Weighted hyperbolic cross polynomial approximation (2407.19442v1)

Abstract: We study linear polynomial approximation of functions in weighted Sobolev spaces $W^r_{p,w}(\mathbb{R}^d)$ of mixed smoothness $r \in \mathbb{N}$, and their optimality in terms of Kolmogorov and linear $n$-widths of the unit ball $\boldsymbol{W}^r_{p,w}(\mathbb{R}^d)$ in these spaces. The approximation error is measured by the norm of the weighted Lebesgue space $L_{q,w}(\mathbb{R}^d)$. The weight $w$ is a tensor-product Freud weight. For $1\le p,q \le \infty$ and $d=1$, we prove that the polynomial approximation by de la Vall\'ee Poussin sums of the orthonormal polynomial expansion of functions with respect to the weight $w^2$, is asymptotically optimal in terms of relevant linear $n$-widths $\lambda_n\big(\boldsymbol{W}^r_{p,w}(\mathbb{R}, L_{q,w}(\mathbb{R})\big)$ and Kolmogorov $n$-widths $d_n\big(\boldsymbol{W}^r_{p,w}(\mathbb{R}), L_{q,w}(\mathbb{R})\big)$ for $1\le q \le p <\infty$. For $1\le p,q \le \infty$ and $d\ge 2$, we construct linear methods of hyperbolic cross polynomial approximation based on tensor product of successive differences of dyadic-scaled de la Vall\'ee Poussin sums, which are counterparts of hyperbolic cross trigonometric linear polynomial approximation, and give some upper bounds of the error of these approximations for various pair $p,q$ with $1 \le p, q \le \infty$. For some particular weights $w$ and $d \ge 2$, we prove the right convergence rate of $\lambda_n\big(\boldsymbol{W}^r_{2,w}(\mathbb{R}^d), L_{2,w}(\mathbb{R}^d)\big)$ and $d_n\big(\boldsymbol{W}^r_{2,w}(\mathbb{R}^d), L_{2,w}(\mathbb{R}^d)\big)$ which is performed by a constructive hyperbolic cross polynomial approximation.

• The paper establishes asymptotic optimality using de la Vallée Poussin sums and hyperbolic cross techniques in both univariate and multivariate settings.
• It derives precise convergence rates and demonstrates norm equivalence between weighted Sobolev and Lp spaces via Freud weight analysis.
• The study extends classical approximation theory, offering practical insights for high-dimensional numerical integration and adaptive computational methods.

Analysis and Insights on Weighted Hyperbolic Cross Polynomial Approximation

The paper investigates the theory of weighted linear polynomial approximation of functions within weighted Sobolev spaces $W_{p,w}(R^d)$ characterized by mixed smoothness. The central objective is to determine the optimal polynomial approximation using hyperbolic cross methodologies, assessed through the lenses of Kolmogorov and linear $n$-widths. This work leverages Freud weights, particularly in contexts of tensor-product representations, to extend existing approximation theories to a weighted, multivariate domain.

Main Contributions

The contributions articulated in the paper encompass several significant theoretical advancements:

1. Optimal Polynomial Approximation for $d=1$: For the univariate case, the authors utilize de la Vallée Poussin sums on orthonormal polynomial expansions. They prove that this method achieves asymptotic optimality regarding linear $n$-widths $\lambda(W^r_{p,w}(R))$ and Kolmogorov $n$-widths $d_n(W^r_{p,w}(R), L_{q,w}(R))$ for $1 \leq q \leq p < \infty$.
2. Hyperbolic Cross Approximation for $d \geq 2$: Extending to multivariate scenarios, the paper constructs linear methods predicated on tensor products of dyadic-scaled successive differences. The result is an upper bound on approximation errors for various $(p, q)$ pairs, showcasing effectiveness across different dimensions and weight scenarios.
3. Right Convergence Rates: A noteworthy highlight is the derivation of convergence rates for specific weights and dimensions when implementing constructive hyperbolic cross polynomial approximations. This includes rigorous results on the convergence rate of the $n$-widths using particular weights $w$.
4. Norm Equivalence Analysis: In a specialized case for parameters $\lambda = 2, 4$, and using Freud-type weights, this work establishes norm equivalences between Sobolev and weighted $L_p$ spaces, thus broadening potential applications in practical computations and theoretical pursuits.

Implications and Future Considerations

The implications of this paper are particularly notable for numerical analysis and computational fields that require robust approximations over high-dimensional spaces. The introduction of hyperbolic cross frameworks into weighted settings is a nontrivial extension that holds potential for more efficient numerical methods, especially in multidimensional integration and approximation contexts.

From a theoretical standpoint, this research further solidifies the connections between classical approximation theory and contemporary computational needs in complex domains. The established convergence rates provide a comprehensive framework for evaluating the efficiency and accuracy of algorithms driven by polynomial expansions in weighted spaces.

Looking ahead, these results pave the way for further exploration into higher dimensional approximations and their applicability in machine learning, data fitting, and beyond. Potential extensions could involve adaptive schemes that leverage the weighted hyperbolic cross structure for more dynamic and real-time applications.

Conclusion

Through meticulous analysis and a structured approach, this paper advances the frontiers of polynomial approximation in weighted Sobolev spaces. By interweaving classical concepts with novel hyperbolic cross methodologies, it delivers profound insights into optimizing approximations in both theoretical and practical landscapes. The established theoretical foundations, coupled with compelling convergence results, signify a vital resource for researchers aiming to exploit weighted polynomial approximation in advanced computational strategies.