In this paper, the authors have obtained \(L_1\)-approximations of functions f in \( {{\,\mathrm{Lip}\,}}(\alpha ,1) \) \( (0 < \alpha \le 1) \) by trigonometrical polynomials \( N_n (f;x)\) whenever the nonnegative and nonincreasing sequence \( (p_n )\) satisfies certain conditions. This enables the authors to approximate \( f \in {{\,\mathrm{Lip}\,}}(\alpha ,p) \) \((0< \alpha \le 1,1\le p < \infty )\) in \( L_p\)-norm by trigonometrical polynomials \( \sigma _n^\beta (f;x)\) \( (\beta > 0)\).

中文翻译：

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$$L_1$$ L 1 -范数中函数的三角近似

在本文中，作者获得了\(L_1\) - 函数f在\( {{\,\mathrm{Lip}\,}}(\alpha ,1) \) \( (0 < \alpha \ le 1) \)由三角多项式\( N_n (f;x)\)只要非负和非递增序列\( (p_n )\)满足某些条件。这使作者能够近似\( f \in {{\,\mathrm{Lip}\,}}(\alpha ,p) \) \((0< \alpha \le 1,1\le p < \infty )\)在\( L_p\) - 三角多项式的范数\( \sigma _n^\beta (f;x)\) \( (\beta > 0)\)。