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One-sided \(L\)-approximation on a sphere of the characteristic function of a layer. (English) Zbl 1451.41001

The problem of one-sided approximation from below of the characteristic function of a spherical layer by the set of algebraic polynomials of given degree \(n\) in \(m\) variables is discussed in the space \(L(\mathbb{S}^{m-1})\) of functions integrable on the unit sphere \(\mathbb{S}^{m-1}\) of the Euclidean space \(\mathbb{R}^{m}\) of dimension \(m\ge 3\). The authors show that a solution for this multi-dimensional problem can be given when a solution of a corresponding reduced, one-dimensional problem is known. In certain cases, corresponding approximations from above can be obtained either as a consequences of results on approximations from below or by a similar scheme.

MSC:

41A10 Approximation by polynomials
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References:

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