The way things were in multivariate splines: A personal view

A personal account of the author's encounters with multivariate splines during their early history. 1 Tensor product spline interpolation My first contact with multivariate splines occurred in August 1960. In my first year at the Harvard Graduate School, working as an RA for Garrett Birkhoff, I had not done too well but, nevertheless, had gotten married and so needed a better income than the RAship provided. On the (very kind and most helpful) recommendation of Birkhoff who consulted for the Mathematics Department at General Motors Research in Warren MI, I had been hired in that department in order to be of assistance to Leona Junko, the resident programmer in that department. Birkhoff and Henry L. Garabedian, the head of that department, had developed a scheme for interpolation to data on a rectangular grid meant to mimic cubic spline interpolation; see [BG]. They would use what they called " linearized spline interpolation " and what is now called cubic spline interpolation, along the meshlines in both directions, in order to obtain values of the first derivative in both directions at each meshpoint, and then fill in each rectangle by a C 1 piecewise low-degree harmonic polynomial function that would match the given information, of value and two first-order derivatives, at each corner and, thereby, match the cubic spline interpolants along the mesh-lines. It occurred to me that the same information could be matched by a scheme that would, say, construct the cubic spline interpolants along all the mesh-lines in the x-direction, and then use cubic spline interpolation to the resulting spline coefficients as a function of y to obtain an interpolant that was a cubic spline in x for every value of y, and C 2 rather than just C 1. Of course, one could equally well start with the cubic spline interpolants along all the mesh-lines in the y-direction, and interpolate the resulting spline coefficients as a function of x and so obtain an interpolant that was a cubic spline in y for every x, and it took me some effort to convince Birkhoff that these two interpolants are the same. This is now known as bicubic spline interpolation [dB0], the tensor-product (I learned that term from Don Thomas there) of univariate cubic spline interpolation, and has become a mainstay