The Gibbs phenomenon for \(n\)-dimensional kernels (Q706213)

This paper discusses the Gibbs phenomenon for kernels in high dimensions. Let \(K: R^{2n}\to R^n\) be a kernel. The authors consider a sequence of kernels \(K_{m,h}, m\in N, h=(h_1, \ldots, h_n)\in N^n\), where \[ K_{m, h}(x_1, \ldots, x_n;y_1, \ldots, y_n):=2^{nm+| h| } K(2^m x_1, \ldots, 2^m x_n; 2^{m+h_1} y_1, \ldots, 2^{m+h_n} y_n). \] The above kernels are not necessarily invariant under rotations and translations. Under some conditions the authors show in Theorems 1 and 2 that the sequence of kernels \(K_{m, h}\) (\(h=0\) in Theorem 1) displays the Gibbs phenomenon in the origin. For periodic kernels (that is, \(K(x+l; y+l)=K(x;y)\) for all \(l\in Z^n\)), under some conditions, it has been shown in Theorem 3 that the sequence of kernels \(K_{m,h}\) displays the Gibbs phenomenon almost everywhere in \(R^n\).