Expansion over a rectangle of real functions in Bernoulli polynomials and applications (Q5952341)

A new expansion of a sufficiently smooth univariate real function in terms of Bernoulli polynomials has been given by \textit{F. Costabile} [Conf. Sem. University Bari, N 273 (1999)]. The authors generalise this expansion for the bivariate case. Using this, some useful kernels are obtained which are substantially different from Sard kernel for a wide class of linear functional that includes the truncated error for cubature formulas.