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I'm trying to approximate a function using a set of piecewise polynomials. For example, perhaps I'd like to uniformly split the domain [-1,+1] 20 times and place Wendland RBF, Gaussian, or maybe a B-Spline in each section and then determine coefficients for the polynomials to best approximate my sample data on the domain.

Normally, to approximate scattered data with an RBF of local support you'd use a Least Squares Approximation, however, this requires you know all your data upfront and this could be quite memory intensive to store all the scattered data points in a matrix and then solve.

Is there a way to project directly onto the polynomials using Monte Carlo? I know if you are using an orthogonal basis you can directly project your sample points but I'm not quite sure how to do it if your basis functions are non-orthogonal.

Thank you

coderdave
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I don't know what you mean by "projection using Monte Carlo". But, to approximate a given function by a polynomial, I would recommend interpolation at Chebyshev nodes. To calculate the interpolant, you can just use Lagrange interpolation formulae -- you don't need to solve a linear system of equations. See these notes or this paper for advice about the right way to do the computations.

bubba
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