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Worst-case $L_p$-approximation of periodic functions using median lattice algorithms

Zexin Pan, Mou Cai, Josef Dick, Takashi Goda, Peter Kritzer

Abstract

We study the worst-case approximation of multivariate periodic functions from the weighted Korobov space $H_{d,α,γ}$ with smoothness $α>1/2$ in the Lebesgue norm $L_p([0,1]^d)$ for $1\le p\le\infty$. We analyze a \emph{median lattice algorithm} that reconstructs a truncated Fourier series by approximating the coefficients on a hyperbolic-cross-type index set using $R$ rank-1 lattice sampling rules with independent randomly chosen generating vectors, and then aggregating the resulting coefficient estimators via the componentwise median. For an odd number of repetitions $R>1$ and an odd prime lattice size $N$, we prove high-probability error bounds in both $L_\infty$ and $L_2$. Interpolation then yields the result for all $1 \le p\le\infty$. In particular, with a high probability, the algorithm satisfies \[ \mathrm{err}(H_{d,α,γ},L_p,A)\ \le\ C_{d,α,β,\boldsymbolγ,p}\, N^{- α+ (\frac12 - \frac1p)_+ + β}, \qquad 1 \le p\le\infty,\ β>0, \] where $(x)_+ = \max\{x, 0\}$, $N$ is the number of function evaluations, and the weights $\boldsymbolγ$ and the constant $C_{d,α,β,\boldsymbolγ,p}$ are independent of $N$. For $p=\infty$, $C_{d,α,β,\boldsymbolγ,\infty}$ is dimension-independent under the summability condition $\sum_{j=1}^\infty γ_j^{1/(2α)}<\infty$. These results extend recent analyses of median-based lattice approximation in $L_2$ and complement related multiple-shift lattice approaches, showing that median aggregation yields nearly optimal $L_p$-approximation rates (up to logarithmic factors and an arbitrarily small loss) in weighted Korobov spaces.

Worst-case $L_p$-approximation of periodic functions using median lattice algorithms

Abstract

We study the worst-case approximation of multivariate periodic functions from the weighted Korobov space with smoothness in the Lebesgue norm for . We analyze a \emph{median lattice algorithm} that reconstructs a truncated Fourier series by approximating the coefficients on a hyperbolic-cross-type index set using rank-1 lattice sampling rules with independent randomly chosen generating vectors, and then aggregating the resulting coefficient estimators via the componentwise median. For an odd number of repetitions and an odd prime lattice size , we prove high-probability error bounds in both and . Interpolation then yields the result for all . In particular, with a high probability, the algorithm satisfies where , is the number of function evaluations, and the weights and the constant are independent of . For , is dimension-independent under the summability condition . These results extend recent analyses of median-based lattice approximation in and complement related multiple-shift lattice approaches, showing that median aggregation yields nearly optimal -approximation rates (up to logarithmic factors and an arbitrarily small loss) in weighted Korobov spaces.
Paper Structure (7 sections, 10 theorems, 58 equations, 1 algorithm)

This paper contains 7 sections, 10 theorems, 58 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. The median algorithm
  4. Approximation error bounds
  5. $L_\infty$-approximation
  6. $L_2$-approximation
  7. $L_p$-approximation

Key Result

Theorem 1.1

Let $\alpha>1/2$ and $d \in \mathbb{N}$, and let $\gamma_j \in (0,1]$ for $j \in \{1, \ldots, d\}$. Let $A$ be the approximation algorithm given by Algorithm alg:median with a parameter $\tau>0$, an odd number $R>1$, and a prime number $N$. Let $\varepsilon_2$ be given by eqn:failprob2 and assume it for any $\beta > 0$, where $(x)_+ = \max\{x, 0\}$, $C_{d, \alpha, \beta, \boldsymbol{\gamma}, R,\ta

Theorems & Definitions (21)

  • Theorem 1.1
  • Remark 1.2
  • Remark 3.2
  • Remark 3.3
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Corollary 4.3
  • proof
  • Lemma 4.4
  • ...and 11 more