  Approximating a function with a finite series, e.g., involving polynomials or
trigonometric functions, is a critical tool in computing and data analysis. The
construction of such approximations via now-standard approaches like least
squares or compressive sampling does not ensure that the approximation adheres
to certain convex linear structural constraints, such as positivity or
monotonicity. Existing approaches that ensure such structure are
norm-dissipative and this can have a deleterious impact when applying these
approaches, e.g., when numerical solving partial differential equations. We
present a new framework that enforces via optimization such structure on
approximations and is simultaneously norm-preserving. This results in a
conceptually simple convex optimization problem on the sphere, but the feasible
set for such problems can be very complex. We establish well-posedness of the
optimization problem through results on spherical convexity and design several
spherical-projection-based algorithms to numerically compute the solution.
Finally, we demonstrate the effectiveness of this approach through several
numerical examples.
