Dear Roshan M. Regy,

In one dimension the representation of your function is

f(x) = ∑ci Hi(x) (Eq 1)

where Hi(x) is the i'th element of an orthogonal polynomial (eg. Hermite, Legendre). We have to use the orthogonal property of the polynomial:

∫ Hi(x)·Hj(x)·w(x) dx = δij (Eq 2)

where w(x) is the corresponding weight for the applied orthogonal polynomial and δij is the Kronecker delta. Use this property for Eq 1! Multiply each side of of Eq 1. with Hj(x)·w(x) and integrate over x:

∫ f(x)·Hj(x)·w(x) dx = ∫ ∑ci Hi(x)·Hj(x)·w(x) dx (Eq 3)

In the right side of Eq 3. we can change the order of integration and the summation (due to the linear nature of the problem):

∫ f(x)·Hj(x)·w(x) dx =∑ci ∫ Hi(x)·Hj(x)·w(x) dx (Eq 4)

The following is straightforward From Eq 2. (only one term won't be vanished in the right hand side of Eq 4.):

∫ f(x)·Hj(x)·w(x) dx = cj (Eq 5)

This is the way how the expansion coefficient can be calculated. Youd need to multiply your function with the orthogonal polynomial and integrate it.

In bigger dimensions you can use a direct-product of orthogonal polynomials to represent the multivariate function.

Peter has succintly described the procedure how one might approximate a function using orthogonal polynomials. The attached notes show how this might be done in Mathematica for two cases (i) given a function; (ii) given a set of data. The PDF of the Mathematica notebook is attached. Enjoy

Polynomial approximation for discrete data basically does not differ from any other nonlinear regression. However, additionally orthogonality is useful in cases when there are no preliminary models of line/surface to be found. However then the discrete data should be first replaced by continuous function (the simplest way is a piecewise linear (broken) line). After such preliminary preparation of the data, we are trying to simplify the function by replacing it by a polynomial of not very high degree; maily with the aim to guess a dependence on a rather low number of parameters covering relatively large class of phenomena of our interests.

Thus, if we are ready to get the final result by some step by step method of better and better approximation, the advantage of orthogonality can be roughly described as follows:

Assume, we are trying the best fitting by a polynomial of degree 8 and it is seen that it gives not fully satisfactory result. Then we can improve the result it by adding next coefficients for the polynomials of degree 9, 10, 11 and 12 (all given by formulas presented by @Peter). All this should be somehow completed by a system of indicators of good fitting.

Above given qualitative remarks require more detailed algorithms, some of them are already presented by @Brian. But the full system depends on the particular goal of the approximation. In many cases orthogonal polynomials are even meaningless, e.g. for estimating probability density functions. In the latter case spline approximation seemingly is much better, since the resulting functions can be positive (very rarely satisfied by polynomials obtained by orthogonal projections).

Regards, Joachim

There are some publications from my colleagues and a Matlab implementation for fitting data using discrete orthogonal polynomials:

https://de.mathworks.com/matlabcentral/fileexchange/41250-discrete-orthogonal-polynomial-toolbox-dopbox-version-1-8