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Trigonometric approximation of functions in \(L_1\)-norm

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Abstract

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 \( \sigma _n^\beta (f;x)\) \( (\beta > 0)\).

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Acknowledgements

We are thankful to the learned referee for giving a number of suggestions which have enabled us to (i) give a comparative study of some of the old and recent results and (ii) include some references for the results obtained in Besov spaces. This has changed the presentation of the present paper to a great extent for the satisfaction of the readers.

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Authors and Affiliations

  1. Formarly Vikram University, C-315, Vivekanand Nagar, Ujjain, 456010, India

    Prem Chandra

  2. Department of Mathematics, Govt.N.P.G.College of Science, Raipur, 492010, India

    Varsha Karanjgaokar

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  1. Prem Chandra
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  2. Varsha Karanjgaokar
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Correspondence to Varsha Karanjgaokar.

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Chandra, P., Karanjgaokar, V. Trigonometric approximation of functions in \(L_1\)-norm. Period Math Hung 84, 177–185 (2022). https://doi.org/10.1007/s10998-021-00397-8

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