Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fourier decay of absolutely and Hölder continuous functions with infinitely or finitely many oscillations

Juhani Nissilä

Abstract

The main result of this paper is, that if we suppose that a function is absolutely continuous and uniformly Hölder continuous and that its finite difference function does not oscillate infinitely often on a bounded interval, then the decay rate of its Fourier coefficients can be estimated exactly. This rate of decay predicts the same uniform Hölder continuity but the two other conditions are not necessary. Several examples from literature and by the author show that none of the assumptions can be relaxed without weakening the decay for some functions. The uniform Hölder continuity of chirps and the decay of their Fourier coefficients are studied. The main result is then applied in the estimation of the error of numerical Weyl fractional derivatives calculated using the discrete Fourier transform. The main result is also extended to Fourier transforms.

Fourier decay of absolutely and Hölder continuous functions with infinitely or finitely many oscillations

Abstract

The main result of this paper is, that if we suppose that a function is absolutely continuous and uniformly Hölder continuous and that its finite difference function does not oscillate infinitely often on a bounded interval, then the decay rate of its Fourier coefficients can be estimated exactly. This rate of decay predicts the same uniform Hölder continuity but the two other conditions are not necessary. Several examples from literature and by the author show that none of the assumptions can be relaxed without weakening the decay for some functions. The uniform Hölder continuity of chirps and the decay of their Fourier coefficients are studied. The main result is then applied in the estimation of the error of numerical Weyl fractional derivatives calculated using the discrete Fourier transform. The main result is also extended to Fourier transforms.

Paper Structure

This paper contains 7 sections, 31 theorems, 99 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Definitions
  3. Known results on smoothness and Fourier decay
  4. Proof of the main result
  5. Chirps
  6. Application to error analysis of numerical Weyl fractional derivatives
  7. Extension to Fourier transforms

Key Result

Theorem 1.1

For some $m \in \mathbb{N}_0$ and $\mu \in (0,1]$, suppose that $f^{(m)}$ is absolutely continuous, $T$-periodic, $f \in C_{m,\,\mu}[0,T]$, and the number of local maxima and minima of $\Delta_h f^{(m)}$ is uniformly bounded for every $0 < h \leq h_0$. Then the Fourier coefficients of $f$ decay like

Figures (5)

  • Figure 1: Function $|x|^{0.7}$ plotted on the interval $[-1,1]$ with $2\cdot 10^5$ samples
  • Figure 2: Absolute values of the DFT of the samples of $|x|^{0.7}$ from figure \ref{['fig:alpha07']} from $k = 0$ to $10^5-1$ on a log-log scale
  • Figure 3: Function $|x|^{0.7}\sin\left(1/|x|^{0.5}\right)$ on the interval $[-1,1]$ plotted with $2\cdot 10^5$ samples
  • Figure 4: Absolute values of the DFT of the samples of $|x|^{0.7}\sin\left(1/|x|^{0.5}\right)$ in figure \ref{['fig:infinitely_oscillating_alpha07_beta05']} from $k = 0$ to $10^5-1$ on a log-log scale
  • Figure :

Theorems & Definitions (90)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 80 more