I'm looking for references on how to construct spline-like functions from a basis that does not include piecewise polynomials.

To be specific, given a class of functions such as "decaying exponentials" or "sines and cosines" (which are parameterized by a single parameter, e.g. the decay rate or the frequency), is there an efficient and numerically stable method to construct a function that is piecewise a linear combination of $N$ such functions (whose parameters are to be determined), interpolates given data $(x_k,f_k)$ [which are assumed to be such that they can be interpolated using the given function class, i.e. e.g. monotonically decreasing for decaying exponentials] and has continuous derivatives at $x_k$ up to order $2N-2$ (in order to fix $N$ linear coefficients and $N$ non-linear parameters)?

I can of course write down the explicit equations needed to satisfy these conditions, but directly solving those using a non-linear equation system solver does not look too promising as an approach.

What literature I could find so far on splines with non-polynomial components referred to spaces spanned by polynomials and some given non-polynomial. Here I'm looking for the case where there are no polynomials and the non-polynomial functions are parameterized by a parameter whose values are to be determined by the interpolation and smoothness conditions.