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Abstract The purpose of the present paper is to introduce and study a sequence of positive linear operators defined on suitable spaces of measurable functions on [ 0 , ∞ ) $[0,\infty )$ and continuous function spaces with polynomial weights. These operators are Kantorovich type generalization of Jakimovski–Leviatan operators based on multiple Appell polynomials. Using these operators, we approximate suitable measurable functions by knowing their mean values on a sequence of subintervals of [ 0 , ∞ ) $[0,\infty )$ that do not constitute a subdivision of it. We also discuss the rate of convergence of these operators using moduli of smoothness.