Moments and Fourier transforms of B-splines

The problem o f determining the moments and the Fourier transforms o f B-splines with arbitrary knots is considered. There exists a simple connect ion between the moments o f such splines and the so-called extended Stirling numbers o f the second kind which are defined in section 2. Some recurrence relations for the moments o f B-splines with arbitrary knots are given in section 3. In the case of equidistant knots we have also further recurrences. For the forward, central and perfect B-splines the explicit formulas for the moments are given in section 3. The Fourier transforms o f B-splines is treated in section 4. The final section is devoted to so-called Stieltjes series nonnegative weight function w(x) and such that f b w(x) dx > 0 in connected w i t h the s o m e closed interval [a, b]. It is proved that such series for the particular values o f the independent variable may be expressed by the finite sums which contain the nodes and coefficients o f the optimal (in the Davies sense) quadrature formulas.