Window functions are among the most expressive features of modern SQL. Surprisingly, relatively little has been written about their optimization. Some techniques exist, such as pushing predicates through a window under ideal conditions, but known optimizations no longer apply when those conditions are even slightly unmet. We show that these limitations are not fundamental, but persist because a reasoning framework for window function optimization has been missing. We provide such a framework, introducing techniques we call Frame Analysis, Partition Analysis, and a new execution strategy called Co-Evaluation. These clarify when and how optimizations can be applied. Co-Evaluation, in particular, allows early evaluation of predicates even when they depend on the window function’s result. We present each technique and organize the re- sults as a table of algebraic equivalences for window functions. We test these optimizations in an open-source engine, where they never hurt performance and make certain common queries up to 40.7× faster, with larger tables yielding larger gains.