Approximation by a Double Sequence of Operators Involving Multivariable q-Lagrange–Hermite Polynomials

In this chapter, we introduce a bivariate extension of the linear positive operators based on multivariate q-Lagrange–Hermite polynomials and determine the degree of approximation with the help of the moduli of continuity and the Peetre’s K-functional for bi-dimensionally continuous functions. Further, we define the corresponding Generalized Boolean Sum (GBS) operators and derive the rate of convergence of these operators for functions in a Bögel space. We also generalize the above operators with the aid of Taylor polynomial and examine the rate of approximation of the generalized operators for functions whose sth derivative is continuous and belongs to the Lipschitz class. Finally, we consider a Kantorovich version of the above bivariate operators involving multivariate q-Lagrange- Hermite polynomials and note that the above study could be extended to these operators.