ABSTRACT
Because of the role of Gaussian functions in the Gaussian test for equality (theorem 29.1 on page 479), it is natural to refer to Gaussians as “test functions” for determining the equality of two formulas. Other functions can be used as test functions (see corollary 26.3 on page 418), but, because Fourier transforms of Gaussians are also Gaussian functions, Gaussians were more convenient for our work in chapter 29. On the other hand, derivatives and linear combinations of Gaussian functions are seldom purely Gaussian. That will make using just Gaussians as our test functions somewhat awkward in the next few chapters. So, in this chapter, we will use Gaussians to generate a larger collection of “test functions” (called, appropriately enough, the “Gaussian test functions”), and we will verify that this expanded set satisfies a number of properties that will be important in our development of the generalized theory.
