The Gibbs phenomenon for Fourier interpolation (Q1332284)

For a real-valued function periodic on \(\mathbb{R}\) with period \(2\pi\) and of bounded variation on \([- \pi, \pi]\) the behaviour of the Fourier interpolation polynomial near an isolated jump discontinuity point is investigated. It is revealed the presence of asymmetry in the local Gibbs phenomenon for left and right continuity in case when the jump point coincides with the interpolation node. To obtain information on the local extremum of the Fourier interpolation polynomial \(S^*_n\) derivatives are used and to obtain boundary values of \(S^*_n\) its convexity is investigated. The cases of odd and even number of nodes of a Fourier interpolation polynomial, as well as its asymptotic behaviour, are treated separately.