Scaled lattice rules for integration on $\mathbb{R}^d$ achieving higher-order convergence with error analysis in terms of orthogonal projections onto periodic spaces

We show that to approximate the integral $\int_{\mathbb{R}^d} f(\boldsymbol{x}) \mathrm{d} \boldsymbol{x}$ one can simply use scaled lattice rules from the unit cube $[0,1]^d$ to properly sized boxes on $\mathbb{R}^d$, achieving higher-order convergence that matches the smoothness of the integrand function $f$ in a certain Sobolev space of dominating mixed smoothness. Our method only assumes that we can evaluate the integrand function $f$ and does not assume a particular density nor the ability to sample from it. In particular, for the analysis we show that the method of adding Bernoulli polynomials to a function to make it ``periodic'' on a box without changing its integral value over the box, is equivalent to an orthogonal projection from a well chosen Sobolev space of dominating mixed smoothness to an associated periodic Sobolev space of the same dominating mixed smoothness, which we call a Korobov space. We note that the Bernoulli polynomial method is often not used because of its computational complexity and also here we completely avoid applying it. Instead, we use it as a theoretical tool in the error analysis of applying a scaled lattice rule to increasing boxes in order to approximate integrals over the $d$-dimensional Euclidean space. Such a method would not work on the unit cube since then the committed error caused by non-periodicity of the integrand would be constant, but for integration on the Euclidean space we can use the certain decay towards zero when the boxes grow. Hence we can bound the truncation error as well as the projection error and show higher-order convergence in applying scaled lattice rules for integration on Euclidean space. We illustrate our theoretical analysis by numerical experiments which confirm our findings.