If a function f is of bounded variation on T<sup>N</sup> (N ≥ 1) and {φ<sub>n</sub>} is a positive approximate identity, we prove that the area of the graph of f * φ<sub>n</sub> converges from below to the relaxed area of the graph of f. Moreover we give asymptotic estimates for the area of the graph of the square partial sums of multiple Fourier series of functions with suitable discontinuities. ©2006 American Mathematical Society.
DE MICHELE, L., Roux, D. (2006). Gibbs' phenomenon and surface area. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 134(12), 3561-3566 [10.1090/S0002-9939-06-08639-4].
Gibbs' phenomenon and surface area
DE MICHELE, LEONEDE;
2006
Abstract
If a function f is of bounded variation on TN (N ≥ 1) and {φn} is a positive approximate identity, we prove that the area of the graph of f * φn converges from below to the relaxed area of the graph of f. Moreover we give asymptotic estimates for the area of the graph of the square partial sums of multiple Fourier series of functions with suitable discontinuities. ©2006 American Mathematical Society.
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