On a class of \(N\)-dimensional trigonometric series
Let \(V\) be a closed bounded polyhedron in \(\mathbb{R}^N\) with vertices at points with rational coordinates, star-shaped with respect to the origin, which is an interior point of \(V\). Let \(nV:= \{x\in \mathbb{R}^N: x/n\in V\}\). Consider the \(N\)-dimensional trigonometric series \[ \lambda_0+ \sum^\infty_{\ell= 1} \lambda_\ell \sum_{k\in\ell V\setminus(\ell- 1)V} e^{i\langle k,x\rangle},\tag{1} \] where \((\lambda_\ell)\) is a sequence of real numbers with \(\lambda_\ell\to 0\) as \(\ell\to\infty\). The following theorem is proved. If \[ \sum^\infty_{\ell= 1} \Biggl\{ 2^{\ell(p- 1)} \sum_{j\in (2^\ell, 2^{\ell+1}]}|\Delta \lambda_j|\Biggr\}^{1/p}< \infty, \] then series (1) converges a.e. to some function \(f\in L(\mathbb{T}^N)\), is its Fourier series, and \[ S_{nV}f(x):= \lambda_0+ \sum^n_{\ell= 1}\lambda_\ell \sum_{k\in\ell V\setminus(\ell- 1)V} e^{i\langle k,x\rangle} \] converges to \(f(x)\) in the norm of \(L(\mathbb{T}^n)\) as \(n\to\infty\) if and only if \(\lim_{n\to\infty} \lambda_n(\log n)^N= 0\).
scientific article; zbMATH DE number 1262563