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Maximally Smooth Functions (MSFs) are a form of constrained functions inwhich there are no inflection points or zero crossings in high orderderivatives. Consequently, they have applications to signal recovery inexperiments where signals of interest are expected to be non-smooth featuresmasked by larger smooth signals or foregrounds. They can also act as a powerfultool for diagnosing the presence of systematics. The constrained nature of MSFsmakes fitting these functions a non-trivial task. We introduce maxsmooth, anopen source package that uses quadratic programming to rapidly fit MSFs. Wedemonstrate the efficiency and reliability of maxsmooth by comparison tocommonly used fitting routines and show that we can reduce the fitting time byapproximately two orders of magnitude. We introduce and implement withmaxsmooth Partially Smooth Functions, which are useful for describing elementsof non-smooth structure in foregrounds. This work has been motivated by theproblem of foreground modelling in 21-cm cosmology. We discuss applications ofmaxsmooth to 21-cm cosmology and highlight this with examples using data fromthe Experiment to Detect the Global Epoch of Reionization Signature (EDGES) andthe Large-aperture Experiment to Detect the Dark Ages (LEDA) experiments. Wedemonstrate the presence of a sinusoidal systematic in the EDGES data with alog-evidence difference of $86.19\pm0.12$ when compared to a pure foregroundfit. MSFs are applied to data from LEDA for the first time in this paper and weidentify the presence of sinusoidal systematics. maxsmooth is pip installableand available for download at: https://github.com/htjb/maxsmooth

maxsmooth: Rapid Maximally Smooth Function Fitting With Applications in Global 21-cm Cosmology