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Starting from an approximation on an inhomogeneous mesh, the idea of splines is to provide a global interpolation that is continuous up to a certain derivative. For example, take a cubic Hermite spline with the 4 parameters , which completely determine all 4 coefficients of the cubic polynomial
It is straight forward to calculate the first and second derives
Starting with only sampled values and no information about the curvature, are usually calculated so as to guarantee a smooth interpolation. Most common is to require that be continuous from one interval to the next, leading to a tridiagonal system that holds for
and can be solved efficiently with operations. Two free parameters at the boundaries are used to impose the boundary conditions by choosing and (1.4.4#eq.2). 1.4.4#fig.1) illustrates the procedure and shows the excellent quality of a cubic approximation until it breaks down at the limit of 2 mesh points per wavelength.
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