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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.

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    $\begingroup$ one strategy that comes to mind is to proceed analogous to the B-Spline technique, i.e. construct basis functions of minimal support with the smoothest possible transition to $f(x)\equiv 0$ and attempt to meet the goal of making the collection of basis functions a partition of unity. Then solvong the interpolation task amounts to evaluating the basis functions at the knots and solving a system of linear equations. $\endgroup$ – Manfred Weis Feb 15 at 8:51
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    $\begingroup$ here is a download link describing exponential splines google.com/… $\endgroup$ – Manfred Weis Feb 15 at 12:59
  • $\begingroup$ Thanks for the comment and the link. I was aware of the McCartin paper, which to my understanding deals with spline interpolation in $\mathrm{span}\{1,x,e^{px},e^{-px}\}$, which is why I stated that existing work seems to deal with mixed polynomial-exponential bases. The problem I'm asking about is different in that the function space to which the splines piecewise belong doesn't contain polynomials of any degree (except possibly constants), and that the non-linear parameter(s) $p$ are to be determined by the interpolation and continuity conditions. $\endgroup$ – gmvh Feb 17 at 8:43
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    $\begingroup$ Could you provide an exemplary example of what kind of functions you have available and which parameters have to be determined as part of the interpolation task? My assumption was that the set of functions is fixed up to multiplication with a constant, but apparently you need something that goes beyond smoothly piecing a linear combination of given functions. $\endgroup$ – Manfred Weis Feb 17 at 12:12
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    $\begingroup$ Just a remark: if the $1$ and $x$ are in the span, then at least granting continuity of the resulting curve poses no problem; otherwise already that would pose severe difficulties for arbitrary non-polynomial function bases. $\endgroup$ – Manfred Weis Apr 9 at 16:52
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This is not a full answer, but I guess it is more than a comment.

One way to reduce the task is to apply the spirit of VARPRO to separate the linear coefficients and the non-linear parameters, i.e. alternatingly solve $N$ of the conditions as a linear equation system for the coefficients of the solution in a given $N$-dimensional subspace, and solve the remaining $N$ conditions as a non-linear system for $N$ parameters giving a different subspace (in which the coefficients, being kept fixed, are those of the solution to those other $N$ conditions). This reduces the problem size by a factor of two.

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  • $\begingroup$ For the case of $N=2$, there is a somewhat canonical-looking way to split the equations by solving the interpolation conditions as linear equations for the coefficients, and the smoothness conditions as non-linear equations for the parameters, but sadly this does not generalize to $N\ge 3$, where the split between the equations appears to be necessarily arbitrary. $\endgroup$ – gmvh Feb 28 at 12:51

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