Piecewise: Flexible piecewise functions for fast integral transforms in Julia

Abstract: A piecewise function of a real variable x returns a value computed from a rule that can be different in each interval of the values of x. The Julia module Piecewise provides an implementation of piecewise functions, where the user is free to choose the rules. A mechanism allows for fitting a piecewise function made of user-defined formulas to a real function of a real variable. With appropriately chosen formulas, various integral transforms of the piecewise function become directly available without relying on quadratures. The module Piecewise defines seven formula that enable the fast calculation of the moments of the piecewise function. The module PiecewiseHilbert supplements these formula with methods enabling a fast Hilbert transform. The module PiecewiseLorentz extends some of these formula to enable what we call a Lorentz transform.