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<p>The recent contribution by Gogoladze and Meskhia (Proc. A. Razmadze Math. Inst. 141:29–40, 2006) presents a generalization of classical results by Bernstein, Szász, Zygmund, and others on the absolute convergence of Fourier series with coefficients raised to a power <i>ξ</i>, analyzed via moduli of smoothness. In this work, we extend their findings to Fourier-Laplace series in the weighted function spaces <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> <mo stretchy="false">(</mo> <msup> <mi>σ</mi> <mrow> <mi>m</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$L_{p}(\sigma ^{m-1})$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <mn>1</mn> <mo>&lt;</mo> <mi>p</mi> <mo>≤</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$1&lt; p\leq 2$</EquationSource> </InlineEquation>, and we also establish the sharpness of our conditions when <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <mi>p</mi> <mo>=</mo> <mn>2</mn> </math></EquationSource> <EquationSource Format="TEX">$p = 2$</EquationSource> </InlineEquation>.</p>

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On the behavior of Fourier-Laplace series in \(L_{p}(\sigma ^{m-1})\) classes

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  • 发表日期：

    2026年4月28日

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  • Salah El Ouadih,
  • Ahmed Jmaiai

摘要

The recent contribution by Gogoladze and Meskhia (Proc. A. Razmadze Math. Inst. 141:29–40, 2006) presents a generalization of classical results by Bernstein, Szász, Zygmund, and others on the absolute convergence of Fourier series with coefficients raised to a power ξ, analyzed via moduli of smoothness. In this work, we extend their findings to Fourier-Laplace series in the weighted function spaces L p ( σ m − 1 ) $L_{p}(\sigma ^{m-1})$ , 1 < p ≤ 2 $1< p\leq 2$ , and we also establish the sharpness of our conditions when p = 2 $p = 2$ .