Hi mathematics7;

You want a basis using Hermite polynomials but there are at least two types of them ( maybe more, I do not know.) Which one do you want?

Also, as far as I know they are used for only one type of integral because they are orthogonal to the exponential function in it.

For this they use the physicists version. You seem to think that they can be used for the general f(x). Why do you think that?

The error term is given as:

where n is the number of points and epsilon is some point in the interval [a,b] that maximizes the error. There is also another type of error estimate that is similar to the Euler Mclaurin Summation Formula.

If you want a specific example I can run of the weights and abscissas ( zeros ).