On the Limiting Profile of Exponentially Decaying Functions

It is shown that, under certain conditions, orthonormalising the positive integer shifts of an exponentially decaying function on the half line by the Gram Schmidt process leads to a limiting profile given by orthonormalising all their integer shifts on the whole line. These results derive from properties of Cholesky factorization of bi-infinite and semi-infinite matrices. An example is provided by the negative exponential function and conjectures are given, supported by numerical evidence, for the Gaussian and inverse multiquadric.