10  Approximation

• Inputs: function values, sampling oracle (values, derivs, integrals, etc)

• Output: Approximating function, often used for other tasks. Plus some notion of error!

• Q to ask: what data is available? Noise? What is assumed about fn (smoothness, derivatives, etc)?

• What is needed from approximator (derivatives?). How fast is fitting? Evaluation? Convergence?

• Discuss general theory and influence of noise later. Focus here on piecewise polynomials and interpolation thereof.

• Polynomial interpolation and regression

• Different forms of the interpolation problem

• Choice of basis and numerical issues, computational speed

• Error analysis

• Piecewise polynomials: splines and more

• Kernel-based approximation

• Which space to use?

• Rational approximation

• Approximation of densities