Sets of uniform convergence of Fourier expansions of piecewise smooth functions
The author considers a homogeneous elliptic operator \[ A(D):= \sum_{|\alpha|= m} a_\alpha D^\alpha \] of order \(m\) with coefficients in \(L_2(\mathbb{R}^n)\). As it is well known this operator is essentially selfadjoint and has a unique selfadjoint extension, which we denote also by \(A\). According to the spectral theorem, to every function \(f\in L_2(\mathbb{R})\) one can associate an eigenfunction expansion, using the corresponding spectral family. In this case it coincides with the Fourier expansion and is of the form \[ E_\lambda f(x)= (2\pi)^{-n/2} \int_{A(\xi)<\lambda}\widehat f(\xi) e^{ix\xi}\,d\xi,\tag{\(*\)} \] where \[ \widehat f(\xi):= (2\pi)^{-n/2}\int_{\mathbb{R}^n} f(x) e^{-ix\xi}\,dx. \] The author proves the following theorem: Let \(n\geq 3\). For any piecewise smooth function \(f\) with smooth corresponding surface \(\Gamma\), the spectral expansion \((*)\) converges uniformly on each compact set \(K\subset\Omega(\Gamma)\setminus\Gamma\), where \(\Omega(\Gamma)\) is the set of all points which are regular relative to \(\Gamma\) (see the definition in the paper).
scientific article; zbMATH DE number 2166811