On Sobolev and Besov Spaces of Hybrid Regularity

TL;DR

The paper tackles nonlinear (best N-term) approximation of functions in $H^q(\square)$ using tensor-product (hybrid) wavelet bases. It develops Jackson and Bernstein inequalities to identify approximation spaces $\mathcal{A}^s(H^q(\square))$ that contain Besov spaces of hybrid regularity $\mathfrak{B}^{q,s,p}_{\tau}(\square)$ and proves embeddings with classical Besov spaces, including the case of negative $q$. By connecting isotropic and tensor-product representations via explicit coordinate transforms, the authors show $\mathfrak{A}^s(H^q(\square))$ aligns with interpolation spaces $(H^q(\square), \mathfrak{B}^{q,r,\tau}_{\tau}(\square))_{s/r,\kappa}$ and establish a sandwich $B^{q+sn,\tau}_{\tau}(\square) \subseteq \mathfrak{B}^{q,s,\tau}_{\tau}(\square) \subseteq B^{q+s,\tau}_{\tau}(\square)$. This demonstrates that all functions classifiable by classical wavelets are at least as well approximable by tensor-product wavelets, and that hybrid-regularity Besov spaces are natural targets for anisotropic approximation with sparse grids. The results extend the Sobolev-Besov correspondence to hybrid regularity in a tensor-product setting and offer a new route to analyze and exploit anisotropic approximations in high dimensions.

Abstract

The present article is concerned with the nonlinear approximation of functions in the Sobolev space H^q with respect to a tensor-product, or hyperbolic wavelet basis on the unit n-cube. Here, q is a real number, which is not necessarily positive. We derive Jackson and Bernstein inequalities to obtain that the approximation classes contain Besov spaces of hybrid regularity. Especially, we show that all functions that can be approximated by classical wavelets are also approximable by tensor-product wavelets at least at the same rate. In particular, this implies that for nonnegative regularity, the classical Besov spaces of regularity q+sn, integrability and weak index t, with 1/t = s + 1/2, are included in the Besov spaces of hybrid regularity with isotropic regularity q and additional mixed regularity s.

Figures (2)

• Figure 1: Primal wavelets (left) and dual wavelets (right) according to the construction in DKU99. In this setting, we have $d = 1$, $\gamma = \frac{1}{2}$, and $\tilde{d} = 3$.
• Figure 2: Primal wavelets (left) and dual wavelets (right) with complementary boundary conditions according to the construction in DS98. In this setting, we have $d = \tilde{d} = 2$, and $\gamma = \frac{3}{2}$. The primal wavelets satisfy zero boundary conditions at $x=0$ and the dual ansatz functions at $x=1$.

Theorems & Definitions (29)

• Lemma 2.1
• proof
• Remark 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof