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In this paper, we establish a new estimate for the degree of approximation of functions f (x , y) belonging to the generalized Lipschitz class L i p ((ξ 1 , ξ 2) ; r) , r ≥ 1 , by double Hausdorff matrix summability means of double Fourier series. We also deduce the degree of approximation of functions from L i p ((α , β) ; r) and L i p (α , β) in the form of corollary. We establish some auxiliary results on trigonometric approximation for almost Euler means and (C , γ , δ) means. [ABSTRACT FROM AUTHOR]

Runge's phenomenon reveals that interpolation methods lack uniform convergence of the sequence of the thus constructed polynomials to the function. The linear positive operators, however, can be relied on to ensure convergence across the whole domain. Further, linear positive operators can be modified suitably to approximate integrable functions. In this paper, we prove theoretical results on the estimates for the rate of approximation of a new sequence of operators. In addition, we find an error estimate for a larger class of functions, the functions of bounded variation. Finally, the theory is supported by suitable numerical examples, and the convergence estimates of the proposed operator are compared with the classical modified Bernstein–Durrmeyer operator. [ABSTRACT FROM AUTHOR]