We give explicit, polynomial–time computable formulas for the number of integer points in any two– dimensional rational polygon. A rational polygon is one whose vertices have rational coordinates. We find that the basic building blocks of our formulas are Dedekind–Rademacher sums, which are polynomial–time computable finite Fourier series. As a by–product we rederive a reciprocity law for these...

We give explicit, polynomial–time computable formulas for the number of integer points in any two– dimensional rational polygon. A rational polygon is one whose vertices have rational coordinates. We find that the basic building blocks of our formulas are Dedekind–Rademacher sums, which are polynomial–time computable finite Fourier series. As a by–product we rederive a reciprocity law for these...

We obtain upper bounds for the fourth and higher moments of short exponential sums involving Fourier coefficients holomorphic cusp forms twisted by rational additive twists with small denominators. conjectured best possible bound in case moment.

Spectral reprojection techniques make possible the recovery of exponential accuracy from the partial Fourier sum of a piecewise-analytic function, essentially conquering the Gibbs phenomenon for this class of functions. This paper extends this result to non-harmonic partial sums, proving that spectral reprojection can reduce the Gibbs phenomenon in nonharmonic reconstruction as well as remove r...

To defeat Gibbs phenomenon in Fourier and Chebyshev series, Gottlieb et al. [D. Gottlieb, C.-W. Shu, A. Solomonoff, H. Vandeven, On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. Comput. Appl. Math. 43 (1992) 81–98] developed a ‘‘Gegenbauer reconstruction’’. The partial sums of the Fourier or other spectral series are ...