Expansions over a simplex of real functions by means of Bernoulli polynomials

In [1] there is an expansion in Bernoulli polynomials for sufficiently smooth real functions in an interval $[a, b]\subset {\mathbb R}$ that has useful applications to numerical analysis. An analogous result in a 2-dimensional context is derived in [2] in the case of rectangle. In this note we generalize the above-mentioned one-dimensional expansion to the case of $C^m$-functions on a 2-dimensional simplex; a method to generalize the expansion on an $N$-dimensional simplex is also discussed. This new expansion is applied to find new cubature formulas for 2-dimensional simplex.