Generally, a Gaussian quadrature of degree $n$ over an interval $I$ is defined so that it integrates exactly polynomials up to degree $2n - 1$. The main tool are the orthogonal polynomials.

When $I$ is unbounded, the approximation of a continuous function $f$ over $I$ by polynomials could be bad. It seems that the vector space of polynomials where the integration over $I$ is exact could be changed to another vector space, so that the approximation over $I$ of $f$ by another family of functions is better than this with polynomials.

For example, the space $\mathbb{R} \left [\dfrac{1}{X} \right ]$ could be useful to develop quadratures over $[1 ; + \infty[$, i.e. a quadrature rule $$ \int _1 ^{+ \infty} \omega(x) f(x) \ dx \approx \sum _{i = 1} ^n \omega_i f(x_i) $$ such that $$ \int _1 ^{+ \infty} \dfrac{\omega(x)}{x^k} \ dx = \sum _{i = 1} ^n \dfrac{\omega_i}{x_i^k} $$ for all $k \in \{0 ; \cdots ; 2n - 1\}$, where $\omega : I \longrightarrow \mathbb{R}^+$ is a nonnegative continuous function, integrable over $[1 ; +\infty[$.

Another example could be to use trigonometric polynomials instead of polynomials.

Does someone know some references using such a change of point of view?

newquadrature rule. $\endgroup$ – MathTolliob Jan 19 at 23:13