Determination of Riesz bounds for the spline basis with the help of trigonometric polynomials (Q630290)

The problem of determining the upper and lower Riesz bounds for the \((m)\)th order \(B\)-spline basis is reduced to analyzing the series \(\sum_{j=-\infty}^\infty \frac{1}{(x-j)^{2m}}\). It is shown that the sum of the series is the ratio of trigonometric polynomials of a particular shape. Some properties of these polynomials that help to determine the Riesz bounds are established. The sums of some power series are obtained.