<p>In this paper, we introduce and study a family of multidimensional mixed exponential Kantorovich operators on the hypercube <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\([0,1]^N\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mi>N</mi> </msup> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(N-1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>-</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

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On mixed multidimensional exponential Kantorovich operators

  • Laura Angeloni,
  • Ali Aral,
  • Firat Ozsarac

摘要

In this paper, we introduce and study a family of multidimensional mixed exponential Kantorovich operators on the hypercube \([0,1]^N\) [ 0 , 1 ] N . 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 \(N-1\) N - 1 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.