Pointwise and directional regularity of nonharmonic Fourier series
The author studies nonharmonic Fourier series, i.e., functions of the form \(\sum_{n\in N}a_ne^{i \lambda_n\cdot x}\) and in particular how the regularity of the series is related to the spacing of the frequencies. The main tool is the Gabor-wavelet transform defined by \(\frac 1{a^d} \int_{ R^d} f(x)e^{-i \lambda\cdot x} \phi_\Omega\left(\frac{x-b}a\right)\, dx\) where \(\phi\) is a smooth function such that its Fourier transform is supported in the unit ball centered at \(0\) and \(\phi_\Omega(x)= \phi(\Omega x)\) where \(\Omega\) is a rotation in \(\mathbb R^d\). Applications to everywhere irregularity of solutions of certain partial differential equations (Schrödinger's equation, simply supported beams) are given. By extending the results to the anistorpic case the author derives directional irregularity criteria as well.
scientific article; zbMATH DE number 5708939
scientific article