"Exponential bases for parallelepipeds with frequencies lying in a pre-scribed lattice."Walnut, DavidWe investigate the problem of finding Riesz bases of exponentials for L2(S) with frequencies in a prescribed lattice. Such results exist for example when S is a subinterval of [0, 1] (Seip, Avdonin) and when S is a disjoint union of intervals contained in [0, 1], or of rectangles with edges parallel to the axes contained in [0, 1]d (Kozma and Nitzan). In each case the prescribed lattice is Z or Zd. In this talk, we discuss the case when S is a parallelepiped in Rd (parallelogram in R2), and when the prescribed lattice is BZd for some invertible d×d matrix B. We give sufficient structural conditions on a d×d matrix A such that L2(A[0, 1]d) permits such a basis, and conditions on the spectral norm of A unrelated to its structure. The latter conditions follow from perturbation results for higher dimensional exponential bases. Related results on the existence of orthonormal bases of exponentials on a prescribed lattice lead us into the realm of cube tilings in Rd and will be discussed in a different talk at this meeting. The work discussed is joint with Dae Gwan Lee (Kumoh National Institute of Technology, South Korea), and Goetz Pfander (Mathematical Institute of Data Science, Ingolstadt). |
« back