Improvement of Fourier Series Convergence on the Basis of Splines and Its Application for Numerical Inversion of Laplaсe Transform

The method of approximation of functions by piecewise continuous polynomials of second degree by means of least squares method is proposed. At that, the finding of functions in the nodal points is reduced to solving the system of linear algebraic equations. The developed approach is used for functions given by Fourier series for which this system is solved in closed form. Thus, the formula for finding the functions in nodal points through modified Fourier series is obtained. There is illustrated the effectiveness of proposed formulas for numerically-analytical finding the original based on an improved approach of Prudnikov, which in general is reduced to calculation of the slowly convergent Fourier series.