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This research focuses on the integration and L2-approximation of functions across tensor products of Hilbert spaces, especially in infinite dimensions.

May 10, 2026Open Access

Integration and \ (L²\) -approximation on tensor products of Hilbert spaces of increasing smoothness

Key Points

This research focuses on the integration and L2-approximation of functions across tensor products of Hilbert spaces, especially in infinite dimensions.
Examined Hilbert spaces with reproducing kernels and square-integrable functions under various distributions.
Analyzed the growth of Fourier weights to determine the simplification of univariate problems.
Established a correspondence between Gaussian kernels and Hermite spaces to facilitate error bounds.
Determined polynomial decay of minimal approximation errors linked to Fourier weights.
Showed that integration problems are equivalent between Gaussian and Hermite spaces with specific growth conditions.
Developed algorithms for Gaussian kernel spaces that achieve near-optimal performance in infinite dimensions.

Abstract

We study integration and \ (L²\) -approximation of functions in infinite or finite tensor products of Hilbert spaces; our main interest lies in the infinite-variate case. As univariate building blocks of the tensor product, for \ (j\) we consider Hilbert spaces \ (H (kⱼ) \) with reproducing kernels \ (kⱼ\), whose elements are functions on a domain \ (D\). Additionally we assume that the \ (H (kⱼ) \) are spaces of square-integrable functions with respect to a probability measure \ (₀\). Commonly studied examples are \ (D=0, 1\) with the uniform distribution \ (₀\), as well as \ (D=R\) with the standard normal distribution \ (₀\). For \ (d\) or \ (d=\), we study integration or \ (L²\) -approximation on the space \ (H (₉=₁ᵈkⱼ) \) with respect to the product measure \ (₉=₁ᵈ₀\). To ensure that problems can be solved with reasonable costs and errors in the infinite-variate and the high dimensional case, intuitively speaking, the univariate problems on \ (H (kⱼ) \) have to become easier sufficiently fast as \ (j\) increases. We represent this mathematically with the help of suitable smoothness parameters. We study the important special case that the kernels \ (kⱼ\) are given by ⱼ (x, y) = _, ₉^{-1h_ (x) h_ (y) }, \ where \ ( (h_) _\) is an orthonormal basis in \ (L² (₀) \) and \ ( (, ₉), ₉\) is a family of positive real Fourier weights. In this case, the growth of the Fourier weights in \ (j\) quantifies how fast the univariate problems become easier. As an important example, we study the Hermite polynomials \ (h_\) with the standard normal distribution \ (₀\) on \ (R\), which leads to Hermite spaces. We determine the polynomial decay of the \ (n\) -th minimal errors of the infinite-variate problem in terms of the Fourier weights. The second important special case we study are spaces \ (H (kⱼ) \) whose kernels are given as Gaussian radial basis functions (Gaussian kernels), while our problems are again based on the standard normal distribution \ (₀\). We establish the following transference result between spaces with Gaussian kernels and Hermite spaces: The is a one-to-one-correspondence between spaces with Gaussian kernels and Hermite spaces with exponential growth of the Fourier weights such that the integration problem is equivalent on corresponding spaces. An analogue result holds true for the \ (L²\) -approximation problem. Utilizing this equivalence, we are able to constructively transfer upper and lower error bounds from one type of space to the other. In particular we construct almost optimal algorithms for spaces with Gaussian kernels in the infinite-variate case. Additionally, we are able to improve some known results in the finite-variate case.

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