# Changing centers in Fourier windows

The formal chapter proves the exact affine change of Fourier parameters, stability of proper PSH limits, transport of collapsed limits, indicator invariance, and a uniform bound throughout expanding logarithmic frequency neighborhoods. The learner chapter contains four worked examples and ten fully solved exercises, including a countable-coordinate extension with its precise quantifiers.

The two original illustrations use exact radius formulas, point-mass profiles and real translates of the PSH model log absolute sine. The latter model is not asserted to be a limiting profile of a particular compact distribution. The geometry description records the coordinates, formulas, display clipping and proof locators.

Run the figure program with Python, NumPy and Matplotlib to reproduce the PNG and SVG figures. Run the numerical program with Python, NumPy, SciPy and mpmath for finite supplementary checks of center changes, logarithmic ratios, derivative profiles, volume factors, smooth integration by parts and translated envelopes. The numerical program reports the finite computations. The analytic proofs, rather than the sampled checks, establish the full conclusions.

Original exposition, proofs, examples, solutions, geometric data and code: GPT-6.1 Sol (OpenAI), Ultra; CC0 1.0. The DejaVu font notice retains its own terms. References: Terence Tao, *246B, Notes 2* (2021); Lars Hörmander, *The Analysis of Linear Partial Differential Operators I* and *II* (1983). No source text or source images are included.