<p>This paper examines the approximation properties of Kantorovich-type generalized sampling operators within the context of Morrey spaces. We first establish point-wise estimates for the Kantorovich sampling operators <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(S_{w}^{\chi }\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>S</mi> <mrow> <mi>w</mi> </mrow> <mi>χ</mi> </msubsup> </math></EquationSource> </InlineEquation>, utilizing the Hardy–Littlewood maximal function. Building on this, we prove the uniform boundedness of these operators in Morrey spaces, particularly under certain smoothness conditions. Further, we obtain the error estimate in the Morrey–Sobolev space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(W^{1}(\mathcal {M}_{p}^{\lambda }(\mathbb {R}))\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>W</mi> <mn>1</mn> </msup> <mrow> <mo stretchy="false">(</mo> <msubsup> <mi mathvariant="script">M</mi> <mrow> <mi>p</mi> </mrow> <mi>λ</mi> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, showing that the approximation error of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(S^{\chi }_{w}f\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>S</mi> <mi>w</mi> <mi>χ</mi> </msubsup> <mi>f</mi> </mrow> </math></EquationSource> </InlineEquation> can be controlled by the Sobolev semi-norm of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(f\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>f</mi> </math></EquationSource> </InlineEquation>. We also establish a convergence rate for the modulus of smoothness of the sampling operator in Morrey spaces. Finally, we estimate the rate of approximation for these sampling operators under the condition of Hölder continuity.</p>

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Approximation by Kantorovich-type generalized sampling operators in Morrey spaces

  • Arash Ghorbanalizadeh,
  • Tahereh Khazaee,
  • A. Sathish Kumar

摘要

This paper examines the approximation properties of Kantorovich-type generalized sampling operators within the context of Morrey spaces. We first establish point-wise estimates for the Kantorovich sampling operators \(S_{w}^{\chi }\) S w χ , utilizing the Hardy–Littlewood maximal function. Building on this, we prove the uniform boundedness of these operators in Morrey spaces, particularly under certain smoothness conditions. Further, we obtain the error estimate in the Morrey–Sobolev space \(W^{1}(\mathcal {M}_{p}^{\lambda }(\mathbb {R}))\) W 1 ( M p λ ( R ) ) , showing that the approximation error of \(S^{\chi }_{w}f\) S w χ f can be controlled by the Sobolev semi-norm of \(f\) f . We also establish a convergence rate for the modulus of smoothness of the sampling operator in Morrey spaces. Finally, we estimate the rate of approximation for these sampling operators under the condition of Hölder continuity.