Modified Trapezoidal Product Cubature Rules. Definiteness, Monotonicity and a Posteriori Error Estimates

We study two modifications of the trapezoidal product cubature formulae, approximating double integrals over the square domain $[a,b{]}^2=[a,b]\times [a,b]$. Our modified cubature formulae use mixed type data: except evaluations of the integrand on the points forming a uniform grid on $[a,b{]}^2$, they involve two or four univariate integrals. An useful property of these cubature formulae is that they are definite of order $(2,2)$, that is, they provide one-sided approximation to the double integral for real-valued integrands from the class $${\mathcal{C}}^{2,2}[a,b]=\{f(x,y):\frac{{\mathrm{\partial }}^{4}f}{\mathrm{\partial }{x}^2\mathrm{\partial }{y}^2}\text{ continuous and does not change sign in }(a,b{)}^2\}\mathrm{.}$$ For integrands from ${\mathcal{C}}^{2,2}[a,b]$ we prove monotonicity of the remainders and derive a-posteriori error estimates.