# Fourier endpoints and the asymptotic density of zeros

Read [the complete lesson](../../AN02-L137.html), or use [the editable source](../../src/fourier-endpoints-and-asymptotic-zeros.md). It includes the full argument, four worked examples and six exercises with complete solutions.

The first original illustration shows how a two-atom measure's scaled zero lattice approaches length on the real axis. The second compares its zero counts, the multiplicities of its third convolution power, and a zero-free point measure. Exact formulas, parameters, Euclidean metrics, sampling conventions and proof locators are in [figures/geometry.json](figures/geometry.json). Each figure is provided as PNG, SVG and a one-page vector PDF.

Use Python with NumPy, ReportLab and pypdfium2, then run `python figures/render_figures.py` from this directory. The program writes the two figures and geometry into its own figures directory. It reads no book pages. Floating-point sampling illustrates the exact formulas; the mathematical limits are proved in the lesson.

Lars Hörmander, The Analysis of Linear Partial Differential Operators II (1983; second revised printing 1990; reprint 2005), Theorem 16.1.9, printed page 313, supplies the mathematical target. The exposition, examples, solutions and figures are original. The lesson names its exact internal compact-support, scalar analytic and dilation proof inputs, including the latter's representation input.

The original material uses [CC0 1.0](LICENSE.txt). Referenced software, fonts and course components retain their own terms.