n,q (t i , x), i = 0, 1, 2 converges uniformly to t i . Thus the result follows by Korovkin's theorem. for all f ∈ C[0, a] and δ n = K n,q ((t − x) 2 , x).

On (p,q)-analogue of divided differences and Bernstein operators

“…For details on q-calculus and (p, q)-calculus, one can refer to [5,8,18,28] and the references therein.…”

STANCU TYPE GENERALIZATION OF MODIFIED GAMMA OPERATORS BASED ON q-INTEGERS

“…In recent years, one of the most interesting areas of research in approximation theory is the application of q-calculus (see [1]). Phillips [15] first introduced the q-analogue of well-known Bernstein polynomials.…”

Approximation properties of the modified Lupas-Kantorovich type operators

“…Later, this idea was applied to some other well-known approximating operators,such as, the Szasz-Mirakjan operators [6], the Baskakov operators [7], the q-operators [8]. Obviously, these operators have a better rate of convergence than the classical operators.…”