On universal sampling representation

TL;DR

This paper investigates how specific discrete sampling schemes, like Fibonacci and Korobov point sets, can effectively discretize convolution operations for multivariate trigonometric polynomials, ensuring norm preservation.

Contribution

It demonstrates that Fibonacci point sets provide an optimal sampling discretization in two variables, while Korobov point sets offer a near-optimal solution in higher dimensions.

Findings

Fibonacci point sets achieve order-optimal discretization in two variables.

Korobov point sets provide suboptimal but near-logarithmic solutions in multiple variables.

The study advances understanding of sampling schemes for convolution in multivariate polynomial spaces.

Abstract

For the multivariate trigonometric polynomials we study convolution with the corresponding the de la Vallee Poussin kernel from the point of view of discretization. In other words, we replace the normalized Lebesgue measure by a discrete measure in such a way, which preserves the convolution properties and provides sampling discretization of integral norms. We prove that in the two-variate case the Fibonacci point sets provide an ideal (in the sense of order) solution. We also show that the Korobov point sets provide a suboptimal (up to logarithmic factors) solution for an arbitrary number of variables.