I am aware of how to compute the cubic spline of a one-variable function with mm control points using a linear equation . However, I am not sure how to do any type of spline on a two-variable function with m knots. Is this even possible?
Consider bi cubic spline B(x,y) = S(x). S(y) where S(x) and S(y) are also cubic spline. May see our papers on bi cubic splines.
If you want to fit a less restricted surface that readily extends to higher dimensions, take a look at Gaussian Processes.
Try key words, please:
multidimensional box-splines
products of B-splines
The following sources may lead you (though, not directly) to the algorithmsyou need
a book:
Bases in Function Spaces, Sampling, Discrepancy, Numerical Integration, by Hans Triebel
a review
https://pdfs.semanticscholar.org/e9d3/1e439cfe3430d3388b09ddf11c526c3453c2.pdf, by C. de Boor
a paper
Tensor products of Sobolev–Besov spaces and applications to approximation from the hyperbolic cross, December 2009, Journal of Approximation Theory 161(2):748-786, DOI: 10.1016/j.jat.2009.01.001 by Winfried Sickel &Tino Ullrich
a paper by G. G. Lorentz & Rudolph A. Lorentz
Article Bivariate Hermite Interpolation and Applications to Algebraic Geometry
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