  Summation-by-parts (SBP) operators are popular building blocks for
systematically developing stable and high-order accurate numerical methods for
time-dependent differential equations. The main idea behind existing SBP
operators is that the solution is assumed to be well approximated by
polynomials up to a certain degree, and the SBP operator should therefore be
exact for them. However, polynomials might not provide the best approximation
for some problems, and other approximation spaces may be more appropriate. In
this paper, a theory for SBP operators based on general function spaces is
developed. We demonstrate that most of the established results for
polynomial-based SBP operators carry over to this general class of SBP
operators. Our findings imply that the concept of SBP operators can be applied
to a significantly larger class of methods than currently known. We exemplify
the general theory by considering trigonometric, exponential, and radial basis
functions.
