Explicit constants for Fejer-type smoothing on finite cyclic groups

TL;DR

This paper provides explicit constants and bounds for Fejer-type smoothing kernels on finite cyclic groups, enabling precise discrepancy estimates with applications in quantitative group problems.

Contribution

It introduces explicit L1 and L2 norms, Fourier transform calculations, and uniform bounds for Fejer-type kernels on finite cyclic groups, with practical discrepancy estimates.

Findings

Explicit L1 and L2 norms for the kernels

Computed Fourier transforms with uniform bounds

Applied to quantitative discrepancy estimates

Abstract

We study a Fejer-type smoothing kernel on the finite cyclic group Z/NZ. For each smoothing radius we give explicit l1 and l2 norms, compute the discrete Fourier transform, and record bounds that are uniform in N. As an application we prove a smoothed discrepancy estimate with explicit constants that can be used in quantitative problems on finite cyclic groups. The arguments are elementary and the note is intended as a self contained reference.