According to the fluctuationlessness theorem the matrix representation of a function can be approximated by the image of independent variable operator's matrix representation under that function. The independent variable operator's action is defined its the multiplication Of the operand by the independent variable. Hence itself and therefore its matrix representation is universal, do not depend on the function. The application of this approximation to numerical integration forms a quadrature whose Structure can be Manipulated by changing the basis set of an n-dimensional Hilbert space. This work focuses Oil reflecting, the effects of a complementary Hilbert space to it restricted Hilbert subspace by forming the basis set as certain linear combinations of some basis functions in order to improve the accuracy of the numerical integration based on fluctuationlessness theorem.