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September 10, 2025Open Access

Study of two dimensional α-modified Bernstein bi-variate operators

NRNadeem RaoChandigarh University MSMohammad ShahzadChandigarh University NJNand Kishor JhaChandigarh University

Key Points

The study demonstrates various approximation results for two dimensional continuous functions using Bernstein bi-variate operators.
Key findings include the evaluation of local approximation results in terms of Lipschitz maximal functions and estimates of some lemmas.
Assessment involved proving uniform convergence and exploring the order of approximation in specific function spaces.
The relevance of approximation results in B?gel-spaces highlights practical applications of mixed-modulus of continuity.

Abstract

This research work is the study of two dimensional ?-Modified Bernstein sequence of operators. Further, to investigate various approximation results in the space of two dimensional continuous functions in terms of these sequence of operators. Next, we estimate some lemmas in terms of test functions and central moments. In subsequent sections, uniform convergence and order of approximation are studied. Moreover, we prove local approximation results in view of Lipschitz maximal functions. In the last section, approximation results in B?gel-spaces are discussed with the help of mixed-modulus of continuity.

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