insight - Computational Complexity - # Constructive representation of functions in N-dimensional Sobolev space and optimal polynomial approximation

Core Concepts

Functions in the Sobolev space with dominating mixed smoothness on an N-dimensional hyperrectangle can be uniquely represented in terms of their highest-order mixed derivative and suitable boundary values. This representation enables optimal polynomial approximation of such functions by projecting the boundary values onto polynomial subspaces.

Abstract

The paper proposes a new representation for functions in an N-dimensional Sobolev space with dominating mixed smoothness. It is shown that these functions can be expressed in terms of their highest-order mixed derivative and their lower-order derivatives evaluated along suitable boundaries of the domain. This expansion is proven to be invertible, uniquely identifying any function in the Sobolev space with its derivatives and boundary values.

The key insights are:

- Any function u in the Sobolev space Sδ

2[Ω] can be expressed as:

u(s) = Σ0≤α≤δ Gδ

α Bα-δ Dα u(s), where Bα-δ extracts the boundary values and Gδ

α are suitable operators. - This representation establishes a bijective relation between the Sobolev space Sδ

2[Ω] and the space Lδ

2[Ω] of boundary values. - Using this bijection, the paper shows how approximation of functions in Sδ

2[Ω] can be performed in the less restrictive space Lδ

2[Ω], by projecting the boundary values onto polynomial subspaces. - Two approximation methods are presented - one using Legendre polynomials and one using step functions. Both exhibit better convergence behavior than direct projection of the function u.

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Quotes

"A new representation is proposed for functions in a Sobolev space with dominating mixed smoothness on an N-dimensional hyperrectangle."
"It is shown that these functions can be expressed in terms of their highest-order mixed derivative, as well as their lower-order derivatives evaluated along suitable boundaries of the domain."
"Using this bijection, it is shown how approximation of functions in Sobolev space can be performed in the less restrictive space L2, reconstructing such an approximation of the function from an L2-optimal projection of its boundary values and highest-order derivative."

Key Insights Distilled From

by Declan S. Ja... at **arxiv.org** 04-30-2024

Deeper Inquiries

The proposed representation in the context provided offers a unique way to express functions in Sobolev spaces in terms of their derivatives and boundary values. This representation can be leveraged to develop efficient numerical methods for solving partial differential equations (PDEs) in Sobolev spaces by providing a structured framework for approximating functions and their derivatives.
One way to utilize this representation is in the numerical solution of PDEs using finite element methods. By approximating the functions in the Sobolev space with polynomials or other suitable basis functions, the PDEs can be transformed into a system of algebraic equations that can be efficiently solved using numerical techniques. The bijective relation between the Sobolev space and the space of boundary values allows for the accurate representation of functions with specific boundary conditions, which is crucial in many PDE problems.
Additionally, the expansion of functions in terms of their derivatives and boundary values can lead to the development of spectral methods for solving PDEs in Sobolev spaces. By leveraging the properties of the basis functions used in the expansion, such as Legendre polynomials or step functions, one can construct efficient numerical schemes that converge rapidly and accurately to the solution of the PDE.
Overall, the proposed representation provides a foundation for the development of numerical methods that can handle the complexities of PDEs in Sobolev spaces, offering a versatile and efficient approach to solving such problems.

The bijective relation between the Sobolev space and the space of boundary values has significant implications for the analysis and approximation of functions with specific boundary conditions.
Analysis of Functions: The bijective relation allows for a unique and invertible representation of functions in the Sobolev space in terms of their derivatives and boundary values. This enables a deeper understanding of the behavior of functions, especially when considering functions with specific boundary conditions. By identifying functions with their derivatives and boundary values, it becomes easier to analyze the properties and characteristics of these functions in a structured manner.
Approximation of Functions: The bijective relation facilitates the approximation of functions with specific boundary conditions. By projecting the functions onto a suitable basis, such as Legendre polynomials or step functions, and reconstructing the approximation using the boundary values, one can develop efficient approximation schemes that accurately capture the behavior of the functions near the boundaries. This is particularly useful in numerical methods for solving PDEs with prescribed boundary conditions.
Boundary Value Problems: The bijective relation is essential for solving boundary value problems in Sobolev spaces. It provides a clear and direct link between the functions and their boundary values, allowing for the formulation and solution of boundary value problems in a systematic and rigorous manner. This is crucial in various fields, including physics, engineering, and mathematical modeling.
In essence, the bijective relation enhances the analysis and approximation of functions with specific boundary conditions by establishing a direct correspondence between the functions in the Sobolev space and their boundary values, leading to more effective and accurate computational methods.

Yes, the ideas presented in this work can be extended to develop constructive representations and approximation schemes for functions in other function spaces beyond Sobolev spaces. The key concepts of representing functions in terms of their derivatives and boundary values, as well as the bijective relation between the function space and the space of boundary values, can be applied to various function spaces to enhance their analysis and approximation.
Hilbert Spaces: The techniques used in this work can be extended to function spaces that are Hilbert spaces, such as $L^2$ spaces. By defining appropriate inner products and basis functions, one can develop constructive representations and approximation schemes for functions in Hilbert spaces, similar to the approach taken for Sobolev spaces.
Function Spaces with Specific Properties: The ideas can also be applied to function spaces with specific properties, such as spaces of continuous functions, piecewise functions, or functions with certain regularity conditions. By adapting the representation and approximation methods to suit the characteristics of these function spaces, one can construct effective numerical schemes for solving problems in these spaces.
Generalized Function Spaces: The concepts can be extended to generalized function spaces, such as spaces of distributions or functionals. By considering the derivatives and boundary values of functions in these spaces, one can develop constructive representations that capture the essential features of the functions and enable efficient approximation techniques.
In conclusion, the principles and techniques introduced in this work can be generalized and adapted to a wide range of function spaces, providing a versatile framework for the analysis and approximation of functions beyond Sobolev spaces.

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