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Estimates for Lebesgue constants in dimension two. (English) Zbl 0778.42008

Fourier series of integrable functions fail to converge in the mean. The divergence is measured by the \(L^ 1\)-norms of the Dirichlet kernel; they are called Lebesgue constants and they provide positive summability results for certain classes of functions [cf. D. I. Cartwright and P. M. Soardi, J. Approximation Theory 38, 344-353 (1983; Zbl 0516.42020)]. An important result of K. Babenko (see the above paper for the reference) shows that for the \(N\)-dimensional torus there are constants \(A\) and \(B\) such that \(AR^{(N-1)/2}\leq\| D_ R\|_ 1\leq BR^{(N-1)/2}\), where \(D_ R\) is the Dirichlet kernel associated to a ball of radius \(R\).
Several extensions of Babenko’s results have been proved. The paper under review is one of them: roughly speaking, it shows that in the 2- dimensional case the disk can be substituted by the interior of a piecewise regular curve with curvature different from zero in at least one point. The proof is direct and the result is best possible since, e.g., the Lebesgue constants associated to a polyhedron have a logarithmic growth.

MSC:

42B08 Summability in several variables

Citations:

Zbl 0516.42020
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References:

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[2] D. I.Cartwright - P. M.Soardi,Best condition for norm convergence of Fourier series, J. Approx. Theory,38 (1983). · Zbl 0516.42020
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