On mixed multidimensional exponential Kantorovich operators

Abstract In this paper, we introduce and study a family of multidimensional mixed exponential Kantorovich operators on the hypercube $$[0,1]^N$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> </mml:msup> </mml:math> . The idea is to define a vector of operators where the integral mean replaces the sample values of the function just with respect to a single variable, letting the exponential sampling-type structure for the other $$N-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>N</mml:mi> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> variables. For such operators, we compute the first multidimensional exponential moments and then we prove both pointwise and uniform convergence: the latter result is obtained by means of Korovkin type theorems. Furthermore, we prove a quantitative estimate of the order of approximation in terms of the modulus of continuity of the approximated function.