<p>Let <i>G</i> be a finite Jordan domain in the complex plane <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {C}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>, bounded by a Carleson curve <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation> and let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> be a weight function given on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Γ</mi> </math></EquationSource> </InlineEquation>. In this work some direct and inverse theorems of approximation theory in the weighted variable exponent Smirnov classes <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$E_{\omega }^{p(\cdot )}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>E</mi> <mrow> <mi>ω</mi> </mrow> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, under some restrictions on the weight <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> and variable exponent <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p(\cdot )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">(</mo> <mo>·</mo> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, are proved.</p>

Direct and Inverse Theorems of Approximation Theory in Weighted Variable Exponent Smirnov Classes

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2026年6月18日

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Ahmet Testici,
Daniyal Israfilov

摘要

Let G be a finite Jordan domain in the complex plane $\mathbb {C}$ $C$ , bounded by a Carleson curve $\Gamma $ $Γ$ and let $\omega $ $ω$ be a weight function given on $\Gamma $ $Γ$ . In this work some direct and inverse theorems of approximation theory in the weighted variable exponent Smirnov classes $E_{\omega }^{p(\cdot )}(G)$ $E_{ω}^{p (\cdot)} (G)$ , under some restrictions on the weight $\omega $ $ω$ and variable exponent $p(\cdot )$ $p (\cdot)$ , are proved.