Construction of optimal quadrature formulas exact for exponentional-trigonometric functions by Sobolev's method
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The paper studies Sard's problem on construction of optimal quadrature formulas in the space W_2^(m,0) 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_2^(m,0) 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/d x^2m-1 is constructed. Further, Sobolev's method of construction of optimal quadrature formulas in the space W_2^(m,0), which based on the discrete analogue D_m(hβ), is described. Finally, for m=1 and m=3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained.
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