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 of 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.