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Fundamental functions vanishing on a given set and division by functions

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Abstract

The space ΨV of fundamental functions (a subspace of S) consisting of functions vanishing together with all their derivatives on a given closed set V⊂Rn is constructed. Multipliers in ΨV are described. In the space ΨV is easily realized the division of unity by an infinitely differentiable function, “vanishing slowly” for approximation to its zero set, (in particular, by a polynomial). In the case of a cone V in Rn, a description of the dual space ΦV consisting of the Fourier preimages of functions of ΨV is given. The problem of multipliers in ΦV is discussed.

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Authors and Affiliations

  1. Rostov State University, USSR

    S. G. Samko

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  1. S. G. Samko
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Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 677–689, May, 1977.

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Samko, S.G. Fundamental functions vanishing on a given set and division by functions. Mathematical Notes of the Academy of Sciences of the USSR 21, 379–386 (1977). https://doi.org/10.1007/BF01788235

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  • DOI: https://doi.org/10.1007/BF01788235

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