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Explicit constants for Fejer-type smoothing on finite cyclic groups

Justin Grieshop

TL;DR

The paper develops a finite-group version of a Fejér-type smoothing kernel on the cyclic group Z/NZ, providing explicit L1 and L2 norms, a complete Fourier symbol, and uniform-in-N bounds. It proves a simple L2→L∞ smoothing estimate, leading to a quantitative smoothed discrepancy bound for mean-zero functions, all with fully elementary, self-contained arguments. The key contributions are the explicit kernel Fr, its nonnegative Fourier multiplier, and a concrete bound that scales as r^{-1/2} N^{1/2}. These results offer a ready-to-use reference for discrete harmonic analysis on finite groups and have potential applications in discrepancy theory and analytic number theory, with natural extensions to higher dimensions and sieve methods.

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.

Explicit constants for Fejer-type smoothing on finite cyclic groups

TL;DR

The paper develops a finite-group version of a Fejér-type smoothing kernel on the cyclic group Z/NZ, providing explicit L1 and L2 norms, a complete Fourier symbol, and uniform-in-N bounds. It proves a simple L2→L∞ smoothing estimate, leading to a quantitative smoothed discrepancy bound for mean-zero functions, all with fully elementary, self-contained arguments. The key contributions are the explicit kernel Fr, its nonnegative Fourier multiplier, and a concrete bound that scales as r^{-1/2} N^{1/2}. These results offer a ready-to-use reference for discrete harmonic analysis on finite groups and have potential applications in discrepancy theory and analytic number theory, with natural extensions to higher dimensions and sieve methods.

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.

Paper Structure

This paper contains 8 sections, 6 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Notation and basic facts
  3. Convolution and the discrete Fourier transform
  4. A triangular Fejér-type kernel on $\mathbb{Z}/N\mathbb{Z}$
  5. Fourier symbol and positivity
  6. An $L^2\to L^\infty$ bound and smoothed discrepancy
  7. A small numerical illustration
  8. Concluding remarks

Key Result

Lemma 3.2

Let $N\ge2$ and $1\le r\le \lfloor N/2\rfloor$. Then $F_r:\mathbb{Z}/N\mathbb{Z}\to\mathbb{R}_{\ge0}$ satisfies:

Theorems & Definitions (18)

  • Definition 2.1: DFT on $\mathbb{Z}/N\mathbb{Z}$
  • Definition 3.1: Triangular kernel on $\mathbb{Z}/N\mathbb{Z}$
  • Lemma 3.2: Basic properties of $F_r$
  • proof
  • Lemma 3.3: $L^2$ norm of $F_r$
  • proof
  • Remark 3.4: Asymptotic behavior
  • Lemma 4.1: Autocorrelation representation
  • proof
  • Proposition 4.2: Fourier symbol and bounds
  • ...and 8 more