We propose an algorithm for construction of quadrature formulas for linear operators acting on periodic functions. For analytic functions, the order of accuracy of quadrature formulas grows unboundedly with the growth of the number of the grid nodes. Within rather general restrictions on the kernels of linear operators, an exponential estimate of a quadrature formula is proved. To exemplify, we find quadrature formulas for the calculation of integral operators with logarithmic singularities which are used in the boundary element method to obtain overconvergent numerical schemes for solution of boundary value problems for harmonic and biharmonic equations on the plane.

我们提出了一种算法，用于构造作用于周期函数的线性算子的正交公式。对于解析函数，正交公式的精度顺序随网格节点数量的增加而无限增加。在线性算子的核的相当一般的限制内，证明了正交公式的指数估计。为了举例说明，我们找到了用于计算具有对数奇异性的积分算子的正交公式，这些公式用于边界元方法中，从而获得了用于求解平面上的调和方程和双调和方程的边值问题的超收敛数值方案。