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<p>In this paper we introduce a new family of Bernstein-type exponential polynomials on the hypercube <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="13398_2024_1693_Article_IEq1.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\([0,1]^d\)</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>d</mi> </msup> </math></EquationSource> </InlineEquation> and study their approximation properties. Such operators fix a multidimensional version of the exponential function and its square. In particular, we prove uniform convergence, by means of two different approaches, as well as a quantitative estimate of the order of approximation in terms of the modulus of continuity of the approximated function.</p>

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Approximation processes by multidimensional Bernstein-type exponential polynomials on the hypercube

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Laura Angeloni,
  • Danilo Costarelli,
  • Chiara Darielli

摘要

In this paper we introduce a new family of Bernstein-type exponential polynomials on the hypercube \([0,1]^d\) [ 0 , 1 ] d and study their approximation properties. Such operators fix a multidimensional version of the exponential function and its square. In particular, we prove uniform convergence, by means of two different approaches, as well as a quantitative estimate of the order of approximation in terms of the modulus of continuity of the approximated function.