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

TL;DR

This paper introduces a median lattice algorithm for approximating multivariate periodic functions in weighted Korobov spaces, achieving near-optimal $L_p$-approximation rates with high probability, using randomized rank-1 lattice sampling and median aggregation.

Contribution

The paper develops a novel median lattice algorithm that extends median-based approximation analysis from $L_2$ to all $L_p$ norms, providing high-probability error bounds in a broad function space.

Findings

Achieves high-probability $L_p$ error bounds for the median lattice algorithm.

Provides dimension-independent constants under certain summability conditions.

Extends median-based lattice approximation results from $L_2$ to all $L_p$ norms.

Abstract

We study the worst-case approximation of multivariate periodic functions from the weighted Korobov space $H_{d,\alpha,\gamma}$ with smoothness $\alpha>1/2$ 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 \[ \mathrm{err}(H_{d,\alpha,\gamma},L_p,A)\ \le\ C_{d,\alpha,\beta,\boldsymbol{\gamma},p}\, N^{- \alpha + (\frac12 - \frac1p)_+ + \beta }, \qquad 1 \le p\le\infty,\ \beta>0, \] where $(x)_+ = \max\{x, 0\}$, $N$ is the number of function evaluations, and the weights $\boldsymbol{\gamma}$ and the constant $C_{d,\alpha,\beta,\boldsymbol{\gamma},p}$ are independent of $N$. For $p=\infty$, $C_{d,\alpha,\beta,\boldsymbol{\gamma},\infty}$ is dimension-independent under the summability condition $\sum_{j=1}^\infty \gamma_j^{1/(2\alpha)}<\infty$. 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.