On constructing an approximation of some arbitrary function, and measuring the badness thereof.

THIS IS CHAOS RIGHT NOW. I need to break out the sampling/interpolation problem for regular data, for one thing.

## Choosing the best approximation

In what sense? Most compact? Most easy to code?

If we are not interpolating, how much smoothing do we do?

We can use cross-validation, especially so-called βgeneralizedβ cross validation, to choose smoothing parameter this efficiently, in some sense.

Or you might have noisy data,
in which case you now have a function approximation *and* inference problem,
with error due to both approximation
and sampling complexity.
Compressive sensing
has some finite-sample guarantees.

To discuss: loss functions.

An interesting problem is how you align the curves that are your objects of study; That is a problem of warping.

## Polynomial spline smoothing of observations

The classic, and not just for functional data, but filed here because thatβs where the action is now.

Special superpowers: Easy to differentiate and integrate.

Special weakness: many free parameters, not easy to do in high dimension.

## Polynomial bases

See polynomial bases.

## Fourier bases

ποΈ

## Radial basis function approximation

I# Radial basis function approximation I actually care about this mostly for densities, so see mixture models, for what information I do have.

## Rational approximation

PadΓ©βs is the method Iβve heard of. Are there others? Easy to differentiate. OK to integrate if you cheat using a computational algebra package.

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