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A criterion for approximability by harmonic functions in Lipschitz spaces

Let X be a compact subset of \( {{\mathbb{R}}^3} \), and let f be a function that is harmonic inside X and belongs to the Lipschitz space C γ(X), 0 < γ < 1. A criterion for approximability of f on X in C γ(X) by functions that are harmonic on neighborhoods of X is obtained in terms of the Hausdorff content of order 1 + γ. The proof is completely constructive, and Vitushkin's method of singularities separation and approximation by parts is applied. Bibliography: 15 titles.

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Correspondence to M. Ya. Mazalov.

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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 401, 2012, pp. 144–171.

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Mazalov, M.Y. A criterion for approximability by harmonic functions in Lipschitz spaces. J Math Sci 194, 678–692 (2013). https://doi.org/10.1007/s10958-013-1557-5

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Keywords

  • Harmonic Function
  • Compact Subset
  • Lipschitz Space
  • Singularity Separation
  • Hausdorff Content