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 $$ \mathcal{C}^{2,2}[a,b]=\{f(x,y)\,:\,\frac{\partial^4 f}{\partial x^2\partial y^2}\ \text{continuous and does not change sign in}\ (a,b)^2\}. $$ For integrands from $\mathcal{C}^{2,2}[a,b]$ we prove monotonicity of the remainders and derive a-posteriori error estimates.

翻译：我们研究了两种修正的梯形乘积求积公式，用于近似方形域 $[a,b]^2=[a,b]\times [a,b]$ 上的二重积分。所提出的修正求积公式采用混合类型数据：除了在 $[a,b]^2$ 上均匀网格点处的被积函数值外，还涉及两个或四个单变量积分。这些求积公式的一个重要性质是具有 $(2,2)$ 阶正定性，即对于属于类 $$ \mathcal{C}^{2,2}[a,b]=\{f(x,y)\,:\,\frac{\partial^4 f}{\partial x^2\partial y^2}\ \text{在}\ (a,b)^2\ \text{上连续且不变号}\} $$ 的实值被积函数，它们能提供二重积分的单侧逼近。对于 $\mathcal{C}^{2,2}[a,b]$ 中的被积函数，我们证明了余项的单调整性，并推导了后验误差估计。