Irregularities of distribution and average decay of fourier transforms

In Geometric Discrepancy we usually test a distribution of N points against a suitable family of sets. If this family consists of dilated, translated and rotated copies of a given d-dimensional convex body D subset of [0, 1)(d), then a result proved by W. Schmidt, J. Beck and H. Montgomery shows that the corresponding L-2 discrepancy cannot be smaller than c(d)N((d-1)/2d). Moreover, this estimate is sharp, thanks to results of D. Kendall, J. Beck and W. Chen. Both lower and upper bounds are consequences of estimates of the decay of parallel to(chi) over cap (D) (rho.)parallel to(L2(Sigma d-1)) for large rho, where (chi) over cap (D) is the Fourier transform (expressed in polar coordinates) of the characteristic function of the convex body D, while Sigma(d-1) is the unit sphere in R-d. In this chapter we provide the Fourier analytic background and we carefully investigate the relation between the L-2 discrepancy and the estimates of parallel to(chi) over cap (D) (rho.)parallel to(L2(Sigma d-1)).