TITLE: Non-polynomial splines, a non-linear problem
QUESTION [5 upvotes]: 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.

REPLY [2 votes]: The interpolant
$$A\left(e^{\frac{a}{A}\xi}-1\right)+B\cdot\left(\cosh\left(\frac{a}{A}\xi\right)-1\right)\ =\  (A+B)\mathbf{e^{\frac{a}{A}\xi}}+B\cdot \mathbf{e^{-\frac{a}{A}\xi}}-(A+B)\\ 
a=\frac{d}{dx}S(x_i),\\ 
\xi=x_{i+1}-x_i,\\ 
\eta=y_{i+1}-y_i,\\ 
B=\frac{\eta-A\left(e^{\frac{a}{A}\xi}-1\right)}{\cosh\left(\frac{a}{A}\xi\right)-1}$$
The construction of the spline $S(x)$ happens from left to right and requires knowldege of the slope at $x_0$ to be able to calculate the Interpolant that connects $(x_0,y_0)$ with $(x_1,y_1)$; knowing the interpolant we can determine the slope at $x_1$ and we are in the same situation as before so that eventually $S(x)$ is determined.
The underlying ideas that led to identifying the interpolant for $C^1$ continuous interpolation can most likely be generalized to higher degrees of smoothness but I'm still on my way.

Addendum:
parameter $A$ provides limited control over the shape of the interpolant:

if we want the "purest" expontial functions, the $A$ should be chosen to minimize $B$
if we strive for preserving shape we can chose $A$ to control the abscissa of the (unique) local extremum in cases of $y_i\lt y_{i+1}\land y_{i+2}\lt y_{i+1}$

align it with the vertex of the parabola through $\left(\,(x_i,y_i),\,(x_{i+1},y_{i+1}),\,(x_{i+2},y_{i+2})\,\right)$
align it with the abscissa of the intersection of the line with slope $a$ that contains $(x_i,y_i)$ and the line through $(x_{i+1},y_{i+1})$ and $(x_{i+2},y_{i+2})$