Approximation on the Voronoi Cells of the Ad Lattice
We study several approximation problems on the Voronoi cells of the Ad lattice. The study depends on discretization of the Voronoi cells induced by a zonotopal algebra and finite quotient groups. The latter provides a natural tool for doing discrete Fourier analysis on the Voronoi cells. The interactions between the groups and their dual groups yield interpolation and quadrature formulas on the Voronoi cells. The zonotope structure allows us to investigate the approximation property of several types of kernels (including the Dirichlet type and the Fejér type) on the Voronoi cells. The convolution operator given by the Fejér type kernel is positive, and consequently an approximate identity. The Fejér type kernel can also be realized as a summability method. As an interesting comparison to the classical (C,1) summability which assigns equal weight to the partial sums of a Fourier series, the Fejér type summability method on the Voronoi cells assigns equal weights to about half of the partial sums and algebraically decaying weights to the rest.
Approximation, Dirichlet kernel, Dual groups, Fejér kernel, Fourier series, Interpolation, Lattice, Quadrature formulas, Voronoi cells, Zonotope
Sun, Xingping. "Approximation on the Voronoi Cells of the A d Lattice." Constructive Approximation 32, no. 3 (2010): 543-567.