The paper studies Sard’s problem on construction of optimal quadrature formulas in the space W ^(m,0)₂ by Sobolev’s method. This problem consists of two parts: first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas. Here the norm of the error functional is calculated with the help of the extremal function. Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space W ^(m,0)₂ is obtained, moreover the existence and uniqueness of the solution of this system are discussed. Next, the discrete analogue D_m(hβ) of the differential operator \({{{d^{2m}}} \over {d{x^{2m}}}} - 1\) is constructed. Further, Sobolev’s method of construction of optimal quadrature formulas in the space W ^(m,0)₂ , which based on the discrete analogue D_m(hβ), is described. Next, for m = 1 and m = 3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained. Finally, at the end of the paper the rate of convergence of the optimal quadrature formulas in the space W ^(3,0)₂ for the cases m = 1 and m = 3 are presented.

论文用Sobolev方法研究了Sard在空间W ^{( m ,0)} _{2 中}构造最优求积公式的问题。这个问题由两部分组成：首先计算误差泛函的范数，然后通过求积公式的系数找到这个范数的最小值。这里误差函数的范数是在极值函数的帮助下计算的。然后利用拉格朗日乘子法得到了空间W ^{( m ,0)} _{2 中}最优求积公式系数的线性方程组，并讨论了该系统解的存在唯一性。接下来，离散模拟D _m ( hβ ) 的微分算子\({{{d^{2m}}} \over {d{x^{2m}}}} - 1\)被构造。此外，还描述了 Sobolev 在空间W ^{( m ,0)} ₂中构造最优求积公式的方法，该方法基于离散模拟D _m ( hβ )。接下来，对于m = 1 和m = 3，获得精确到指数三角函数的最佳正交公式。最后，在论文的最后，对于m = 1 和m的情况，最优求积公式在空间W ^(3,0) _{2 中}的收敛速度 = 3 呈现。