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(1-dimensional) Splines are functions defined over contiguous intervals delimited by knots. Between those knots, a spline is a low-degree polynomial function - but not the same function in every interval. Thus a spline can be characterized by its polynomial function coefficients in each segment.

Now, suppose instead of coefficients for a polynomial basis of functions, I choose some arbitrary functions. Again, their linear combination may differ between intervals.

What do I call such functions? That is, is there a commonly-used term for them? If not, would "generalized splines" do?

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People sometimes use splines whose basis functions are trigonometric or exponential functions. And these are still called splines.

Here is a paper by Schoenberg on trigonometric splines.

See here for some early work on exponential "splines under tension".

In computer-aided design, it's very common to use splines whose basis functions are rational functions (quotients of polynomials). CAD folks call these things NURBS (Non-Uniform Rational B-Spline) curves and surfaces.

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