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

The problem of determining the upper and lower Riesz bounds for the mth order $B$-spline basis is reduced to analyzing the series $\sum_{j=-\infty}^\infty\frac1{(x-j)^{2m}}$. The sum of the series is shown to be a ratio of trigonometric polynomials of a particular shape. Some properties of these polynomials that help to determine the Riesz bounds are established. The results are applied in the theory of series to find the sums of some power series.