I want to understand why, according to several sources, Gauss's discovery of Gaussian quadrature in his 1814 article was "the most significant event of the 19th century in the field of numerical integration and perhaps in all of numerical analysis" (this quotation is taken from the article "Construction and applications of Gaussian quadratures with nonclassical and exotic weight functions").The level of my knowledge at numerical analysis is very low, and the most advanced method that i know is Newton-Cotes method for approximate integration by using an $n-1$ degree interpolating polynomial (where $n$ is the number of sample points on the function).
According to several sources, Gauss proved that for an unknown function $f(x)$, it's better (in the sense of the size of remainder) to integrate the function in the interval $(-1,+1)$ by using a weighted sum of it's values at abscisas which are roots of Legendre-polynomials, rather than uniformly spaced abscisaas. Gauss originally proved this by using his method of continued fractions derived from hypergeometric series, and later (1826) Jacobi gave an elegant interpretation using orthogonal polynomials.
Therefore my questions are:
- Did Gauss prove that for arbitrary function, the best way to numerically integrate it (i.e the difference between the area under the function and the weighted sum will be minimal) is to use Gauss-Legendre quadrature? obviously for some functions this will be a better method while for others this will be less-suitable, so if this statement is true, than it must be true "in average". This seems to me like a very general functional statement, and a very difficult theorem to prove... so do i understand Gauss's statement correctly?
- If the answer to the previous question is yes, than obviously i understand the historic significance of Gauss's article on numerical integration. If the answer is no, than i'd also like to know what was so significant in Gauss's article?