Dyadic splines are a simple and efficient function representation that supports multiresolution design and analysis. These splines are defined as limits of a process that alternately doubles and perturbs a sequence of points, using B-spline subdivision to smoothly perform the doubling. An interval-query algorithm is presented that efficiently and flexibly evaluates a limit function for points and intervals. Methods are given for fitting these functions to input data, and for minimizing the energy and redundancy of the representation. Several methods are given for designing dyadic splines by controlling the perturbations of the limit process. Several applications are explored, including shape design, synthesis of terrain and other natural forms, and compression.